This document summarizes a new polynomial-time algorithm for linear programming.
1) The algorithm reduces the general linear programming problem to a canonical form and solves it through repeated application of projective transformations and optimization over spheres.
2) Each projective transformation followed by optimization reduces the objective function value by a constant factor, allowing the optimal solution to be found in polynomial time.
3) The algorithm runs in O(n3.5L0.5lnLlnlnL) time, an improvement over the ellipsoid method's O(n6L2lnLlnlnL) time.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
Presentation of the paper "An output-sensitive algorithm for computing (projections of) resultant polytopes" in the Annual Symposium on Computational Geometry (SoCG 2012)
Recent developments in control, power electronics and renewable energy by Dr ...Qing-Chang Zhong
The document provides an overview of Qing-Chang Zhong's research activities in control theory and engineering. It outlines his work in areas such as robust control, time-delay systems, process control, and applying infinite-dimensional systems theory to time-delay systems. It also lists publications, research projects solved, teaching responsibilities, funding sources, and future research plans.
Efficient Edge-Skeleton Computation for Polytopes Defined by OraclesVissarion Fisikopoulos
This document summarizes algorithms for computing the edge skeleton of a polytope defined by oracle functions. It first describes an existing algorithm for vertex enumeration in the oracle model that works by computing an initial simplex and recursively querying the oracle. It then presents a new algorithm for computing the edge skeleton that takes as input the oracle functions and a superset of edge directions, and works by generating candidate edge segments and validating them with the oracle. The runtime of this edge skeleton algorithm is polynomial in parameters of the polytope representation.
Polyhedral and algebraic method in computational geometrySpringer
This chapter introduces geometric fundamentals that will serve as the basis for topics covered later. It lays the foundations of projective geometry, which allows for a simpler formulation of statements compared to affine geometry. Projective spaces are defined by extending affine spaces so that parallel lines intersect at a point at infinity. Linear transformations between vector spaces induce projective transformations between the corresponding projective spaces. Any two maximal flags in a projective space can be mapped to each other by some projective transformation.
A note on arithmetic progressions in sets of integersLukas Nabergall
This document presents a new upper bound on r3(n), the maximum size of a set of integers between 1 and n that contains no three elements in arithmetic progression. The author proves that r3(n) = O(n/log^h n) for any arbitrarily large h, improving on previous bounds. The proof uses the fundamental theorem of discrete calculus and the pigeonhole principle to show that any sufficiently dense set of integers must contain arbitrarily long arithmetic progressions. This verifies a 1936 conjecture of Erdos and improves understanding of a major problem in combinatorics.
The document introduces the concept of generalized quasi-nonexpansive (GQN) maps. Some key results are:
1) GQN maps generalize quasi-nonexpansive maps but the fixed point set may not always be closed or convex.
2) If a subset satisfies certain conditions, it is a GQN-retract of the space.
3) Under these conditions, the class of GQN-retracts is closed under intersection and the common fixed point set of an increasing sequence of GQN maps is a GQN-retract.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
Presentation of the paper "An output-sensitive algorithm for computing (projections of) resultant polytopes" in the Annual Symposium on Computational Geometry (SoCG 2012)
Recent developments in control, power electronics and renewable energy by Dr ...Qing-Chang Zhong
The document provides an overview of Qing-Chang Zhong's research activities in control theory and engineering. It outlines his work in areas such as robust control, time-delay systems, process control, and applying infinite-dimensional systems theory to time-delay systems. It also lists publications, research projects solved, teaching responsibilities, funding sources, and future research plans.
Efficient Edge-Skeleton Computation for Polytopes Defined by OraclesVissarion Fisikopoulos
This document summarizes algorithms for computing the edge skeleton of a polytope defined by oracle functions. It first describes an existing algorithm for vertex enumeration in the oracle model that works by computing an initial simplex and recursively querying the oracle. It then presents a new algorithm for computing the edge skeleton that takes as input the oracle functions and a superset of edge directions, and works by generating candidate edge segments and validating them with the oracle. The runtime of this edge skeleton algorithm is polynomial in parameters of the polytope representation.
Polyhedral and algebraic method in computational geometrySpringer
This chapter introduces geometric fundamentals that will serve as the basis for topics covered later. It lays the foundations of projective geometry, which allows for a simpler formulation of statements compared to affine geometry. Projective spaces are defined by extending affine spaces so that parallel lines intersect at a point at infinity. Linear transformations between vector spaces induce projective transformations between the corresponding projective spaces. Any two maximal flags in a projective space can be mapped to each other by some projective transformation.
A note on arithmetic progressions in sets of integersLukas Nabergall
This document presents a new upper bound on r3(n), the maximum size of a set of integers between 1 and n that contains no three elements in arithmetic progression. The author proves that r3(n) = O(n/log^h n) for any arbitrarily large h, improving on previous bounds. The proof uses the fundamental theorem of discrete calculus and the pigeonhole principle to show that any sufficiently dense set of integers must contain arbitrarily long arithmetic progressions. This verifies a 1936 conjecture of Erdos and improves understanding of a major problem in combinatorics.
The document introduces the concept of generalized quasi-nonexpansive (GQN) maps. Some key results are:
1) GQN maps generalize quasi-nonexpansive maps but the fixed point set may not always be closed or convex.
2) If a subset satisfies certain conditions, it is a GQN-retract of the space.
3) Under these conditions, the class of GQN-retracts is closed under intersection and the common fixed point set of an increasing sequence of GQN maps is a GQN-retract.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.
1. The document discusses linear wire antennas, specifically infinitesimal dipoles and finite dipoles. It derives expressions for the electric and magnetic fields, power, and directivity of infinitesimal and finite length dipoles.
2. For infinitesimal dipoles (wire lengths much less than the wavelength), the document derives near, intermediate, and far field approximations. It also calculates the radiation resistance and power radiated.
3. For small dipoles (wire lengths between λ/50 and λ/10), the document compares the field expressions to those of an infinitesimal dipole. For finite length dipoles (wire lengths comparable to the wavelength), it discusses current distribution and far-field
This document outlines methods and results for analyzing the singularity behavior of functions in function spaces. It introduces concepts like distribution functions, maximal functions, and growth envelope functions. There are three cases discussed - subcritical, critical, and supercritical - depending on parameter values. For subcritical and critical cases, sharp inequalities are derived relating a function's maximal function to its norm in a function space. These inequalities involve Lorentz or Lorentz-Zygmund spaces and establish growth envelopes for the function spaces. Hardy inequalities are also discussed and related to rearrangement inequalities via integral transforms. Examples involving distance functions from fractal sets are provided.
Double Robustness: Theory and Applications with Missing DataLu Mao
When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
p-adic integration and elliptic curves over number fieldsmmasdeu
The document discusses a conjectural p-adic construction of elliptic curves attached to modular forms over number fields. It proposes using p-adic path integrals to define periods of the elliptic curves in terms of cohomology classes of certain arithmetic groups. Specifically, the period lattice would be generated by integrals of a modular form's cohomology class against cycles representing the boundary of the Bruhat-Tits tree. This aims to make the modularity conjecture constructive in a non-archimedean setting.
In this presentation, a more accurate expression of the zeta zero-counting function is developed exhibiting the expected step function behavior and its relation to the primes is demonstrated.
This document discusses properties of injective modules over various non-commutative algebras. It is noted that injective hulls of simple modules over Down-Up algebras may or may not be locally Artinian depending on the algebra, with examples given of algebras where the injective hulls are and are not locally Artinian. The document also examines related properties for Weyl algebras, quantum planes, Heisenberg algebras, and other algebras.
The document discusses algorithms for finding the convex hull of a set of points in two dimensions. It describes the Jarvis march (also called the gift wrapping algorithm) and the Graham scan algorithm. The Jarvis march finds the convex hull by iterating through points and finding the next point that forms the smallest interior angle with the last two points on the hull. The Graham scan sorts points by angle and then iterates through, eliminating any point that forms an obtuse angle with the last three points. Both algorithms run in O(n log n) time.
Realizations, Differential Equations, Canonical Quantum Commutators And Infin...vcuesta
1) The document discusses finding different realizations of quantum operators q and p that obey the canonical commutator [q,p]=iħ. It considers cases where p is defined as -iħf(q)∂/∂q and solves for the corresponding q operator.
2) This leads to an infinite number of possible representations, as f(q) can be any function of q. Three specific cases are analyzed.
3) For each case, the Schrodinger equation is derived and solutions are found for a free particle and infinite square well potential. However, some cases cannot yield normalizable wavefunctions or satisfy boundary conditions.
I am Travis W. I am a Computer Science Assignment Expert at programminghomeworkhelp.com. I hold a Master's in Computer Science, Leeds University. I have been helping students with their homework for the past 9 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
This document outlines the thesis presentation of Frank Romascavage III on deriving an explicit formula for the mean square of a Dirichlet L-function of prime power modulus. The introduction reviews previous work on power moments of the Riemann zeta function and Dirichlet L-functions. It also motivates studying power moments and notes the main theorem will not include an error term. The outline lists the introduction, main theorem, and sections on the diagonal sum, near-diagonal sums, and off-diagonal sums.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
Regular Convolutions Care progressions in one – to – one correspondence with sequences
p p P
of decompositions of
N
into arithmetical progressions (finite or infinite)and this is represented by
writing C
p p P
. In this paper we proved that any regular convolution C gives rise to a structure of meet
semilattice on
,
C
Z
and the convolution C is completely characterized by certain lattice theoretic
properties of
, . C
Z
In particular we prove that the only regular convolution including a lattice structure
on
,
C
Z
is the Dirichlit's convolution
On complementarity in qec and quantum cryptographywtyru1989
The document discusses complementarity in quantum error correction and quantum cryptography. It begins with an introduction and outline. It then covers the Stinespring dilation theorem, purification of mixed states, conjugate/complementary channels, private quantum codes including for single qubits, and the complementarity of quantum codes. An example is also provided to illustrate complementarity between a swap channel and its conjugate.
