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This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.
Efficient Computation ofRegret-ratio Minimizing Set:A Compact Maxima Repres...Abolfazl Asudeh
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The document presents a method for efficiently computing a regret-ratio minimizing set (RRMS) from multi-dimensional data. For 2D data, it proposes a dynamic programming algorithm that runs in O(r*s*log(s)*log(c)) time, where r is the set size, s is the skyline size, and c is the convex hull size. For higher dimensions, it transforms the problem into a discretized min-max problem (DMM) and then solves a series of set cover problems to find an approximate RRMS in O(n*log(n)) time for fixed dimensions, where n is the number of data points. The method is evaluated on both synthetic and real-world datasets
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
This document provides a full numerical solution to the incompressible Couette flow problem using the Crank-Nicolson implicit finite difference method. The Navier-Stokes equations are simplified for this problem and non-dimensionalized. A tridiagonal matrix equation is derived and solved using both the Gauss method and Thomas algorithm. Results show the numerical solution converging to the known analytical solution and demonstrate the effects of varying time step, Reynolds number, and grid density on the velocity field.
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
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We describe a new approach to solve the problem to ๏ฌnd the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the ef๏ฌciency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
2014 vulnerability assesment of spatial network - models and solutionsFrancisco Pรฉrez
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This document proposes models and optimization problems to assess the vulnerability of spatial networks to disruptions. It defines problems such as the Max-Cost Single-Failure Shortest Path Problem (MCSFSPP) and Max-Cost Multiple-Failure Shortest Path Problem (MCMFSPP) to determine the maximum increase in shortest path length between two nodes due to failures. Failure is modeled by disruption disks of a given radius that can be placed throughout the network space. The problems are formulated as mixed integer programs to find vulnerabilities. Computational results on realistic networks are used to show how the models can characterize network robustness.
Subproblem-Tree Calibration: A Unified Approach to Max-Product Message Passin...Varad Meru
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Max-product message passing algorithms are commonly used for MAP inference in MRFs. Recent work showed these algorithms can be viewed as performing block coordinate descent in a dual objective. However, existing algorithms are limited by the restricted ways they select blocks to update. The paper proposes a "Subproblem-Tree Calibration" framework that subsumes MPLP, MSD, and TRW-S as special cases and allows more flexible block selection. The algorithm represents the problem as a subproblem multi-graph and calibrates potentials on randomly selected subproblem trees via message passing, achieving dual optimality with respect to the tree's block of variables. Experimental results show the approach converges to different dual objectives than existing methods.
This document describes a numerical study involving the solution of Poisson's equation using the finite volume method. It presents results from solving Laplace's equation on square domains with mixed boundary conditions, as well as solving the pressure Poisson equation derived from incompressible flow equations. Gaussian elimination, successive over-relaxation, and second-order accuracy are discussed. Numerical experiments demonstrate that over-relaxation improves convergence and the method achieves second-order accuracy based on grid refinement studies.
Efficient Computation ofRegret-ratio Minimizing Set:A Compact Maxima Repres...Abolfazl Asudeh
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The document presents a method for efficiently computing a regret-ratio minimizing set (RRMS) from multi-dimensional data. For 2D data, it proposes a dynamic programming algorithm that runs in O(r*s*log(s)*log(c)) time, where r is the set size, s is the skyline size, and c is the convex hull size. For higher dimensions, it transforms the problem into a discretized min-max problem (DMM) and then solves a series of set cover problems to find an approximate RRMS in O(n*log(n)) time for fixed dimensions, where n is the number of data points. The method is evaluated on both synthetic and real-world datasets
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This document provides a full numerical solution to the incompressible Couette flow problem using the Crank-Nicolson implicit finite difference method. The Navier-Stokes equations are simplified for this problem and non-dimensionalized. A tridiagonal matrix equation is derived and solved using both the Gauss method and Thomas algorithm. Results show the numerical solution converging to the known analytical solution and demonstrate the effects of varying time step, Reynolds number, and grid density on the velocity field.
