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Unformatted text preview: Final Review IE417 In the Beginning... In the beginning, Weierstrass's theorem said that a continuous function achieves a minimum on a compact set. Using this, we showed that for a convex set S and y not in the set, there is a unique point in S with minimum distance from y . This allowed us to show that we can separate a convex set S from any point not in the set . Finally, we arrived at Farkas' Theorem which is at the heart of all optimization theory. Convex Functions Recall that if f :S → R n is twicedifferentiable, then f is convex if and only if the Hessian of f is positive semidefinite at each point in S . If f is convex and S is a convex set, the point x * ∈ S is an optimal solution to the problem min x ∈ S f ( x ) if and only if f has a subgradient ξ such that ξ T ( x x * ) ≥ 0 2200 x ∈ S. Note that this is equivalent to there begin no improving, feasible directions . Hence, if S is open, then x * is an optimal solution if and only if there is a zero subgradient of f at x * . Characterizing Improving Directions Unconstrained Optimization Consider the unconstrained optimization problem min f ( x ) s.t. x ∈ X where X is an open set (typically R n ) . If f is differentiable at x * and there exists a vector d such that ∇ f ( x * ) T d < 0 , then d is an improving direction. If ∇ f ( x * ) T d > 0 2200 d ∈ R n , then there are no improving directions. Optimality Conditions Unconstrained Optimization If x * is a local minimum and f is differentiable at x * , then ∇ f ( x * ) = 0 and H( x * ) is positive semidefinite. If f is differentiable at x * , ∇ f ( x * ) = 0 , and H( x * ) is positive definite, then x * is a local minimum. If f is convex and x * is a local minimum, then x * is a global minimum. If f is strictly convex and x * is a local minimum, then x * is the unique global minimum. If f is convex and differentiable on the open set X , then x * ∈ X is a global minimum if and only if ∇ f ( x * ) = 0 . Constrained Optimization Now consider the constrained optimization problem min f ( x ) s.t. g i ( x ) ≤ 0 2200 i ∈ [1, m ] h i ( x ) = 0 2200 i ∈ [1, l ] x ∈ X where X is again an open set (typically R n ). Feasible and Improving Directions Constrained Optimization Definition : Let S be a nonempty set in R n and let x * ∈ cl S . The cone of feasible directions of S at x * is given by D = { d : d ≠ 0 and x * + λ d ∈ S, 2200λ ∈ (0, δ ), 5δ > 0} Definition : Let S be a nonempty set in R n and let x * ∈ cl S . Given a function f : R n → R , the cone of improving directions of f at x * is given by F = { d : f ( x * + λ d) < f ( x * ) , 2200λ ∈ (0, δ ), 5δ > 0} Necessary Conditions Constrained Optimization If x * is a local minimum, then F ∩ D = ∅ ....
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 Spring '08
 Linderoth
 Optimization, Mathematical optimization, x*

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