1. The document discusses fuzzy topological spaces, where open sets have fuzzy boundaries defined by a complete lattice L.
2. It defines cl∞-monoids which are used to measure the degree of membership of points in sets, and L- or fuzzy sets. Collections of fuzzy sets form L-topological spaces.
3. The main result is a version of the Tychonoff theorem for L-topological spaces, which gives necessary and sufficient conditions on L for all collections of a given cardinality of compact L-spaces to have a compact product.
4. It is shown that every product of α-compact L-spaces is compact if and only if 1 is α-isolated in L
In this presentation, I will demonstrate an alternative novel proof of the prime-counting function, by utilizing the Heaviside function, the Laplace transform, and the residue theorem.
1) The document describes a canonical transformation from the original (x, p) coordinates to new canonical coordinates (X, P) for the harmonic oscillator Hamiltonian.
2) An explicit transformation is found using a generating function of type 1. The new coordinates (X, P) correspond to an action-angle pair where P is the action (energy) and X is the cyclic coordinate proportional to time.
3) In the new coordinates, the Hamiltonian and equations of motion simplify greatly, with P being a constant of motion and X varying linearly with time.
Here are slides from the presentation Kenny Ascher and I gave at Brown University. The work was part of research done by Caleb Holtzinger, Kenny Ascher, Professor Michael Falk and myself at Northern Arizona University, funded by the National Science Foundation.
It should be noted that an edge-joint partition of a graph is some partition of a graph's edges such that if an edge, e (between vertex a and b), is in a partition then if for any vertex c not equal to a or b, c is in the neighborhood of a AND b then ac and bc are also in the same partition as e. If a graph is simple this is the same as saying that if an edge is in a partition any K_3's that contain this edge are also in said partition.
A graph is edge-joint maximal if there exists no non-trivial edge-joint partition (i.e. the only edge-joint partition is of G with the empty set).
A maximal edge-joint component, A, of a graph, G, is an edge-joint maximal sub-graph, A, such that the partition {E(A), E(G-A)} is an edge-joint partition.
The document summarizes conditions for the Fredholm property of matrix Wiener-Hopf plus/minus Hankel operators with semi-almost periodic symbols. Key points:
1) Such operators involve Wiener-Hopf and Hankel operators defined using Fourier transforms and reflection operators on function spaces.
2) Symbols are from the algebra of semi-almost periodic functions, which contains almost periodic and continuous functions.
3) A matrix function is said to admit a right/left almost periodic factorization if it can be written as the product of almost periodic matrix functions and a diagonal matrix.
4) The Fredholm property holds if the symbol and its almost periodic parts admit canonical factorizations and the geometric mean
The document discusses using Plücker coordinates to determine if a set of homogeneous polynomials generate the vector space of higher degree polynomials when projected onto a quotient ring. It begins by defining Plücker coordinates and showing that for two quadratic generators, the cubic vector space has zero image if the Plücker quantity P1P3 - P22 is non-zero. It then aims to generalize this to three cubic generators and the quartic vector space.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.
1. The document discusses linear wire antennas, specifically infinitesimal dipoles and finite dipoles. It derives expressions for the electric and magnetic fields, power, and directivity of infinitesimal and finite length dipoles.
2. For infinitesimal dipoles (wire lengths much less than the wavelength), the document derives near, intermediate, and far field approximations. It also calculates the radiation resistance and power radiated.
3. For small dipoles (wire lengths between λ/50 and λ/10), the document compares the field expressions to those of an infinitesimal dipole. For finite length dipoles (wire lengths comparable to the wavelength), it discusses current distribution and far-field
This document outlines methods and results for analyzing the singularity behavior of functions in function spaces. It introduces concepts like distribution functions, maximal functions, and growth envelope functions. There are three cases discussed - subcritical, critical, and supercritical - depending on parameter values. For subcritical and critical cases, sharp inequalities are derived relating a function's maximal function to its norm in a function space. These inequalities involve Lorentz or Lorentz-Zygmund spaces and establish growth envelopes for the function spaces. Hardy inequalities are also discussed and related to rearrangement inequalities via integral transforms. Examples involving distance functions from fractal sets are provided.
Double Robustness: Theory and Applications with Missing DataLu Mao
When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
p-adic integration and elliptic curves over number fieldsmmasdeu
The document discusses a conjectural p-adic construction of elliptic curves attached to modular forms over number fields. It proposes using p-adic path integrals to define periods of the elliptic curves in terms of cohomology classes of certain arithmetic groups. Specifically, the period lattice would be generated by integrals of a modular form's cohomology class against cycles representing the boundary of the Bruhat-Tits tree. This aims to make the modularity conjecture constructive in a non-archimedean setting.
In this presentation, a more accurate expression of the zeta zero-counting function is developed exhibiting the expected step function behavior and its relation to the primes is demonstrated.
This document discusses properties of injective modules over various non-commutative algebras. It is noted that injective hulls of simple modules over Down-Up algebras may or may not be locally Artinian depending on the algebra, with examples given of algebras where the injective hulls are and are not locally Artinian. The document also examines related properties for Weyl algebras, quantum planes, Heisenberg algebras, and other algebras.
The document discusses algorithms for finding the convex hull of a set of points in two dimensions. It describes the Jarvis march (also called the gift wrapping algorithm) and the Graham scan algorithm. The Jarvis march finds the convex hull by iterating through points and finding the next point that forms the smallest interior angle with the last two points on the hull. The Graham scan sorts points by angle and then iterates through, eliminating any point that forms an obtuse angle with the last three points. Both algorithms run in O(n log n) time.
Realizations, Differential Equations, Canonical Quantum Commutators And Infin...vcuesta
1) The document discusses finding different realizations of quantum operators q and p that obey the canonical commutator [q,p]=iħ. It considers cases where p is defined as -iħf(q)∂/∂q and solves for the corresponding q operator.
2) This leads to an infinite number of possible representations, as f(q) can be any function of q. Three specific cases are analyzed.
3) For each case, the Schrodinger equation is derived and solutions are found for a free particle and infinite square well potential. However, some cases cannot yield normalizable wavefunctions or satisfy boundary conditions.
I am Travis W. I am a Computer Science Assignment Expert at programminghomeworkhelp.com. I hold a Master's in Computer Science, Leeds University. I have been helping students with their homework for the past 9 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
This document outlines the thesis presentation of Frank Romascavage III on deriving an explicit formula for the mean square of a Dirichlet L-function of prime power modulus. The introduction reviews previous work on power moments of the Riemann zeta function and Dirichlet L-functions. It also motivates studying power moments and notes the main theorem will not include an error term. The outline lists the introduction, main theorem, and sections on the diagonal sum, near-diagonal sums, and off-diagonal sums.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
Regular Convolutions Care progressions in one – to – one correspondence with sequences
p p P
of decompositions of
N
into arithmetical progressions (finite or infinite)and this is represented by
writing C
p p P
. In this paper we proved that any regular convolution C gives rise to a structure of meet
semilattice on
,
C
Z
and the convolution C is completely characterized by certain lattice theoretic
properties of
, . C
Z
In particular we prove that the only regular convolution including a lattice structure
on
,
C
Z
is the Dirichlit's convolution
On complementarity in qec and quantum cryptographywtyru1989
The document discusses complementarity in quantum error correction and quantum cryptography. It begins with an introduction and outline. It then covers the Stinespring dilation theorem, purification of mixed states, conjugate/complementary channels, private quantum codes including for single qubits, and the complementarity of quantum codes. An example is also provided to illustrate complementarity between a swap channel and its conjugate.
1. The document discusses fuzzy topological spaces, where open sets have fuzzy boundaries defined by a complete lattice L.
2. It defines cl∞-monoids which are used to measure the degree of membership of points in sets, and L- or fuzzy sets. Collections of fuzzy sets form L-topological spaces.
3. The main result is a version of the Tychonoff theorem for L-topological spaces, which gives necessary and sufficient conditions on L for all collections of a given cardinality of compact L-spaces to have a compact product.
4. It is shown that every product of α-compact L-spaces is compact if and only if 1 is α-isolated in L
In this presentation, I will demonstrate an alternative novel proof of the prime-counting function, by utilizing the Heaviside function, the Laplace transform, and the residue theorem.
1) The document describes a canonical transformation from the original (x, p) coordinates to new canonical coordinates (X, P) for the harmonic oscillator Hamiltonian.
2) An explicit transformation is found using a generating function of type 1. The new coordinates (X, P) correspond to an action-angle pair where P is the action (energy) and X is the cyclic coordinate proportional to time.
3) In the new coordinates, the Hamiltonian and equations of motion simplify greatly, with P being a constant of motion and X varying linearly with time.
Here are slides from the presentation Kenny Ascher and I gave at Brown University. The work was part of research done by Caleb Holtzinger, Kenny Ascher, Professor Michael Falk and myself at Northern Arizona University, funded by the National Science Foundation.
It should be noted that an edge-joint partition of a graph is some partition of a graph's edges such that if an edge, e (between vertex a and b), is in a partition then if for any vertex c not equal to a or b, c is in the neighborhood of a AND b then ac and bc are also in the same partition as e. If a graph is simple this is the same as saying that if an edge is in a partition any K_3's that contain this edge are also in said partition.
A graph is edge-joint maximal if there exists no non-trivial edge-joint partition (i.e. the only edge-joint partition is of G with the empty set).
