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Unformatted text preview: IE521 Advanced Optimization Lecture 5 Dr. Zeliha Akc ¸a October 2011 1 / 22 Reading I Bertsimas 3.13.3. 2 / 22 Recall I There is a onetoone correspondence between the extreme points of a polyhedron and the basic feasible solutions . I Constructing a basic solution : I Choose a basis B of m linearly independent columns of A . I Set the remaining columns to 0, x N = (nonbasic solutions). I Solve the system Bx B = b to obtain the values of the basic variables. I Can move between adjacent (nondegenerate) basic solutions by removing one column of the basis and replacing it with another. I In the presence of degeneracy , we might stay at the same extreme point. I These are the building blocks we need to construct algorithms for solving LPs. 3 / 22 Iterative Search Algorithms I Start from an initial point, move to a new point in a determined search direction . I A search direction is defined as a feasible (inside the feasible region) and improving (improve the objective function value). I Stop when a feasible and improving direction can not be found (a local optimal is found). I Many optimization algorithms are iterative . I For linear programs, local optimal means the global optimal solution . I This is because we are optimizing a convex function over a convex set. I This makes linear programs ’easy’ to solve. 4 / 22 Feasible and Improving Directions I Assume we have point ˆ x ∈ P . We want to move in a direction d , but want to stay inside P . Definition Let ˆ x be an element of a polyhedron P . A vector d ∈ R n is said to be a feasible direction if there exists θ ∈ R + such that ˆ x + θ d ∈ P . Definition Consider a polyhedron P and the associated linear program min x ∈P c &gt; x for c ∈ R n . A vector d ∈ R n is said to be an improving direction if c &gt; d &lt; . I When we move along an improving direction d from ˆ x , we would get a better objective function value c &gt; ˆ x + c &gt; d . I Move along that the feasible and improving direction as far as possible . I Recall that we are interested in extreme points . I The algorithm moves between adjacent extreme points using improving directions. 5 / 22 Constructing Feasible Search Directions I Consider a BFS ˆ x ....
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 Fall '11
 ZelihaAkça
 Optimization, Mathematical optimization, basic variables

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