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We describe a new approach to solve the problem to ๏ฌnd the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the ef๏ฌciency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
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This document proposes models and optimization problems to assess the vulnerability of spatial networks to disruptions. It defines problems such as the Max-Cost Single-Failure Shortest Path Problem (MCSFSPP) and Max-Cost Multiple-Failure Shortest Path Problem (MCMFSPP) to determine the maximum increase in shortest path length between two nodes due to failures. Failure is modeled by disruption disks of a given radius that can be placed throughout the network space. The problems are formulated as mixed integer programs to find vulnerabilities. Computational results on realistic networks are used to show how the models can characterize network robustness.
My talk at the International Conference on Monte Carlo Methods and Applications (MCM2032) related to advances in mathematical aspects of stochastic simulation and Monte Carlo methods at Sorbonne Universitรฉ June 28, 2023, about my recent works (i) "Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing" (link: https://doi.org/10.1080/14697688.2022.2135455), and (ii) "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities" (link: https://arxiv.org/abs/2003.05708).
This document discusses methods for performing sparse time-frequency representation (STFR) on signals in 2 dimensions. STFR decomposes signals into intrinsic mode functions (IMFs) by finding the sparsest representation of the signal over a redundant dictionary. The 1D version has been implemented successfully, but extending to 2D is challenging due to the need to update in two directions simultaneously. Several attempted algorithms are described, including applying the 1D algorithm to slices of the 2D signal in different directions and averaging the results. The most recent attempt uses bi-directional slicing to overcome issues with previous global approaches.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
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This document contains the course calendar for a machine learning course covering topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, independent component analysis, and clustering algorithms. The calendar lists 15 classes over the semester, the topics to be covered in each class, and any dates where there will be no class. It also includes lecture plans and slides on principal component analysis, linear discriminant analysis, and comparing PCA and LDA.
The document provides an introduction to various MATLAB fundamentals including:
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- Plotting the velocity-time solution and customizing graphs.
- Describing algorithms using flowcharts and pseudocode.
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This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It finds that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than directly calculating the matrix inverse and products. For large, sparse datasets, the sparse matrix classes in Matrix produce the fastest results of all. Retaining intermediate computations allows for very fast repeated calculations.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approaches, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are common in statistics. A small example is used to demonstrate calculating the least squares solution manually and with the lm.fit function. For a large, sparse example, using crossprod and the Matrix package classes is much faster than directly calculating the inverse and transpose. The Matrix package retains information about matrix structure and factorization to optimize least squares calculations.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It finds that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than directly calculating the matrix inverse and products. For large, sparse datasets, the sparse matrix classes in Matrix produce the fastest results of all. Retaining intermediate computations allows for very fast repeated calculations.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approaches, especially for large, sparse model matrices. The Matrix package classes also allow reuse of factorizations without recalculation. For very sparse matrices, the sparse matrix methods are fastest of all.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approaches, especially for large, sparse model matrices. The Matrix package classes also allow reuse of factorizations without recalculation. For very sparse matrices, the sparse matrix methods are fastest of all.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approaches, especially for large, sparse model matrices. The Matrix package classes also allow reuse of factorizations without recalculation. For very sparse matrices, the sparse matrix methods are fastest of all.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It finds that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than directly calculating the matrix inverse and products. For large, sparse datasets, the sparse matrix classes in Matrix produce the fastest results of all. Retaining intermediate computations allows for very fast repeated calculations.
The document provides instructions for requesting writing assistance from a website. It outlines a 5-step process: 1) Create an account with a password and email. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and select one based on qualifications. 4) Review the completed paper and authorize payment if satisfied. 5) Request revisions to ensure satisfaction, with a refund offered for plagiarized work.
Political Science Research Paper Outline. Free PoliticaKarla Adamson
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False. Cultural relativism and moral relativism are not exactly the same. Cultural relativism is the view that cultural practices and beliefs should be understood based on the cultural context from which they arise rather than being judged against the standards of another culture. Moral relativism is the view that moral truths are relative to the culture or society holding them rather than being absolute. So cultural relativism focuses on understanding other cultures, while moral relativism focuses on whether moral claims can be objectively true or false.