A maximal edge-joint component, A, of a graph, G, is an edge-joint maximal sub-graph, A, such that the partition {E(A), E(G-A)} is an edge-joint partition.
The document summarizes conditions for the Fredholm property of matrix Wiener-Hopf plus/minus Hankel operators with semi-almost periodic symbols. Key points:
1) Such operators involve Wiener-Hopf and Hankel operators defined using Fourier transforms and reflection operators on function spaces.
2) Symbols are from the algebra of semi-almost periodic functions, which contains almost periodic and continuous functions.
3) A matrix function is said to admit a right/left almost periodic factorization if it can be written as the product of almost periodic matrix functions and a diagonal matrix.
4) The Fredholm property holds if the symbol and its almost periodic parts admit canonical factorizations and the geometric mean
The document discusses using Plücker coordinates to determine if a set of homogeneous polynomials generate the vector space of higher degree polynomials when projected onto a quotient ring. It begins by defining Plücker coordinates and showing that for two quadratic generators, the cubic vector space has zero image if the Plücker quantity P1P3 - P22 is non-zero. It then aims to generalize this to three cubic generators and the quartic vector space.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
1. The doubling map on the circle has 2n-1 periodic points of period n. Its periodic points are dense.
2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
Section 18.3-19.1.Today we will discuss finite-dimensional.docxkenjordan97598
Section 18.3-19.1.
Today we will discuss finite-dimensional associative algebras and their representations.
Definition 1. Let A be a finite-dimensional associative algebra over a field F . An element
a ∈ A is nilpotent if an = 0 for some positive integer n. An algebra is said to be nilpotent if
all of its elements are.
Exercise 2. Every subalgebra and factor algebra of a nilpotent algebra are nilpotent. Con-
versely, if I ⊂ A is a nilpotent ideal, and the quotient algebra A/I is nilpotent, then so is
A.
Example. The algebra of all strictly lower- (or upper-) triangular n×n matrices is nilpotent.
Proposition 3. If the algebra A is nilpotent, then An = 0 for some n ∈ Z+, that is the
product of any n elements if the algebra A equals 0.
Proof. Let B ⊂ A be the maximal subspace for which there exists n ∈ Z+ such that Bn = 0.
Note that B is closed under multiplication, i.e. B ⊃ Bk for any k ∈ Z+. Assume B 6= A
and choose an element a ∈ A\B. Since aBn = 0, there exists k ∈ Z+ such that aBk 6⊂ B.
Replacing a with a non-zero element in aBk we obtain aB ⊂ B. Recall that there exists
m ∈ Z+ so that am = 0. Now, let us set C = B ⊕〈a〉. Then we have
Cmn = 0
which contradicts the definition of the subspace B. �
Unless A is commutative, the set of all nilpotent elements of A does not have to be an
ideal (in general, not even a subspace). On the other hand, if I,J are nilpotent ideals in A,
then so is
I + J = {x + y |x ∈ I,y ∈ J} .
Therefore, there exists the maximal nilpotent ideal which contains every other nilpotent ideal
of A.
Definition 4. The radical of A is the maximal nilpotent ideal of A it is denoted rad(A).
The algebra A is semisimple if rad(A) = 0.
If char(F) = 0, there exists an alternative description of semisimple algebras. Consider
the regular representation of the algebra A
ρ: A → L(A), ρ(a)(b) = ab,
and define a “scalar product” on A via
(a,b) = tr(ρ(ab)) = tr(ρ(a)ρ(b)).
One can see that (·, ·) is a symmetric bilinear function on A satisfying (ab,c) = (a,bc).
Definition 5. For any ideal I ⊂ A, its orthogonal complement I⊥ is defined as
I⊥ = {a ∈ A |(a,i) = 0 for all i ∈ I} .
Proposition 6. For any ideal I ⊂ A, its orthogonal complement I⊥ ⊂ A is also an ideal.
Proof. For any a ∈ A, x ∈ I⊥, and y ∈ I we have
(xa,y) = (x,ay) = 0 and (ax,y) = (y,ax) = (ya,x) = 0.
�
1
2
Proposition 7. If A is an algebra over a field F with char(F) = 0, then every element a ∈ A
orthogonal to all of its powers is nilpotent.
Proof. Let a ∈ A be such that
(a,an) = tr ρ(a)n+1 = 0 for all n ∈ Z+.
Let L ⊃ F be the splitting field of the characteristic polynomial f(x) of the operator ρ(a).
Then, over L we have
f(x) = tk0
s∏
i=1
(t−λi)ki,
where λi are distinct for i = 1, . . . ,s, and
tr ρ(a)n+1 =
s∑
i=1
kiλ
n+1
i = 0.
Letting n run through the set 1, . . . ,s, the above equation yields a system of s homogeneous
linear equations in variables k1, . . . ,ks. The determinant of this system equals
λ21 . . .λ
2
nV (λ1, . . . ,λn),
where V (λ1.
Section 18.3-19.1.Today we will discuss finite-dimensional.docxrtodd280
Section 18.3-19.1.
Today we will discuss finite-dimensional associative algebras and their representations.
Definition 1. Let A be a finite-dimensional associative algebra over a field F . An element
a ∈ A is nilpotent if an = 0 for some positive integer n. An algebra is said to be nilpotent if
all of its elements are.
Exercise 2. Every subalgebra and factor algebra of a nilpotent algebra are nilpotent. Con-
versely, if I ⊂ A is a nilpotent ideal, and the quotient algebra A/I is nilpotent, then so is
A.
Example. The algebra of all strictly lower- (or upper-) triangular n×n matrices is nilpotent.
Proposition 3. If the algebra A is nilpotent, then An = 0 for some n ∈ Z+, that is the
product of any n elements if the algebra A equals 0.
Proof. Let B ⊂ A be the maximal subspace for which there exists n ∈ Z+ such that Bn = 0.
Note that B is closed under multiplication, i.e. B ⊃ Bk for any k ∈ Z+. Assume B 6= A
and choose an element a ∈ A\B. Since aBn = 0, there exists k ∈ Z+ such that aBk 6⊂ B.
Replacing a with a non-zero element in aBk we obtain aB ⊂ B. Recall that there exists
m ∈ Z+ so that am = 0. Now, let us set C = B ⊕〈a〉. Then we have
Cmn = 0
which contradicts the definition of the subspace B. �
Unless A is commutative, the set of all nilpotent elements of A does not have to be an
ideal (in general, not even a subspace). On the other hand, if I,J are nilpotent ideals in A,
then so is
I + J = {x + y |x ∈ I,y ∈ J} .
Therefore, there exists the maximal nilpotent ideal which contains every other nilpotent ideal
of A.
Definition 4. The radical of A is the maximal nilpotent ideal of A it is denoted rad(A).
The algebra A is semisimple if rad(A) = 0.
If char(F) = 0, there exists an alternative description of semisimple algebras. Consider
the regular representation of the algebra A
ρ: A → L(A), ρ(a)(b) = ab,
and define a “scalar product” on A via
(a,b) = tr(ρ(ab)) = tr(ρ(a)ρ(b)).
One can see that (·, ·) is a symmetric bilinear function on A satisfying (ab,c) = (a,bc).
Definition 5. For any ideal I ⊂ A, its orthogonal complement I⊥ is defined as
I⊥ = {a ∈ A |(a,i) = 0 for all i ∈ I} .
Proposition 6. For any ideal I ⊂ A, its orthogonal complement I⊥ ⊂ A is also an ideal.
Proof. For any a ∈ A, x ∈ I⊥, and y ∈ I we have
(xa,y) = (x,ay) = 0 and (ax,y) = (y,ax) = (ya,x) = 0.
�
1
2
Proposition 7. If A is an algebra over a field F with char(F) = 0, then every element a ∈ A
orthogonal to all of its powers is nilpotent.
Proof. Let a ∈ A be such that
(a,an) = tr ρ(a)n+1 = 0 for all n ∈ Z+.
Let L ⊃ F be the splitting field of the characteristic polynomial f(x) of the operator ρ(a).
Then, over L we have
f(x) = tk0
s∏
i=1
(t−λi)ki,
where λi are distinct for i = 1, . . . ,s, and
tr ρ(a)n+1 =
s∑
i=1
kiλ
n+1
i = 0.
Letting n run through the set 1, . . . ,s, the above equation yields a system of s homogeneous
linear equations in variables k1, . . . ,ks. The determinant of this system equals
λ21 . . .λ
2
nV (λ1, . . . ,λn),
where V (λ1.
I am Luther H. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of Illinois, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Diffusion.
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This document discusses representations and operations for polynomials. It begins by defining polynomials and their coefficient and point-value representations. It then summarizes common operations like addition, multiplication, evaluation, and converting between representations. A key part discusses using the Fast Fourier Transform (FFT) to multiply polynomials much more efficiently than brute force. The FFT takes advantage of evaluating polynomials at complex roots of unity to perform the multiplication in O(n log n) time versus O(n^2) for brute force.
Math 511 Problem Set 4, due September 21Note Problems 1 tAbramMartino96
Math 511 Problem Set 4, due September 21
Note: Problems 1 through 7 are the ones to be turned in. The remainder of the problems are
for extra functional analytic goodness.
1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a linearly independent subset of
C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly independent set in C[a,b]. Give
an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . . .}.
2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 .
3. Consider C[0, 2] with the function ‖ ·‖1 defined by
‖f‖1 =
∫ 2
0
|f(x)|dx, for f ∈ C[0, 2].
(a) Prove that ‖ ·‖1 is a norm.
(b) Prove that the normed linear space (C[0, 2],‖·‖1) is not complete (and thus not a Banach
space) by considering the sequence of functions
fn(x) =
1, x ≤ 1 − 1
n
n−nx, 1 − 1
n
< x < 1 + 1
n
−1, x ≥ 1 + 1
n
.
