MIT6_235S10_hw05_sol - ¯ 6.253: Convex Analysis and...

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Unformatted text preview: ¯ 6.253: Convex Analysis and Optimization Homework 5 Prof. Dimitri P. Bertsekas Spring 2010, M.I.T. Problem 1 Consider the convex programming problem minimize f ( x ) x subject to x ∈ X, g ( x ) ≤ , of Section 5.3, and assume that the set X is described by equality and inequality constraints as X = { x | h i ( x ) = 0 , i = 1 ,..., ¯ = r + 1 ,..., r ¯ } . m, g j ( x ) ≤ , j Then the problem can alternatively be described without an abstract set constraint, in terms of all of the constraint functions h i ( x ) = 0 , i = 1 ,..., ¯ g j ( x ) ≤ , r. m, j = 1 ,..., ¯ We call this the extended representation of (P). Show if there is no duality gap and there exists a dual optimal solution for the extended representation, the same is true for the original problem. Solution . Assume that there exists a dual optimal solution in the extended representation. Thus there exist nonnegative scalars λ ∗ 1 ,...,λ ∗ ,λ ∗ m 1 ,...,µ ∗ ,µ ∗ r ¯ m +1 ,...,λ ∗ ¯ and µ ∗ r +1 ,...,µ ∗ such that m r ⎧ ⎨ ⎫ ⎬ r ¯ m f ∗ = inf f ( x ) + λ ∗ i h i ( x ) + µ ∗ j g j ( x ) ⎭ x ∈ R n ⎩ , i =1 j =1 from which we have m ¯ r ¯ f ∗ ≤ f ( x ) + λ i ∗ h i ( x ) + µ ∗ j g j ( x ) , ∀ x ∈ R n . i =1 j =1 For any x ∈ X , we have h i ( x ) = 0 for all i = 1 ,..., m ¯ , and g j ( x ) ≤ for all j = r + 1 ,..., r ¯, so that µ ∗ j g j ( x ) ≤ for all j = r + 1 ,..., r ¯. Therefore, it follows from the preceding relation that r f ∗ ≤ f ( x ) + µ ∗ j g j ( x ) , ∀ x ∈ X. j =1 1 Taking the infimum over all x ∈ X , it follows that ⎧ ⎨ ⎫ ⎬ ⎭ r f ( x ) + µ ∗ j g j ( x ) f ∗ ≤ inf x ∈ X ⎩ j =1 ⎧ ⎨ ⎫ ⎬ ≤ inf x ∈ X,g j ( x ) ≤ , j =1 ,...,r ⎩ f ( x ) + r j =1 µ ∗ j g j ( x ) ⎭ inf f ( x ) ≤ x ∈ X, h i ( x )=0 , i =1 ,...,m g j ( x ) ≤ , j =1 ,...,r = f ∗ . Hence, equality holds throughout above, showing that the scalars λ ∗ 1 ,...,λ ∗ , µ ∗ 1 ,...,µ ∗ constitute m r a dual optimal solution for the original representation. Problem 2 Consider the class of problems minimize f ( x ) x subject to x ∈ X, g j ( x ) ≤ u j , j = 1 , . . . , r, where u = ( u 1 ,...,u r ) is a vector parameterizing the right-hand side of the constraints. Given two distinct values u ¯ and ˜ u of u , let f ¯ and f ˜ be the corresponding...
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MIT6_235S10_hw05_sol - ¯ 6.253: Convex Analysis and...

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