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This document summarizes the propagation of error bounds due to active subspace reduction in computational models. It presents two algorithms for performing active subspace reduction: one that is gradient-free and reduces the response or state space, and one that is gradient-based and reduces the parameter space. It then develops a theorem for propagating error bounds across multiple reductions, both in the parameter and response spaces. Numerical experiments on an analytic function and a nuclear reactor pin cell model are used to validate the error bound approach.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
This document compares different methods for performing least squares calculations, which are commonly used in statistical analysis. It shows that using the crossprod and Matrix functions from the Matrix package provides faster and more numerically stable solutions than the naive matrix multiplication and inverse approach, especially for large, sparse model matrices. The Matrix package classes also allow reusing factorizations to solve multiple least squares problems more efficiently.
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A Comparison Of Methods For Solving MAX-SAT Problems
1. A Comparison of Methods for Solving MAX-SAT
Problems
Steve Joy, John E. Mitchell
Mathematical Sciences,
Rensselaer Polytechnic Institute,
Troy NY 12180 USA.
email: joys@rpi.edu, mitchj@rpi.edu
http://www.math.rpi.edu/~mitchj
Brian Borchers
Mathematical Sciences,
New Mexico Tech,
Socorro, NM 87801 USA
email: borchers@rpi.edu
http://www.nmtech.edu/~borchers
Seattle INFORMS Conference
October 25{28, 1998
Session MD29
2. Seattle INFORMS MD29 1
Abstract
We compare the performance of 3 approaches to the solu-
tion of MAX-SAT problems, including a version of the Davis-
Putnam-Loveland algorithm extended to solve MAX-SAT, an
integer programming branch-and-cut algorithm and an algo-
rithm for MAX-2-SAT problems based on a semide nite pro-
gramming relation.
1 Overview
De nition of MAXSAT.
Expressing MAXSAT as an integer program.
Cutting planes.
Extended Davis-Putnam-Loveland algorithm.
Semide nite programming relaxation of MAX2SAT.
Computational results.
Conclusions.
3. Seattle INFORMS MD29 2
2 De nition of MAXSAT
A propositional logic formula F can always be written in
conjunctive normal form with m clauses and n vari-
ables.
Each clause is a disjunction of a set of literals.
Each literal is either a variable or the negation of a vari-
able.
The formula is the conjunction of the clauses.
For example:
F = (X1 _
X2) ^ (X1 _ X2 _
X3) ^ (X2 _ X3):
The satis ability problem, SAT:
Given a propositionallogic formulaF in conjunctive
normal form, is there an assignment of the logical
variables that makes F true?
The maximum satis ability problem, MAXSAT:
Given a propositional logic formula F in conjunc-
tive normal form, nd an assignment of the logical
variables that maximizes the number of clauses in
F that are true.
4. Seattle INFORMS MD29 3
Computational complexity
SAT is NP-Complete.
Even if each clause has exactly three literals, SAT is still
NP-complete.
Some special cases are solvable in polynomial time, includ-
ing 2-SAT and HORN-SAT.
MAXSAT is NP-Hard even if each clause has only two
literals.
There isno polynomialtime approximationscheme(PTAS)
for MAXSAT unless P=NP.
It is possible to approximate MAXSAT to within a factor
of 1.32 and MAX2SAT to within a factor of 1.14 in poly-
nomial time, using a semide nite programming approach
(Goemans and Williamson, 1995).
5. Seattle INFORMS MD29 4
3 An integer programming approach
Any weighted MAXSAT instance can be expressed as an
equivalent integer programming problem.
For example:
C1 = v1 _v2 weight 1
C2 = v1 _v2 _v3 weight 4
C3 = v1 _v2 weight 3
Equivalent to:
min w1 + 4w2 + 3w3
s:t: (1 ,x1) + (1 ,x2) + w1 1
x1 + x2 + x3 + w2 1
x1 + (1 ,x2) + w3 1
xi = 0 or 1; i = 1; : : : ; 3; wi = 0 or 1; i = 1; : : : ; 3
We call the added variables w1; : : : ; w3 weighted vari-
ables.
6. Seattle INFORMS MD29 5
Branch-and-cut
Use a primal heuristic similar to GSAT (Selman et al.,
1992, 1996) or the heuristics of Gu (1992,1993,1994) to nd
a good solution.