Show these are continuous functions, this sequence is a Cauchy sequence in the metric
derived from ‖ ·‖1, but that this sequence does not converge in C[0, 2] with this metric.
4. Let V be a vector space over R or C. A subset A ⊆ V is convex if for any v,w ∈ A and any
λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting v and w is also in A.
(a) Let W be a vector subspace of V . Show that W is convex.
(b) Let X be a normed linear space. Show that the unit ball B1(0) is convex.
5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the definition of c) and so is c0, the
set of all sequences (xn) so that limn→∞ xn = 0.
6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth place and 0 in all othe coordinates.
Show that {en : n ∈ N} is a Schauder basis for lp.
7. Now if X is a Banach space and (yn) a sequence in X, prove that
∑∞
n=1 ‖yn‖ < ∞ does imply
the convergence of
∑∞
n=1 yn. Thus in Banach spaces, absolute convergence implies convergence
of the series.
The following questions are for you to think about and not to be turned in.
1001. What is the completion of (0, 1) as a metric subspace of R with the euclidean metric?
Explain.
1002. Show that the discrete metric on a nontrivial vector space cannot be obtained from a norm.
1003. Show that if a normed vector space has a Schauder basis, then the space is separable. (You
can use a similar argument to your proof that lp is separable for 1 ≤ p < ∞.)
1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p < ∞, and assume that
1
p
+
1
q
=
1
r
.
Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we define the componentwise product
xy = (x1y1,x2y2, . . .), then
‖xy‖r ≤‖x‖p‖y‖q.
You may assume that x ∈ lp and y ∈ lq, although this is not necessary. (Hint: 1 = 1p
r
+ 1q
r
, and
use the regular Hölder inequality on particular sequences).
(Note: We can extend this to let p = r, and in this case q = ∞. The result will still hold.)
1005. Give an example of a subspace of l∞ which is not closed. Repeat for l2. (Hint: Look at
problem 3, p. 70)
1006. Let X be a normed vector space. Show that the convergenc ...
The document discusses complex eigenvalues and eigenvectors for systems of linear differential equations. It shows that if the matrix A has complex conjugate eigenvalue pairs r1 and r2, then the corresponding eigenvectors and solutions will also be complex conjugates. This leads to real-valued fundamental solutions that can express the general solution. An example demonstrates these concepts, finding the complex eigenvalues and eigenvectors and expressing the general solution in terms of real-valued functions. Spiral points, centers, eigenvalues, and trajectory behaviors are also summarized.
This document discusses several mathematical topics related to analysis, algebra, and geometry. It defines p-adic numbers as the completion of rational numbers with respect to the p-adic valuation. It also defines the p-adic valuation, chain complexes, derived functors, partial differential operators, and Picard's theorems regarding complex functions taking on every value. Additionally, it provides brief definitions for concepts like parabolic subgroups, the Picard variety, and the Picard-Lefschetz theory.
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- Eigenvalues are values that satisfy the equation AX = λX, where X is a non-zero eigenvector. The eigenspace associated with λ contains all eigenvectors for that eigenvalue.
- The characteristic polynomial of a matrix A is defined as det(xI - A). The eigenvalues of A are the roots of its characteristic polynomial.
- Similar matrices have the same eigenvalues, determinant, rank, trace, and characteristic polynomial. Two matrices are similar if one can be obtained from the other by conjugation via an invertible matrix.
- The trace of a matrix is the sum of its diagonal entries
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(ii) It works through proofs for various statements about rings, ideals, nilpotent and Jacobson radicals, and the prime and Zariski spectra of rings.
(iii) The solutions cover topics such as when a sum of a nilpotent element and unit is a unit, when a polynomial is a unit or zero divisor based on its coefficients, and properties of the prime and Zariski topologies on the prime spectrum of a ring.
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It first provides background on polyhedra, recession cones, and asymptotic cones. It then shows that the set of convex polytopes in Rn forms a semiring under these operations. The proof establishes that the union operation yields a closed convex set, and the sum and other axioms hold.
Subsequent sections will consider semirings of other closed convex sets in Rn, such as compact sets and those with a fixed recession cone
This document discusses Taylor's theorem, which approximates functions using polynomials.
It begins by introducing Taylor's theorem for functions of one and two variables, and its applications for finding maxima and minima. Taylor's theorem states that a function can be approximated by its Taylor polynomial plus a remainder term.
Several examples are provided to demonstrate calculating Taylor polynomials for different functions like e^x, sin(x), and (1-x)^-1. The key steps are computing the derivatives at the given point and substituting them into the Taylor polynomial formula.
Taylor's theorem precisely relates a function to its Taylor polynomials, stating that the value of the function is equal to its nth order Taylor polynomial plus a
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The document discusses quadratic functions and their graphs. It defines quadratic functions as functions of the form f(x)=ax^2+bx+c, where a is not equal to 0. The graph of a quadratic function is a parabola with certain characteristics: it is symmetrical about an axis of symmetry and has a vertex which is either a maximum or minimum point. The axis of symmetry is the line x=0 for functions of the form f(x)=ax^2 and the vertex is at (0,0). For functions of the form f(x)=ax^2+k, the graph is a translation of f(x)=ax^2, so the vertex is (0,k) and the
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The document discusses algorithms for computing volumes of polytopes. It notes that exactly computing volumes is hard, but randomized polynomial-time algorithms can approximate volumes with high probability. It describes two algorithms: Random Directions Hit-and-Run (RDHR), which generates random points within a polytope via random walks; and Multiphase Monte Carlo, which approximates a polytope's volume by sampling points within a sequence of enclosing balls. RDHR mixes in O(d^3) steps and these algorithms can compute volumes of high-dimensional polytopes that exact algorithms cannot handle.
Reproducing Kernel Hilbert Space of A Set Indexed Brownian MotionIJMERJOURNAL
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Similar to A New Polynomial-Time Algorithm for Linear Programming (20)
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إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
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The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
A New Polynomial-Time Algorithm for Linear Programming
1. COMBINATOR1CA4 (4) (1984) 373--395
A N E W POLYNOMIAL-TIME ALGORITHM
FOR LINEAR P R O G R A M M I N G
N. K A R M A R K A R
Received 20 August 1984
Revised 9 November 1984
We present a new polynomial-time algorithm for linear programming. In the worst case,
the algorithm requires O(tf'SL) arithmetic operations on O(L) bit numbers, where n is the number of
variables and L is the number of bits in the input. The running,time of this algorithm is better than
the ellipsoid algorithm by a factor of O(n~'~). We prove that given a polytope P and a strictly in-
terior point a E P, there is a projective transformation of the space that maps P, a to P', a' having
the following property. The ratio of the radius of the smallest sphere with center a', containing
P ' to the radius of the largest sphere with center a' contained in P ' is O(n). The algorithm consists
of repeated application of such projective transformations each followed by optimization over an
inscribed sphere to create a sequence of points which converges to the optimal solution in poly-
nomial time.
1. Informal outline
In this Section we give an informal discussion of the main ideas involved in
the algorithm. A more rigorous description will be given in the next Section.
1.1. Problem definition
We shall reduce the general linear programming problem over the rationals
to the lollowing particular case:
minimize crx
Ax=O
subject to ~ ~'xi = 1
(x=>O.
Here x = ( x l . . . . , x.)TER ", cEZ", A E Z " ~ ' .
This is a substantially revised version of the paper presented at the Symposium on Theory
of Computing, Washington D. C., April 1984.
AMS subject classification (1980): 90 C 05
2. 374 N. KARMARKAR
Let 12 denote the subspace {xlAx=0}. Let A denote the simplex A =
= {xlx >=O, .~xi = 1} and let H be the polytope defined by
1I = ONA.
Thus the problem we consider is
minimize crx
subject to xE/7.
1.2. Optimization over a sphere
If we replace the constraint xEA by a constraint of the form x~S, where S
is a sphere, then it is trivial to see that the optimum can be found by solving a
linear system of equations.
The intersection of a sphere and an affine space is a lower dimensional
sphere inside that affine space. Hence the problem becomes:
min c'rx, subject to xES" (a sphere)
where c' was obtained by projecting c orthogonally onto t2.
But this problem is trivial: From the center of the sphere, take a step along
the direction - c ' , of length equal to the radius of the sphere.
1.3. Bounds on the objective function
Let a0 be strictly interior point in the polytope P. Suppose we draw an ellip-
soid E with center a0, that is contained in the polyt ope and solve the optimization prob-
lem over the restricted region E instead of P. How good a solution do we get as com-
pared to the optimal solution? To derive a bound, we create another ellipsoid E'
by magnifying E by a sufficiently large factor v so that E' contains P.
E c P c E', E" = rE.
Let rE, A , fr, denote the minimum values of objective function f ( x ) on E, P, and
E' respectively.
f ( a o ) - f ~ <----f(ao)--fe ~--f(ao)-fw = vtf(ao)-fe].
The last equation follows from the linearity off(x).
f(a0) - f ~ 1
f(ao)-fp- v
~-~ ~ (1_1).
f(ao) - f e
Thus by going from a0 to the point, say a', that minimizesf(x) over E we come closer
to the minimum value of the objective function by a factor I - v -1. We can repeat
the same process with a' as the center. The rate of convergence of this method depends
on v, the smaller the value of v, the faster the convergence.
3. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR P R O G R A M M I N G 375
1.4. Projective transformations
Projective transformations of R" are described by formulas of the form
Cx+d
x~ frx+g
where C6R *X", d, f6R", g~R and the matrix
is non-singular.
We shall be interested in projective transformations of the hyperplane
2:= { IZx, = 1}.
