L7_optim - Lecture 7: Existence of Solutions February 7,...

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Lecture 7: Existence of Solutions February 7, 2007
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Lecture 7 Outline I Optimization Terminology I Issues in Optimization I Existence of Solutions I Implications and Applications I Optimality Condition Convex Optimization 1
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Lecture 7 Optimization Terminology Consider the following constrained optimization problem minimize f ( x ) subject to x C Example: C = { x R n | g ( x ) 0 , x X } Terminology : The set C is referred to as feasible set We say that the problem is feasible when C is nonempty We refer to the value inf x C f ( x ) as optimal value and denote it by f * We say that a vector x * is optimal (solution) when x * is feasible and attains the optimal value f * , i.e., x * C and f ( x * ) = f * Convex Optimization 2
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Lecture 7 Important Issues in Optimization The problem minimize f ( x ) subject to x C Correctness of the model needs to be addressed in any optimization problem before attempting to solve the problem I Correctness of the optimization model includes Problem feasibility: Is C is empty? Problem “meaningful”: Is f * = + ? (equiv. to dom f C = ?) Finitness of the optimal value: Is f * = -∞ ? Convex Optimization 3
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Lecture 7 Model Correctness Examples I A constraint set in Geometric Programming C = ± ( x,y,z ) | x 2 + 3 y z y, x y = z 2 , 2 x 3 ² I Stability in Linear Time Invariant Systems often reduces to determin- ing whether there exist matrices P ± 0 and Q ± 0 such that A T P + PA = - Q Determining feasibility of a problem can be a hard problem on its own Convex Optimization 4
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Lecture 7 Feasibility Problem C = { x R n | g ( x ) 0 , x X } A feasibility problem can be posed as an optimization problem with objective function f ( x ) = 0 for all x R n . I The feasibility problem: minimize 0 subject to g ( x ) 0 , x X Convex Optimization 5
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Lecture 7 Existence of Solutions in Convex Optimization minimize f ( x ) subject to x C with a closed convex function f and a closed convex set C In the rest of the development: we assume that the problem is feasible and meaningful , i.e., C 6 = and dom f C 6 = Existence of solutions in convex optimization is intimately related to the
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L7_optim - Lecture 7: Existence of Solutions February 7,...

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