Solve the linear programming relaxation obtained
by requiring 0 xi 1 for i = 1;:::;n, 0 wi 1 for
j = 1;:::;m.
If the solution to the relaxation is non-integral, modify it
by adding cutting planes, or branch.
Use three di erent types of cutting planes:
{ Resolution cuts (Hooker, 1988.)
{ Odd cycle inequalities (Cheriyan et al., 1996.).
{ Clique cuts.
The cuts are added locally, so we don't lift them to be
valid throughout the branch-and-cut tree.
Branching: split the relaxation into two problems, in one
of which a speci c variable is required to equal zero and in
the other it is required to equal one.
The algorithm is written in C. It uses MINTO to handle
the branch-and-cut tree, which uses CPLEX 4.0 to solve
the linear programming relaxations.
7. Seattle INFORMS MD29 6
Clique cuts
Motivated bypigeonholeproblems(Hooker): holen asks
if it is possible to place n + 1 pigeons in n holes without
two pigeons being in the same hole.
We describe these cuts for 2-SAT problems:
Example: Assume our instance of 2SAT contains the four
literals l1; l2; l3; l4 and the six clauses:
l1 _ l2; l1 _ l3; l1 _ l4; l2 _ l3; l2 _ l4; l3 _ l4:
We can think of this as giving a clique, with vertices la-
belledbythe four literals, andedgesgivenbythe sixclauses.
Any satisfying truth assignment must have at least three
of these literals true. This gives a potential cutting plane.
Thiscanbeextended toMAX2SATandgeneral MAXSAT
by regarding some edges in the clique as missing and mod-
ifying the cutting plane appropriately. This is really a lift-
ing procedure.
We seek inequalities that are violated by at least 0.1. The
separation routine looks for literals with large values and
tries to nd cliques involvingsuch variables. The cliques are
built vertex-by-vertex. We reject inequalities that contain
too many weighted variables.
8. Seattle INFORMS MD29 7
Resolution cuts
A classical logical procedure.
Also implemented in branch-and-cut by Hooker and Fedjki,
1990.
Example: Given two clauses, one containing a variable and
the other containing its complement:
x1 _ x3 _ x7 _ w9
x2 _ x3 _ x7 _ w11
we can resolve to generate the following:
x1 _ x2 _ x7 _ w9 _ w11
Hookershowedthat thisisequivalentto a Chvatal-Gomory
cutting planeprocedure in the integer programmingsetting.
Our resolution routine consists of a sequence of passes, each
designed to generate resolvents satisfying certain criteria.
Only accept resolved clauses that are unit clauses and
that are violated by at least 0.3.
9. Seattle INFORMS MD29 8
Odd cycle inequalities
Derived by Cheriyan et al., 1996, motivated by similar in-
equalities for MAXCUT problems.
Example:
x1 +x2 +w10 +w11 1
+x2 ,x3 +w11 0
,x3 ,x4 +w12 ,1
,x1 ,x4 +w13 ,1
The rst, third, and fourth constraints are odd; the second
is even. Adding these constraints together,dividing by 2
and rounding coecients we obtain:
x2 ,x3 ,x4 + w10 + w11 + w12 + w13 0:
Use a modi ed version of Dijkstra's algorithm to nd cut-
ting planes.
Look for cuts violated by at least 0.3, and of length at
most 2.
Odd cycle cuts combine together clauses of length 2.
Thus, only e ective if the current relaxation containsa large
number of clauses of length 2, eg, MAX2SAT problems.
For problems with longer clauses, can only use these cuts
deep in the tree.
10. Seattle INFORMS MD29 9
Branching strategy
After generating cuts and applying bounds, we branch if
there are still fractional variables in the LP solution.
Thebranchingschemewe useisa modi cationof theJeroslow-
Wang (1990) scheme.
Jeroslow-Wang examines the probability of satisfying all
the constraints if we randomly assign values to the remain-
ing un xed variables. It attempts to pick a variable whose
assignment will have the greatest change in this probability.