It is easy to see that these transformations are described by
Cx
X ~" eTfx
where e=(1, 1. . . . . 1)r and C is a non-singular n×n matrix.
For each interior point a of A there is a unique projective transformation
T(a, ao) of the hyperplane 2: fixing all vertices of the simplex A and moving a =
= (al .... , a,) r to the center a0=(1/n)e. This transformation is given by
t _ x_i / a f .
x~ = Z xJaj
J
Observe that the corresponding matrix C is now diagonal: C =
,1
= d i a g [ 1 .... 7,,)" The transformation C(a, a0) has the following properties :
1. T is one-to-one and maps the simplex onto itself. Its inverse is given by
a i XPt
Xi = ,~ adxj i= 1,2,...,n.
J
2. Each facet of the simplex given by xi=0 is mapped onto the corresponding
facet xi=0.
1
3. Image of the point x = a is the center of the simplex given by X-' ~ - - .
I
n
4. Let Ai denote the ith column of A. Then the system of equations
z~ Aixi = 0
i
now becomes
.~ Aiaix~
t
=0
Z a,x;
i
4. 376 N. K A R M A R K A R
or
Z A;x; = o
i
where
A~ = aiAi.
We denote the affine subspace {x'lA'x'=0} by f2".
5. If a~ f2 then the center of the simplex aoC f2'.
Let B(ao, r) be the largest ball with center a0, the same as the center of the
simplex, and radius r that is contained in the simplex. Similarly let B(ao, R) be
the smallest ball containing A. It is easy to show that R / r = n - 1 .
B(a0, r) ~ A ~ B(ao, R)
B(a0, r)O~2' =c A (q (2" ~ B(a0, R)N f2'.
But A fqf2' is the image 17' of the polytope H=Af~f2. The intersection of a ball
and an affine subspace is a ball of lower dimension in that subspace and has the
same radius if the center of the original ball lies in the affine subspace, which it
does in our case. Hence we get
R
B'(ao, r ) ~ H ' ~ B ' ( a o , R), -r- = n - 1
which proves that v =dimension of the space can always be achieved by this method.
(Note: A is an n - 1 dimensional simplex.)
1.5. Invariant potential functions
The algorithm creates a sequence of points x (°), x (1), ..., x (k), ... having dec-
reasing values of the objective function. In the k th step, the point x (k) is brought
into the center by a projective transformation. Then we optimize the objective
function over the intersection of the inscribed ball and the affine subspace to find
the next point x (k+l). Based on the results of the previous Sections, we expect the
objective function to be reduced by a factor of (1 - n - 0 at least, in each step, but
there is one more technicality we have to deal with. The set of linear functions is
not invariant under a projective transformation, but ratios of linear functions are
transformed into ratios of linear functions. With every linear objective function
/(x) we associate a "potential function" g(x) expressed in terms of logarithms of
ratios of linear functions, in order to measure the progress of the algorithm.
It has the following properties:
1. Any desired amount of reduction in the value o f / ( x ) can be achieved by sufficient
reduction in the value of g(x).
2. g(x) is mapped into a function of the same.form by projective transformations
T(a, a0) described in 1.4.
3. Optimization of g(×) in each step can be done approximately by optimizing a
linear function (but a different linear function in different steps).
5. A NEW POLYNOMIAL-TIME A L G O R I T H M FOR LINEAR P R O G R A M M I N G 377
1.6. Complexity of the algorithm
The value of the objective function is reduced by a constant factor in O(n)
steps. As in the ellipsoid algorithm we define
L = log (1 + ID,..~!)+log (1 +~)
where Dm=x = max {Idet(X)lIX is a square submatrix of constraint matrix A}
: max {[c;I, Ib,[ i = l .... , n}.
Clearly, L is not greater than the length of the input.
During the course of the algorithm, we periodically check for optimality by
converting the interior point solution to an extreme point solution without degrading
the objective function value, and testing the extreme point for optimality. The inter-
val between two successive checks can be adjusted so that time spent between checks
is comparable to tinae spent in checking for optimality. Any two distinct values of
the objective function on the vertices of the polytope have to differ by at least 2 -kz
for some constant k [3]. Therefore, in at most O(nL) steps we reach a situation in
which any extreme point better than the current interior point has to be an exact
optimal solution.
In each step, we have to solve a linear system of equations which takes O(n 3)
arithmetic operations in the worst case. However, if we solve the equations at each
step by modifying the solution to the previous step, we can s a v e a factor of 1/n.
This method is described in Section 6.
For each arithmetic operation, we need a precision of O(L) bits. Note that this
much precision is required for any method that solves the equations A x = b if
det (A) :O(2L), e.g., the complexity of gaussian elimination is O(n3L In L lnln L),
although it is normally stated as O(n~). This factor L must appear in the conlplexity
of any algorithm for linear programming, since representing the output can require
O(L) bits/variable in the worst case.
The overall complexity of our algorithm is O(n3-SL"~In L Inln L) as compared
to O(n6L 2 In L lnln L) for the ellipsoid algorithm.
2. Main algorithm
2.1. Problem definition
In order to focus on the ideas which are new and the most important in our
algorithm, we make several simplifying assumptions. In Section 5 we will remove
all these restrictions by showing how a linear programming problem in standard
form can be transformed into our canonical form.
Canonical form of the problem
Input: ACZ '~×', c~Z".
Let g2={xlAx=0}, A={xlx=>0, ~ ' x i = l } and ao=e/n.
Assumptions: crx=>0 for every x~ ~2NA; ao is a feasible starting point, i.e. A a o - 0 ;
crao#O.
Output:either Xo~QNA such that crxo=0 or a proof that min{erxlxCflAA}>0,
6. 378 N. KARMARKAR
2.2. Comments
1. Note that the feasible region is the intersection of an affine space with a
simplex rather than the positive orthant. Initially it takes one projective transfor-
mation to bring a linear programming problem to this form.
2. The linear system of equations defining f2 is homogeneous i.e., the right
hand side is zero. The projective transformation that transforms positive orthant
into simplex also transforms inhomogeneous system of equations into an homo-
geneous one.
3. The target minimum value of the objective function is zero. This will be
justified by combining primal and dual into a single problem.
2.3. Description of the Algorithm
The algorithm creates a sequence of points x (°), x (1), ..., x (~) by these steps:
Step 0. Initialize
x (°) : center of the simplex
Step 1. Compute the next point in the sequence
x(k+ 1) .~_ ~D(x(k)).
Step 2. Check for infeasibility
Step 3. Check for optimality
go to Step 1.
Now we describe the steps in more detail.
Step 1. The function b=q~(a) is defined by the following sequence of operations.
Let D = d i a g {al, a2 . . . . . a,} be the diagonal matrix whose fh diagonal
entry is a~.
1. Let B = ~
i.e., augment the matrix AD with a row of all l's. This guarantees that Ker B ~ 2;.
2. Compute the orthogonal projection of Dc into the null space of B.
cp = [ I - BT (BBr)-IB] Dc.
^ Cp
3. c - ~ 7 I , i.e. ~ is the unit vector in the direction of %.
4. b'=a0-~tr~, i.e. take a step of length ctr in the direction ~, where r is
the radius of the largest inscribed sphere
1
r=
t/n(n- 1)
and ~E(O, l) is a parameter which can be set equal to 1/4.
7. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 379
5. Apply inverse projective transformation to b"
Db"
h~--~
erDb , •
Return b.
Step 2. Check for infeasibility.
We define a "potential" function by
cTx
f(x) = Z ln--
i Xt
We expect certain improvement 6 in the potential function at each step. The value
of 3 depends on the choice of parameter ~ in Step 1.4• For example if ~ - 1/4 then
6=1/8. If we don't observe the expected improvement i.e. if f(x(k+l))>f(x(k))--3
then we stop and conclude that the minimum value of the objective function must
be strictly positive. When the canonical form of the problem was obtained by trans-
formation on the standard linear program, then this situation corresponds to the
case that the original problem does not have a finite optimum i.e. it is either infea-
sible or unbounded•
Step 3. Check for optimality.
This check is carried out periodically• It involves going from the current
interior point to an extreme point without increasing the value of the objective
function and testing the extreme point for optimality• This is done only when the
time spent since the last check exceeds the time required for checking•
3. Underlying conceptual algorithm
In this Section we show how the computations in step 2 represent an underlying
conceptual sequence of operations. We will also introduce more notation and state
theorems about the performance of the algorithm. All proofs will be given in Sec-
tion 4. (A note on notation : In choosing variable names we use the same philosophy
as in the design of a large program: Some variables are local to a theorem while
others are global, some theorems are stated in terms of "formal" parameters and
in order to apply them to the algorithm, one has to substitute actual values for the
formal parameters e.g•, a result about one step of the algorilhm may be stated in
terms of input point a and output point b, and can be applied to the k th step of the
algorithm by substituting a = x_ (k) and b : x ( k + I) •)
•
3.1. The top level
Theorem 1. In O(n(q+log n)) steps the algorithm finds a feasible point x such that
cTx
_ _ :
-~ 2--q •
e Ta 0
We associate the objective function c r x with a "potential function" f ( x )
8. 380 N. KARMARKAR
given by
f(x) y
7 ~ x; J"
Theorem 2. Either (i) f(xtk+l))~f(x(k))--6 or
(ii) the minimum value o f objective function is strictly positive, where 6 is a
constant and depends on the choice o f the value o f the parameter ~ o f Step 1, Sub-
step 4.
A particular choice that works: If ~ = 1/4, then 6~1/8.
3.2. The basic step
One step of the algorittun is of the form b=~0(a) which consists of a sequ-
ence of three steps:
1. Perform the projective transformation T(a, a0) of the simplex A that
maps input point a to thecenter a0.
