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CSE 203B: Convex Optimization Week 7 Discuss Session

Explore techniques for proving optimality in unconstrained minimization problems with strong convexity and upper bound considerations. Learn about Taylor expansion, Cauchy-Schwarz inequality, and first-order conditions.

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CSE 203B: Convex Optimization Week 7 Discuss Session

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  1. CSE203B:ConvexOptimizationWeek7DiscussSession Xinyuan Wang 02/20/2019

  2. Unconstrained Minimization Problems • Assume that the objective function is strongly convex on , there exists an such that (1) for all . • Upper bound on , there exist a constant such that (2) for all . In class we are given two inequality condition for the optimal How to prove?

  3. Unconstrained Minimization Problems • Taylor expansion with any . for on the line segment between and . • Consider the strong convexity for any • It holds for any , let be the optimal

  4. Unconstrained Minimization Problems • Taylor expansion with any . for on the line segment between and . • Consider the upper bound for any try to minimize the rhs, the minimum is achieved at • Let be the optimal

  5. Unconstrained Minimization Problems • Taylor expansion with any . for on the line segment between and . • Consider the strong convexity for any , we have Cauchy-Schwarz inequality • Sincefor all

  6. Unconstrained Minimization Problems • Taylor expansion with any . for on the line segment between and . • Consider the first-order condition for any and is optimal Cauchy-Schwarz inequality • We already prove that

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