1) The document discusses generalized inequality constraints in convex optimization problems, where the constraints are defined by proper cones instead of just non-negativity constraints.
2) It provides examples of conic forms of optimization problems including second-order cone programming, semidefinite programming, and moment problems.
3) Vector optimization problems aim to optimize multiple objectives simultaneously and the concepts of Pareto optimality and scalarization techniques for finding Pareto optimal solutions are introduced. Trade-off analysis is discussed for analyzing the trade-offs between objectives.
This is the second lecture in the CS 6212 class. Covers asymptotic notation and data structures. Also outlines the coming lectures wherein we will study the various algorithm design techniques.
This is the second lecture in the CS 6212 class. Covers asymptotic notation and data structures. Also outlines the coming lectures wherein we will study the various algorithm design techniques.
it contains the detail information about Dynamic programming, Knapsack problem, Forward / backward knapsack, Optimal Binary Search Tree (OBST), Traveling sales person problem(TSP) using dynamic programming
Euclid's Algorithm for Greatest Common Divisor - Time Complexity AnalysisAmrinder Arora
Euclid's algorithm for finding greatest common divisor is an elegant algorithm that can be written iteratively as well as recursively. The time complexity of this algorithm is O(log^2 n) where n is the larger of the two inputs.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into subproblems and stores the results of subproblems to avoid computing the same results again.
QUESTION BANK FOR ANNA UNNIVERISTY SYLLABUSJAMBIKA
first of all i am very happy that the only university that keeps its blog updated. the habit of using algorithm analysis to justify design decisions when you write implement new algorithms and to compare the experimental performance .
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
it contains the detail information about Dynamic programming, Knapsack problem, Forward / backward knapsack, Optimal Binary Search Tree (OBST), Traveling sales person problem(TSP) using dynamic programming
Euclid's Algorithm for Greatest Common Divisor - Time Complexity AnalysisAmrinder Arora
Euclid's algorithm for finding greatest common divisor is an elegant algorithm that can be written iteratively as well as recursively. The time complexity of this algorithm is O(log^2 n) where n is the larger of the two inputs.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into subproblems and stores the results of subproblems to avoid computing the same results again.
QUESTION BANK FOR ANNA UNNIVERISTY SYLLABUSJAMBIKA
first of all i am very happy that the only university that keeps its blog updated. the habit of using algorithm analysis to justify design decisions when you write implement new algorithms and to compare the experimental performance .
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
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Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
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1. Nov 19, 2020
Kitsuregawa Lab. (M2)
Koki Isokawa
4.6 Generalized inequality constraints
4.7 Vector optimization
Reading circle on Convex Optimization - Boyd & Vandenberghe
2. Generalized inequality constraints
Standard form convex optimization problem (review)
Standard form convex optimization problem with
generalized inequality constraints
2
where are proper cones, are -convex
f0 : Rn
→ R, Ki ⊆ Rki fi : Rn
→ Rki Ki
3. Relationship with convex optimization
• Convex optimization problem is a special case with
• Some results in convex opt. problem can be diverted
• The feasible set, any sublevel set, and the optimal set are
convex
• Any point that is locally optimal for the problem is globally
optimal
• The optimality condition for differentiable (see 4.2.3) holds
without any change
Ki = R+, i = 1,…, m
f0
3
4. Basic example1: Conic form problem
One of the simplest convex optimization problems with
inequality constraints
A generalization of linear programs in which componentwise
inequality is replaced with a generalized linear inequality
4
• A linear objective
• One inequality constraint
5. Standard and inequality form conic form problem
Conic form problem in standard form
Conic form problem in inequality form
5
Both forms are derived using the analogy of linear programming
6. Basic example2: Semidefinite programming
(SDP): conic form problem when is
(the cone of positive semidefinite matrices)
If are all diagonal, the SDP reduces to a linear
program
Semidefinite program K
Sk
+ k × k
G, F1, …, Fn
6
where , and
G, F1, …, Fn ∈ Sk
A ∈ Rp×n
7. Standard and inequality form SDP
A standard form SDP
An inequality form SDP
7
where C, A1, …, Ap ∈ Sn
where B, A1, …, An ∈ Sk
8. Multiple LMIs and linear inequalities