Our approach di ers from Jeroslow-Wang in that:
1. We give considerably less weight to singleton clauses.
2. We consider the e ect on both branches.
11. Seattle INFORMS MD29 10
4 Extended Davis-Putnam-Loveland
Davis-Putnam-Loveland (1960,1978) is a classical branch-
ing and backtracking procedure for solving SAT.
Borchers andFurman (1995) extendedthisto solve MAXSAT
problems. (AlsodoneindependentlybyWallace andFreuder
(1996).)
We maintain an upper bound, ub, given by the number
of unsatis ed clauses in the best known solution.
We keep track of unsat, the number of clauses left unsat-
is ed by the current partial solution.
If unsat ub, backtrack from the current partial solu-
tion.
Written in C.
Code uses primal heuristic to nd good solution.
Incorporates unit clause trackingwhen unsat = ub,1.
Uses the variable selection rule of Dubois et al..
12. Seattle INFORMS MD29 11
5 Semide nite programming
This formulation is due to Goemans and Williamson, 1995.
MAX2SAT as a QP
For each logical variable yi, introduce a 1 variable xi.
Add a variable x0. We'll set yi =TRUE if x0 = xi, and
FALSE otherwise.
Let
v(yk _yl) := 0:25(1 + x0xk + 1 + x0xl + 1 ,xkxl):
We obtain v(yk _ yl) = 0 if the clause is FALSE, and 1
otherwise.
Similar formulas can be constructed for clauses involving
negated literals.
The MAX2SAT problem becomes:
maxf
m
X
j=1
v(Cj) : xi 2 f,1; 1g; i = 0; : : : ; ng:
13. Seattle INFORMS MD29 12
Formulation as an SDP
Express QP as:
maxfxT
Ax + c : xi 2 f,1;1g;i = 0;:::;ng
for a symmetric matrix A and a scalar c.
Note that xTAx = tr(XA), where X = xxT and tr de-
notes the trace of a matrix.
Rewrite problem as:
max tr(XA) + c
subject to X = xxT
xi 2 f,1;1g; i = 0;:::;n:
Notice that Xii = 1.
Relax the requirement that X be a rank 1 matrix to the
requirement that it be a positive semide nite matrix.
Write this X 0.
Obtain the semide nite programming (SDP) relaxation of
MAX2SAT:
max tr(XA) + c
subject to X 0
Xii = 1; i = 0;:::;n:
This relaxation can be solved in polynomial time using an
interior point method.
14. Seattle INFORMS MD29 13
Cutting planes
Can strengthen the SDP relaxation by adding linear in-
equalities, as shown by Helmberg and Rendl (1998) for
MAXCUT.
Add constraints of the form:
Xij + Xjk + Xik ,1
Xij ,Xjk ,Xik ,1
,Xij + Xjk ,Xik ,1
,Xij ,Xjk + Xik ,1:
These constraints are derived by considering possible values
for xixj, xjxk, and xixk for 1 variables.
Add these constraints selectively, as cutting planes.
15. Seattle INFORMS MD29 14
Our SDP implementation
Written in C.
Uses the primal-dual interior point semide nite program-
mingalgorithmof Helmberg, Rendl, Vanderbei, andWolkow-
icz (1996) to solve the SDP relaxations.
Uses a primal heuristic due to Goemans and Williamson,
1995.
Adds cutting planes of the type described above.