2. Optimize (approximately) the transformed objective function over an
inscribed sphere to find a point b'.
3. Apply the inverse of T to point b' to obtain output b.
We describe each of the steps in detail.
3.3. Projective transformation T(a, ao)
Let a = (al, a2 . . . . , a,)
Let D = d i a g {at, a2 . . . . , a,} be the diagonal matrix with diagonal entries
al, a2 . . . . . a,. Then T(a, ao) is given by
D-1X
x'- where e=(1,1,.. 1).
eTD -ix "'
Its inverse is givenby
Dx"
X -- eTDx .
L e t f ' be the transformed potential function defined by
f ( x ) = f ' (T(x)).
Then
c'Ty
f'(Y) = Z In ~ - ~ Inaj
J Yj J
where c ' = Dc. Let f2' be the transformed affine space f2.
A x = O ¢~, AD x" =0.
Thus 12' is the null-space of AD.
9. A NEW POLYNOMIAL-TIME A L G O R I T H M FOR LINEAR P R O G R A M M I N G 381
3.4. Optimization over a ball
Let r be the radius of the largest ball with center a0 that can be inscribed in
the simplex A. Then
1
t-
en(n-i)
We optimize over a smaller ball B(ao, c~r) 0 < ~ < 1 for two reasons:
1. It allows optimization of f '(y) to be approximated very closely by opti-
mization of a linear function.
2. If we wish to perform arithmetic operations approximately rather than
by exact rational arithmetic, it gives us a margin to absorb round-off errors without
going outside the simplex.
One choice of the value of ~ that works is ~ = 1/4 and corresponds to 3 > 1/8
in Theorem 2. We restrict the affine space ~ ' = {Yl ADy=0} by the equation ~'y~=
= 1. Let O" be the resulting at'fine space. Let B be the matrix obtained by adding
a row of all l's to AD. Then any displacement in ~ " is in the null space of B,
i.e., for u E ~ " ~ ? ' = u + K e r B .
We are interested in optimizing f ' ( y ) over B(a0, ~r)NO". First, we prove
the existence of a point that achieves a constant reduction in the potential function.
Theorem3. There exists a point b'EB(a0, c o ' ) n ~ " such that f ' ( b ' ) = < f ' ( a 0 ) - 3 ,
where 3 is a constant.
Then we prove that minimization of f '(x) can be approximated by minimi-
zation of the linear function c'rx.
Theorem 4. Let b" be the point that minimizes c ' r x over B(ao, e r ) N f U . Then
f ' ( b ' ) ~ = f ' ( a o ) - 6 where 3 is a positive constant depending on ~. For c~=1/4 we
may take 6 = 1/8.
Finally we describe an algorithm to minimize c'rx, over B(a0, e'r).
Algorithm A.
1. Project c' orthogonally hIto the null space o f B:
c, = [ I - Br(BB r) -1B]c'.
2. Normalize cv:
Cp
: lop---L
3. Take a step o f length ~tr along the direction - c v
h p _~ a o - ~ r c p ,
Theorem 5. The point b" returned by algorithm A minimizes c ' r x over B(a0, ~r)OfY'.
10. 382 N. KARMARKAR
4. Analysis of the algorithm
Proof of Theorem 5. Let z be any point such that zEB(ao, ~r)7) Q"
b', z E Q" ~ B (b" - z) = 0
=~ B T ( B B r ) - a B ( b " - Z) = 0
= (c' -- ce)T(b" -- Z) = 0
=~ c ' T ( b ' - - Z ) : c ~ ( b ' - - z )
ceT(b' - Z) : IceTIcer[ao-- o~r~e - z]
= [cpr[{~.er(ao-- z) -- ~r}
~-er(ao-z) --<--[~:pllao--Z[ --<--~r since zEB(ao, ~r)
c~(b'-z) <_- 0
c ' r ( b " - z) <= 0
c ' T h ' ~ c tT z, for all zEB(ao, a r ) N Q " , l
Proof of Theorem 3. Let x* be the point in ( 2 " N S where the m i n i m u m of c t T x
is achieved. Clearly x* ~[B(ao, ~r).
Let b" be the point where the line segment aox* intersects the b o u n d a r y
of the sphere B(ao,~r). h ' = ( 1 - 2 ) a o + 2 x * for some 2E[0, 1]. By definition,
b ' E B ( a o , ~ r ) and ao, x*EQ" imply b'EgT'.
Hence h'EB(ao, ~r)Og2"
c ' r b ' = (1 - 2 ) c ' r a o + A c ' r x * = (1--2)c'Tao
cPTao ]
c'rb ' 1- 2
and
b~ = (1 - 5~)aoj + 2x*.
Hence
c ' r ao b~ (1 - )0 a. r + 2x,*-
f'(ao)-f'(b') = Z In = ~ In
c ' r b ' aoj i (1 - 2)aoj
=Zln[14 2 x~.]
j l--2 aojJ"
11. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 383
Now we use the following inequality: If P~_->O then /-/(1 +P3=>I + z ~ P t . There-
i
fore, z~ In (1 + P 3 =>ln (1 + z~ P~).
i
f f ( a o ) - f ' ( b ' ) => In lq 1--2 aoi • x = In 1+
h" = ( 1 - 2 ) a o + 2 X *
b'-ao = 2(x*-ao)
c{r = ]b'-aol = 2[x*-aol -<- XR
where R is the radius of the smallest circumscribing sphere and R/r=n- 1.
r
)~ -->_cx -- - -c~
R n--1
2n n~
14 1~-2 >- 1 ' na/(n -1) = 12r - - >- 1 +
-- -I- 1 - c ~ / ( n - 1) n-l--~
f ' ( a o ) - f ' ( b ' ) ->_ ln(1 + c 0.
Taking 6 = I n ( 1 +c0, f'(h')<=f'(ao)-3. I
Proof of Theorem 4. First we prove a few lemmas.
X2
Lemma 4.1. If Ix1-<__13<1 then [ln(l+x)-xl~--
2(1 - / ~ )
Proof. A simple exercise in calculus. I
Lemma 4.2. Let fl=ct n Thenforany xEB(ao,~r), In "'J ~ f12
• aoj 2 (1 - fl) "
Proof. xEB(ao, ~r) implies ]x-ao/2_-<~2r~, Therefore
y
7" ~ aoj / (l/n) 2 n(n--1)
Xj -- aoj =
i-<13 for j 1. . . . , n
I aoj
Iln(iNxj--aollaoj = 2(1-I 13) ~ aoi )' by Lemma 4.1
ao.~ J
xi--ao,]< txi--ao,l'
Zln = Z l n [ l + xj-a°J]
xj
j j aoj aoj J
~[lnXJ x,--aoy<= ------'-szl (xi--aoji' - fl~
• ao~. aoj 2(I--13__#~ aoj. / 2(1--13)
but
xy--aoj_ 1 [~xj--~aoj]=-~ol/[1--1]=O
J aoj -- aoj " j
~- 2(1-13)" I
12. 384 N. KARMARKAR
Now we prove Theorem 4.
Definef(x)=n In c'Tx Let bm be the point in B (ao, czr)A f2" where f(x)
CpT ao •
achieves its minimum value.
f(ao) - f ( b ' ) = f(ao) -fib.,) +f(h,,,) - f ( b ' )
= [f(a0)-f(b,~)]
+ [f(b,.)-(f(ao) +f(b,.))]
-[f(b')-(f(ao) +f(b'))]
+ [f(b,,) -f(b')]
By Theorem 3,
(1) f(ao)-f(b,,,) --> In (I +a).
For xEB(ao, :¢r)(q f2",
f(x) - (f(ao) +f(x)) = Z In - - - -
cPTx In e ' r a ° - n ~ ' In e'r--''~x
j Xj J ao j cp T ao
=-Z In xj
j aoj
I ~ In xj f12
(2~ If(x)- (f(a°)+f(x))l = aoj <= 2(i= D
by Lemwa 4.2.
Since f(x) depends on c'rx in a monotonically increasing manner, fix)
aud e'rx achieve their minimum values over B(a0, ~r) (3 Q" at the same point, viz b',
(3) f(b,,,) ~ f(b').
By (1), (2) and (3),
fl ~ ~2 ~2 H 2
f(ao) - f ( b ' ) >= In (1 + 00
(l-fl) - 2 n ( n _ 1) [1 _ ~/n__Z_~_]
Define
0¢2 ~2//
6=a
2 (n-l)
[
1--c~
V ]
f ( a o ) - f ( b ' ) => 6.
Observe that
~2
cx~ l
1. .lira 3(n)=~z-~-
... • If ~=~-, then lira 6 (n) => I
1-c¢
>1
2. If" n=>4, c~=1 then ~ = T o " |
Proof of Theorem 2. By Theorem 4, f " (b' ) = f '(a o)-6. Applying the inverse trans-
<
form T-l(a, ao) and substituting x(aJ=T-l(ao), we obtain x(k+l)=T-l(b'),
f(xC~+.)~f(x%-6. |
13. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 385
Proof of Theorem 1. For each k we have
f ( x (k+x)) ~_ f ( x ok))-- 6
f ( x (k)) _~ f ( x C°))-- k6
crx(k) In era° -
T xF" J aoj
In cr x(k)
n ~_ 2 In (x5 k)) -- ~ In a0j - k3.
crao j J
c r x tk) k
But x~k)~l, aoj= 1 , so lnc---~ao<-ln(n)--~'n Therefore there exists
a constant k" such that after m=k'(n(q+ln (n))) steps.
cTx
Crao =< 2 - L II
5. Transformation of a linear program to canonical form
Consider a linear programming problem in the standard form:
minimize cTx
~Ax_~ b
subject to t x _-> O.