Following problem is common to be referred as an SDP
These problems can be transformed to an SDP
8
linear objective several LMI constraints
linear equality and inequality
9. Examples: Second order cone programming 9
in which,
⇔ (Aix + bi, cT
i x + di) ∈ Ki
SOCP can be expressed as a conic form problem
10. Examples: matrix norm minimization
Suppose an unconstrained convex problem
Let , where
where denotes the spectral norm (maximum singular value)
Problem with matrix inequality constraint
A(x) = A0 + x1A1 + ⋯ + xnAn Ai ∈ Rp×q
∥ ⋅ ∥2
10
⇔
SDP semidefinite matrix
(see A.5.5)
特異値
11. Examples: moment problems 1/3
The (power) of the distribution of :
The expected value ( be a random variable in )
Moments satisfy following constraint:
(Proof) Let
moments t
xk = Etk
t R
y = (y0, y1, …, yn) ∈ Rn+1
11
Hankel matrix
12. Examples: moment problems 2/3
Let : a given polynomial in
Suppose is a random variable on
not knowing the distribution
knowing some bounds on the moment as follow
The expected value of :
p(t) = c0 + c1t + ⋯ + c2nt2n
t
t R
p(t)
12
13. Examples: moment problems 3/3
Upper and lower bound for :
Rewritten as following SDP by using moments
Ep(t)
13
14. 4.7 Vector optimization
A general :
:
• is -convex
• are convex
• are affine
Here, the two objective values need not be comparable:
we can have neither
vector optimization problem
Convex vector optimization problem
f0 K
f1, …, fm
h1, …, hp
f0(x), f0(y)
14
f0 : Rn
→ Rq
fi : Rn
→ R, hi : Rn
→ R
K ⊆ Rp
15. Optimal points and values
Here we consider the set of :
If it has a minimum element (a feasible such that
for all feasible ), is called and
- A point is optimal iff it is feasible and
- Most vector optimization problems do not
have an optimal point and an optimal value
achievable objective values
x
y x optimal f0(x) optimal value
x⋆
15
with K = R2
+
16. Pareto optimal points and values
A feasible point is if is a minimal
element and is called a
- A point is Pareto optimal iff it is feasible and
- The set of Pareto optimal values satisfies
x Pareto optimal f0(x)
f0(x) pareto optimal value
x
16
f0(x) − K
with K = R2
+
17. Scalarization
A standard technique for finding Pareto optimal points
Choose any , and consider the opt. problem and
let be an optimal point
(proof)
• If were not Pareto optimal, then there is a feasible point
which satisfies and
• Since and is nonzero, we have
scalar
x
x y
17
Contradict the assumption that is optimal for the scalar problem
x
18. Properties on scalarization
• The vector is called
• By varying , we obtain different Pareto optimal
solutions
• Scalarization cannot find every Pareto optimal point
weight vector
18
is Pareto optimal but cannot
be found by scalarization
x3
19. Scalarization of convex vector opt. problems
• When is , any solution is Pareto optimal
• For every Pareto optimal point , there is some
nonzero , such that is a solution of the
scalarization problem
e.g.,
λ
xPO
xPO
19
20. Multicriterion optimization
When a vector optimization problem involves the cone
, it is called - optimization problem
• are interpreted as different scalar objectives
• a multicriterion opt. problem is convex iff. : convex,
: affine, and : convex
K = Rq
+ multi objective
f0 = (F1, …Fq) q
f1, …, fm
h1, …, hp F1, …, Fq
20
21. An optimal point in a multicriterion problem
An optimal point satisfies for
every feasible
In other words, is optimal for each of the -th scalar
problems
If there is an optimal point, the objectives are said
x*
y
x* j
noncompeting
21
22. Trade-off analysis(1/4)
Suppose and are Pareto optimal points, say,
Here and must be both empty or nonempty
We want to compare to
x y
A C
x y
22
where A ∪ B ∪ C = {1,…, q}
25. Trade-off analysis(4/4) 25
by small amount of increase in F1
we obtain small large reduction in F2
by large amount of increase in F2
we obtain small reduction in F1
• A point of large curvature in one objective is called -
• In many applications represent a good compromise solution
knee of the trade off curve
26. Scalarizing multicriterion problems
Scalarizing multicriterion problem is shown in the form
of weighted sum objective
• we can interpret as the attached to the -th objective
• when we want to be small, we should take large
• : , or or relative importance of -th
objective compared to -th
λi weight i
Fi λi
λi/λj excahnge rate relative weight i
j
26
27. Examples: risk-return trade-off in portfolio optimization
Objectives: negative mean return and the variance of
the return
scalarize with :
λ1 = 1, λ2 = μ > 0
27
: the amount of asset
: price change of asset
xi i
pi i
(QP)
28. Examples: risk-return trade-off in portfolio optimization
e.g.,
28
opt. problem Optimal risk-return trade-off curve
x
scalarized opt. problem
known asset properties
29. Examples: risk-return trade-off in portfolio optimization
e.g.,
29
opt. problem Optimal risk-return trade-off curve
x
scalarized opt. problem
known asset properties