16. Seattle INFORMS MD29 15
6 Computational results
MAX2SAT
-
6
CPU
Clauses
0 500 1000 2000 3000 4000
1 sec
1 min
10 min
1 hr
4 hrs
y
y
yy
y
y
y
y
y
y
y
y
y
y
y
y
}
}
}
~~
~
~
~
~
~
~
~
~
y Branch-and-cut
} EDPL
~ SDP (successful)
SDP (gap=1)
Log(CPU time) vs number of clauses for random 50
variable MAX-2-SAT problems
17. Seattle INFORMS MD29 16
Results for 100 variable MAX-2-SAT problems
problem clauses unsat. B+C EDPL SDP
name clauses CPU nodes CPU backtracks CPU lower upper
p2180 5 180 0 0.62 1 0.1 1 366.85 180 180
p2180 8 180 1 4.96 1 0.7 2 1015.99 179 179
p2180 6 180 1 4.89 1 0.7 47 911.65 178 178
p2180 3 180 2 5.18 1 0.7 40 1417.08 178 178
p2180 9 180 2 5.23 1 0.7 34 2497.35 178 179
p2180 2 180 2 5.29 1 0.7 10 2152.11 178 179
p2180 1 180 3 4.99 1 0.7 162 2299.32 177 178
p2180 7 180 4 5.25 1 0.8 964 1322.69 176 176
p2180 4 180 4 5.19 1 0.9 1773 2315.90 176 177
p2180 10 180 4 5.31 1 0.9 1933 1106.76 176 176
p2300 2 300 13 7.18 1 2229.1 21334438 2307.46 287 288
p2300 3 300 13 7.50 1 2497.9 23764177 2399.92 287 288
p2300 4 300 14 7.29 1 4618.5 47629271 2202.70 286 287
p2300 9 300 15 7.51 1 8746.0 85235320 1220.04 285 285
p2300 1 300 15 8.18 1 10564.3 104553113 2086.05 285 286
p2300 10 300 15 8.24 1 12693.1 125325067 2292.71 285 286
p2300 5 300 15 11.35 3 29738.0 309999407 2263.76 285 287
p2300 6 300 17 9.67 1 69325.3 704246333 2200.03 283 284
p2300 8 300 17 9.42 1 Not Run Not Run 2123.03 283 284
p2300 7 300 20 32.33 11 Not Run Not Run 2266.63 280 281
p2400 3 400 25 13.36 1 Not Run Not Run 2097.99 375 376
p2400 7 400 26 30.69 11 Not Run Not Run 2127.00 374 375
p2400 6 400 27 18.70 3 Not Run Not Run 2134.13 373 374
p2400 4 400 28 13.58 1 Not Run Not Run 640.15 372 372
p2400 2 400 28 18.29 3 Not Run Not Run 2134.42 372 373
p2400 1 400 29 21.75 3 Not Run Not Run 2157.36 371 372
p2400 5 400 29 45.13 15 Not Run Not Run 2167.33 371 373
p2400 10 400 30 64.82 11 Not Run Not Run 2141.70 370 372
p2400 8 400 33 71.13 11 Not Run Not Run 2166.69 367 369
p2400 9 400 34 121.73 19 Not Run Not Run 2074.66 365 368
19. Seattle INFORMS MD29 18
MAX3SAT
-
6
CPU
Clauses
200 250 300 350 400 450 500
5 sec
1 min
10 min
1 hr
4 hrs
y
y
y
y
y
y
y
}
}
}
}
}
y Branch-and-cut
} EDPL
Log(CPU time) vs number of clauses for random 50
variable MAX3SAT problems
22. Seattle INFORMS MD29 21
Pigeonhole problems Hooker.
holen asks if it is possible to place n + 1 pigeons in n holes
without two pigeons being in the same hole.
-
6
CPU
Holes
0 5 10 15 20 25
1 sec
5 sec
30 sec
1 min
5 min
y
y
y
y
y
y
y
y
y
y
y
y
y
y
y
y
}
}
}
y Branch-and-cut
} EDPL
Log(CPU time) vs Number of Pigeons for pigeon
hole problems
24. Seattle INFORMS MD29 23
7 Conclusions
Branch-and-cut generates smaller trees than EDPL,
but spends more time at each node.
Branch-and-cut is very e ective for MAX2SAT prob-
lems with a reasonable number of clauses.
OncetheMAX2SATinstancehasalargenumber ofclauses,
the semide nite programming approach is superior.
The SDP relaxation appears to improve as the number of
clauses increases.
EDPL is superior for MAX3SAT. For MAX3SAT, the
branch-and-cut tree can expand in almost a binary fashion
until a sucient number of clauses of length 2 have been
generated.
Branch-and-cut is superior for the Steiner problems
and for the Pigeonhole problems.
The clique inequalities are essential for e ective solution
of the Pigeonhole problems.
25. Seattle INFORMS MD29 24
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