There are many ways by which this problem can be converted to our canonical
form. We describe one such method.
Step 1. Combine primal and dual problems.
Ax=>h
AZu =< c
eTx--bTu = 0
x_~O u~O.
By duality theory, this combined problem is feasible if and only the original problem
has finite optimum solution.
Step 2. bltroduee slack variables.
Ax-y =b
Aru+v = c
crx-bru = 0
x_~O u~O y=>O v~O.
14. 386 N. KARMARKAR
Step 3. Introduce an artificial variable to create an interior starthlg pohTt. Let Xo, Yo,
uo, vo be m i c t l y interior points in the positive orthant
minimize ~.
Ax-y+(b-Axo+Yo)• = b
JAru+v+(c-Aruo)~ = c
subject to ] e r x _ b r u + ( _ c r x o + b r v o ) X = 0
Ix_-->0 y=>0, u=>0, v:>0, )~=>0.
Note that x - x 0 , Y=Yo, u = u o , V=Vo, ;~=1 is a strictly interior feasible solution
which we can take as a starting point. The minimum value of X is zero if and only
if the problem in step 2 is feasible.
Step 3. Change o f Notation. For convenience of description we rewrite the problem
in Step 3 as
minimize c r x , c, xER"
subject to Ax = b, x => 0
x = a is a known strictly interior starting point, and the target value of the objective
function we are interested in is zero.
Step 4. A projective transformation o f the positive orthant into a simplex. Consider
the transformation x ' = T ( x ) where xER", x ' E R "+1 are defined by
xl/ai i = 1, 2, ..., n
x" -- X (xflaj) + 1
J
Xn+ 1 ~- 1 - X i.
i=1
Let P+ denote the positive orthant
P+ = {xER"lx =~ 0}
and A denote the simplex
n+I
A={xER"+IIx->0, z~x,=l}- i=l
The transformation T(x) has the following properties:
I. It maps P+ onto A.
2. It is one-to-one and the inverse transformation is given bv
a;x~ i = 1, 2 . . . . . n.
"k"i _ _ Xn+l
3. The inaage of point a is the center of the simplex.
4. Let Ai denote ith column of A. I f
~ Aix i = b
i=1
15. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 387
then
.~, Aialx; = b
Xn+l
or
Aiaix; - b x , + t 0.
i=l
We define the columns o f a matrix A" by these equations:
A ~ = a i A i, i= 1,...,n
A~+I : - b .
Then
n+l
Z A x; = O.
i=1
Let I2" denote this affine subspace, i.e. f2'={x'ER"+IIA'x'=0}. Thus we
get a system of homogeneous equations in the transformed space.
5. Since we are interested in target value of zero, we define Z to be the afflne
subspace corresponding to the zero set of the objective function in the original space
Z = {xER"IcTx = 0}.
Substituting
a i x~
Xi = . ,
Xn+ l
we get
n+l
ciaixi•
Y I
- O.
l=l Xn+l
Define c'ER "+1 by
c;=aici i=1,2 .... ,n
c L 1 = 0.
Then c T x = 0 implies c ' T x ' = 0 .
Therefore the image of Z is given by Z ' = {x'ER"+l[c'rx'=0}.
The transformed problem is
minimize crTx ~
A'x' = 0
subject to n+l
×' > 0, Z x'i = 1
i=l
The center of the simplex is a feasible starting point. Thus we get a problem
in the canonical form.
Application of our main algorithm to this problem either gives a solution
x' such that c ' T x ' = 0 or we conclude that minimum value of c ' r x ' is strictly posi-
tive. In the first case, the inverse image of x' gives an optimal solution to the original
problem. In the latter case we conclude lhat the original problem does not have
a finite optimum, i.e. it is either infeasible or unbounded.
9*
16. 388 N. KARMARKAR
6. Incremental computation of inverse
6.1. A modification to the main algorithm
The computational effort in each step of the main algorithm is dominated
by the computation of cv:
cv = [I--Br(BBr)-~B]Dc.
We have to find the inverse of (BB r) or solve a linear system of equations
of the form (BBr)u=v.
(1)
The only quantity that changes from step to step is the diagonal matrix D,
since D}k) =x} k). In order to take advantage of the computations done in previous
steps, we exploit the following facts about matrix inserve:
Let D, D' be n × n diagonal matrices.
1. If D and D" are "close" in some suitable norm, the inverse of AD'2A r
can be used in place of the inverse of AD2A r.
2. If D and D' differ in only one entry, the inverse of AD'2A r can be com-
puted in O(n') arithmetic operations, given the inverse of AD2A r.
(Note: Instead of finding the inverse, one could also devise an algorithm
based on other techniques such as LU decomposition. In practice, it may be better
to update the LU decomposition directly, but this will not change the theoretical
worst-case complexity of the algorithm.)
We define a diagonal matrix D "ok), a "working approximation" to D t*) in
step k, such that
1 [D','k']
(2) ~I_D~) I -<2 for i-- 1,2,...,n.
We use (A(D'(k))2Ar) -1 in place of (A(D(k))2Ar) -1. Initially, D'(°)=D(°)= 1 I.
n
We update D' by the following strategy:
1. D'(k+lJ=a(k)D "Ck), where a (k) is a scaling factor whose significance will
be explained in Section 6.3. For the purpose of this section take a (k)= 1.
2. For i = l t o n d o
if
'
then let D~[k+I)=D~(~+I); and update (A(D'(k+x))~Ar) -~ by rank-one modification.
Thus, this strategy maintains the property in equation (2). Now we describe
the rank-one modification procedure. Suppose a symmetric square matrix M is
17. A N E W P O L Y N O M I A L - T I M E A L G O R I T H M FOR L I N E A R P R O G R A M M I N G 389
modified by a rank-one matrix expressed as the outer product of two vectors u and
v. Then its inverse can be modified by the equation
(3) (M+uvr)-i - - 1 (M-Xu)(M-Xv) r
=M -- l + u r M - l v "
Given M -x, computation of (M+uvr) -x can be done in O(n 2) steps. In order to
apply equation (3), note that if D" and D" differ only in the ith entry then
(4) AD'2Ar=AD"ZAr+[D~ -D~Z] alaT
where a~ is the ith column of A. This is clearly a rank-one modification. If D' and
D" differ in l entries we can perform I successive rank-one updates in O(n21) time
to obtain the new inverse. All other operations required for computing cv, the
orthogonal projection of c, involve either multiplication of a vector by a matrix,
or linear combination of two vectors, hence can be performed in O(n 2) time.
6.2. Equivalence of the modified and original problems
In this Section we analyze the effect of the replacement of D (k) by D '(k).
Consider the following generalization of the optimization problem over an inscribed
sphere in the transformed space:
Problem P" :
Minimize c'rx }
(5) subject to Bx = 0 .
and x r Q x --<_~'r
(Taking Q = I and ~'=ct corresponds to the original problem.)
We are going to show that replacing D ~k) by D "tk~ corresponds to solving
the problem P , for appropriate choice of Q. Let h(x)=xrQx. Its gradient is given
by Vh(x)=2Qx.
At the optimal solution x to the problem P',
(6) c' = BTZ+# .2Qx
where the vector 7, and the scalar # are undetermined Lagrange multipliers
BQ-lc" = BQ-aBr~,+21~Bx = BQ-1BrZ
(7) k = (BQ-1BT)-IBQ-Xe ".
Substituting for k in equation (6), x, which we now denote by cv, is given (up to
a scale factor) by
(8) x = cv = [I-Q-XBT(BQ-1Br)-IB]Q-lc ".
This requires the inverse of BQ-IB r. Substituting B = 7 [A° 1 we get
[ADQ-1DA r ADQ-le]
(9) BQ-XB r = [ (ADO_Xe)r erQ_Xe | .
18. 390 N. KARMARKAR
It is enough to know the inverse of the submatrix ADQ-aDA r, because then
(BQ-~Br) -1 can be computed in O(n 2) additional steps by the formula
(10) [M ~]-~_ 1 [(c-arM-xa)M-~+(M-~a)(M-la) T -M-Xa]
aT c-arM-la - (M-la) r 1
where M, a and c are m × m , m × l , 1)<I matrices, respectively. But we know
(AD'2Ar) -~ where D ' = D .E, and E is a diagonal "error matrix" such that E~E
E[1/2, 2]. Setting Q = E -2 we get
AD,2A r = ADEeDAr = ADQ-1DA r.
In the modified algorithm we keep (AD'eAr) -~ around, update it whenever
any entry in D' changes and use it in equation (8) to compute cp and thereby solve
the modified problem P'.
Now we relate the solution of problem P' to the main optimization problem.
Since Qu=Eff2E[]/2, 2]
(11) L x T x <=
2 xrQx ~ 2xrx.
Hence
B ao, ~--r ~ {xlxrQx ~ c~'r} ~ B(ao, 2~'r).
Taking ~'=~/2, and taking intersection with f2"
(12) B(a0, ~r/4)0 f2" c= {xlxTQx ~ ~'r, Bx = O} c B(ao, ar)N g2".
Because of the first inclusion,
(13) rain {f'(x)[xrQx ~ c(r, Bx = 0}
rain {f'(x)lxEB(a0, c~r/4)f-) f2"} ~f'(a,,) - l n (1 +c~/4).
The last inequality follows from Theorem 3. Thus for the modified algorithm, the
claim of Theorem 3 is modified as follows
(]4) f ' ( b ' ) <=f'(a0)-ln (I +a/4).
