Consider a differentiable convex optimization problem minimizesubjectto?f0?(x)fi?(x)?0,i=1,,mhi?(x)=0,i=1,,p? Let x? and (??,??) be primal and dual optimal points, respectively. Suppose that the strong duality holds. 1 (a) [10pt] Prove that ?f0?(x?)+?i=1m??i???fi?(x?)+?i=1p??i???hi?(x?)=0. (b) [5pt] Prove that ?i??fi?(x?)=0 for i=1, solutionspile.com

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Consider a differentiable convex optimization problem minimizesubjectto?f0?(x)fi?(x)?0,i=1,,mhi?(x)=0,i=1,,p? Let x? and (??,??) be primal and dual optimal points, respectively. Suppose that the strong duality holds. 1 (a) [10pt] Prove that ?f0?(x?)+?i=1m??i???fi?(x?)+?i=1p??i???hi?(x?)=0. (b) [5pt] Prove that ?i??fi?(x?)=0 for i=1,,m. solutionspile.com

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