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MIT6_235S10_hw05

# MIT6_235S10_hw05 - 6.253 Convex Analysis and Optimization...

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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 | m, g j ( x ) ≤ , j r } . l i ( x ) = 0 , i = 1 ,..., ¯ = r + 1 ,..., ¯ Then the problem can alternatively be described without an abstract set constraint, in terms of all of the constraint functions l 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. 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...
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MIT6_235S10_hw05 - 6.253 Convex Analysis and Optimization...

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