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L8_cprob - Lecture 8 Convex Problems Lecture 8 Outline I...

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Unformatted text preview: Lecture 8: Convex Problems February 12, 2007 Lecture 8 Outline I Revisit Optimality Condition I Unconstrained Problems I Problems with Linear Constraints I Problems with Inequalities and Equalities I Linear-Fractional Programming I Second-Order Cone Programming Convex Optimization 1 Lecture 8 Optimality Condition for Differentiable f General Case Let f be a differentiable convex function with dom f R n and let C be a nonempty closed convex set Theorem A vector x * is optimal if and only if x * C dom f and f ( x * ) T ( z- x * ) for all z C dom f In the following, unless otherwise stated, the function f is assumed to be differentiable and convex, with dom f = R n Convex Optimization 2 Lecture 8 Unconstrained Optimization minimize f ( x ) subject to x R n I An optimal solution exists if R f S f , i.e., there is no nonzero direction v and a vector x for which the scalar function g v ( t ) = f ( x + tv ) is strictly decreasing in t as t increases from 0 to I A vector x * is optimal if and only if f ( x * ) = 0 Convex Optimization 3 Lecture 8 Linear Equality Constrained Problem minimize f ( x ) subject to Ax = b with A R m n and b R m I When does an optimal solution exist? I A vector x * is optimal if and only if f ( x * ) y = 0 for all y N A Using N A = Im A T , we have that x * is optimal if and only if there exists * R m such f ( x * ) + A T * = 0 This is Primal-Dual (Lagrange Multiplier) Optimality Condition Convex Optimization 4 Lecture 8 Minimization over the Nonnegative Orthant minimize f ( x ) subject to x I When does an optimal solution exist? I A vector x * is optimal if and only if f ( x * ) T x * = 0 This known as Complementarity Condition in Lagrangian duality....
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L8_cprob - Lecture 8 Convex Problems Lecture 8 Outline I...

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