Because of the second inclusion, Lemma 4.2 continues to be valid and we can appro-
ximate minimization of f "(x) by minimization of a linear function. The claim (as in
Theorem 2) about one step of the algorithm f ( x (k+t))~f(x(k))-6 also remains
valid where 6 is redefined as
(15) 6= In(l+e/4) 2(1--fl)"
This affects the number of steps by only a constant factor and the algorithm still
works correctly.
19. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 391
6.3. Performance of the modified algorithm
In this Subsection we show that the total number of rank-one updating ope-
rations in m steps of the modified algorithm is O(mJ/n). Since each rank-one modi-
fication requires 0@21 arithmetic operations, the average work per step is O(n 2"5)
as compared to O(n 3) in the simpler form of the algorithm.
In each step lh'-a0]_=c~r. Substituting h'=T(x(k+t)); a 0 = n - l e and T ( x ) =
D-1X I
- - r - -
eTD-lx " 1/n(n - 1)
[ x}k+i)lx}k) 1.] 2 < ~_2
n] = n ( n - l ) '
j
Let
a(k) 1 j~ (k+l) .(k)
(16) : -- Xj /Xj ,
El
Then
f i]2<=
Let
..(k+l)
(17/ A!k) - .tl
X!k) ~(k) •
Therefore • [A~ k ) - 1]2_-<fla. Recall that D (k) = Diag {x(~'), x~ k). . . . . x(,k)}, D(k+l)=
i
Diag {x}k+l), ,,(~,+11 x,(k+l)}. D"(k) is updated in two stages.
First we scale D "(k) by a factor a (k) defined in equation (16)
(18) D ' ( k + l ) _-- a(g)D'(k).
Then for each entry i such that
D,(k+ 11]~
we r e s e t D ; I ( g + I ) = D ! / k + l ) .
Define discrepancy between D uo and D '(k) as
(19) 5}k) = In f D;i('' "l
t D}p 1"
Just after an updating operation for index i,
6} k) = 0.
Just before an updating operation for index i,
20. 392 N. KA.RMARKAR
Between two successive updates, say at steps kx and k~,
~}~')-,~1 ~'> = " Z ' [z}~+'- ,~?>1.
k:k I
Let A~ k) be the change in the discrepancy at step k.
(20) z16}~) = 6}~+x)_ 6}~).
Then
(21) k•l
k~k I
Now we analyze the change in discrepancy in step k, assuming that no updating
operation was performed for index i.
D{,(k+l> D}p> a(~)x}k>
Af}k) = &}k+x)_6[~) = In --.ntk+l) D~ff) = In ~ = - l n A [ k)
(22) IzJ6}~)I = IlnzJ}~>
1.
Let nL= the number of updating operations corresponding to index i in m
n
steps of the algorithm and N = Z nL, be the total number of updating operations
1=1
in m steps.
By equations (20) and (21),
k=l
(23) In ~ N ~ Z ~ Iln Atk)[.
i=1 k=l
We use equation (17) to bound the R.H.S. of equation (23)
Z [A?>- 1]' ~_ t~' =, IA?) - 11 ~-
i
=~ [In A}a - (A/k>- 1)] ~_ (A[k)-- 1)3
2-0 ---~ by Lemma 4.1
Z [A[~)- 1]' _ p' =, Z [a~~>- if ~_ ¢n~.
l i
21. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 393
Hence
Iln A}~)I = Z ]lnA[ k) - (A}~)- 1)-4-(A!k)-- 1)1
i=l i=l
-~ ~ [lnA}~)-(A}*'-I)I÷ ~ [A}~)- ll
i=i i=1
< y [Aif)- 1]2
2 (l
(24) ~ IlnA}k)] -< m "2(i-- B) ~-gn-/~ = O (m Vn-).
i=I k:l
From equations (22) and (24)
(25) N : O(ml/n).
7. Concluding remarks
7.1. Global analysis of optimization algorithms
While theoretical methods for analyzing local convergence of non-linear
programming and other geometric algorithms are well-developed the state-of-the-art
of global convergence analysis is rather unsatisfactory. The algorithmic and analy-
tical techniques introduced in this paper may turn out to be valuable in designing
geometric algorithms with provably good global convergence properties. Our method
can be thought of as a steepest descent method with respect to a particular metric
space over a simplex defined in terms of "cross-ratio", a projective invariant. The
global nature of our result was made possible because any (strictly interior) point in
the feasible region can be mapped to any other such point by a transformation that
preserves affine spaces as well as the metric. This metric can be easily generalized to
arbitrary convex sets with "well-behaved" boundary and to intersections of such con-
vex sets. This metric effectively transforms the feasible region so that the boundary of
the region is at infinite distance from the interior points. Furthermore, this tians-
formation is independent of the objective function being optimized. Contrast this
with the penalty function methods which require an ad hoc mixture of objective
function and penalty function. It is not clear a priori in what proportion the two
functions should be mixed and the right proportion depends on both the objective
function and the feasible region.
7.2. Comments on the factor "L" in running time
Since representing the output of a linear programming problem requires
O(L) bits per variable in the worst case, the factor L must appear in the worst-case
complexity of any algorithm for linear programming. In practice L is much smaller
than n. Therefore, whenever it is possible to replace a factor of n by L, one should
always do so if one is interested in an algorithm which is efficient in practice.
22. 394 N. KARMARKAR
7.3. Performance in practice
Each step of the algorithm requires optimization of a linear function over an
ellipsoid or equivalently, solution of a system of linear equations of the type
(ADAr)x=b, where A is a fixed matrix and D is a diagonal matrix with positive
entries which changes by small amount from step to step. We devise a method based
on successive rank-one modifications and prove a worst-case bound of O(n 2"5)
arithmetic operations per step.
In practice, the matrices encountered are sparse, hence more efficient methods
for solving the above problem are possible. Another feature of the algorithm which
can lead to computational saving is that it is not necessary to find the exact solution
to the optimization problem stated above. Here it is useful to distinguish between
two types of approximate solutions. Let x0 be the exact solution and x an approxi-
mate solution and let cTx be the objective function. A strong approximation (or
approximation in the solution space) requires that x be close to x0 in some suitable
norm. A weak approximation (or approximation in objective function space)requires
that c r x be close to crx0. A weak approximation is sufficient for our method, and
is easier to achieve numerically. The optimization problem in each step is a good
candidate for applying iterative methods to obtain approximate solutions. The
worst-case bound is based on finding an exact solution.
The number of steps of the algorithm depends on the "R/r" ratio, i.e. the
ratio of radius of the sphere circumscribing the polytope to the radius of the inscribed
sphere. We prove an O(n) upper bound on this ratio. However it is likely to be smaller
than n in typical problems of praclieal interest and also in the worst-case if the cir-
cumscribing sphere is chosen to include only that portion of the polytope having
values of objective function better than the current value, and after computing lhe
direction of one step (as described in section 2) its length is chosen to be ~r" ralher
than ar where r' is the largest possible step length without going outside the feasible
region. The possibility of improvement in R/r ratio gives rise to a number of research
problems for further investigation, such as probabilistic analysis of the problem,
experimental measurements on problems arising in practical applications, analysis
of special cases of practical interest such as "box" type of constraints and finally,
the most difficult oi all, obtaining a better worst-case bound.
7.4. Numerical stability and round-off errors
In the ellipsoid algorithm, the round-off errors in numerical computation
accumulate from step to step. The amount of precision required in arithmetic ope-
rations grows with the number of steps. In our algorithm, errors do not accumulate.
On the contrary, the algorithm is self-correcting in the sense that the round-off
error made in one step gets compensated for by future steps. If we allow 1% error
in computation of the vector x (~ in each step, we can compensate for the errors
by running the algorithm 2 % more steps.
23. A NEW POLYNOMIAL-TIME ALGORITHM FOR LINEAR PROGRAMMING 395
7.5. Multiple columns and parallel computation
In the simplex method, the current solution is modified by introducing a non-
zero coefficient for one of the columns in the constraint matrix. Our method allows
the current solution to be modified by introducing several columns at once. In the
algorithm described in this paper, all columns are "active", but a variation in which
a suitably chosen subset of columns is active at one time is possible. It seems that
on a highly parallel architecture a method that uses many columns at once may be
preferable to the simplex method.
7.6, Other areas for further research
Since many polynomial-time solvable combinatorial problems are special
cases o f linear programming, this work raises the possibility o f applying the new
techniques to such special cases. Problems solved by the column generation method
require a special variant of the algorithm.
Acknowledgement. The author wishes to thank Arjen K. Lenstra, Mark Manasse,
Joan Plumstead, Alon ltai, L~iszl6 Babai, Mike Garey and Ron Graham for pointing
out errors and making suggestions for improvement.
References
[1] H. S, M. COXETER,Introduction to Geometry, Wiley (1961).
[2] G. B. DANTZIG, Linear Programming and Extensions, Princeton University Press, Princeton,
NJ (t963).
[3] M. GR6rSCHEL, L. LOVASZand A. ScHmJvzg, The Ellipsoid Method and its Consequences in
Combinatorial Optimization, Combinatorica 1 (1981), 169--197.
[4] L. G. KHACHIVAr~,A polynomial Algorithm in Linear Programming, Doklady Akademii Nauk
SSSR 244:S (1979), 1093--1096, translated in Sov&t Mathematics Doklady 20:1 (1979),
191--194.
[5] V. KLEEand G. L. MINTV,How good is the simplex algorithm? in Inequalities 111, (ed. O. Shisha)
Academic Press, New York, 1972, 159-179.
[6] O. VEBLENand J. W. YOUNG,Projective Geomeo:v, I--2, Blaisdell, New York, (1938).
[7] R. J. WAL~Ea, Algebraic Curves, Princeton University Press (1950).
N. Karmarkar
A T & T Bell Laboratories
Murray Hill, N J 07974
U.S.A.