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Convex Optimization Problems and Equivalent Transformations

Explore convex optimization problems and their solution techniques, including linear and quadratic optimization. Learn about equivalent transformations that preserve convexity and various examples of optimization problems.

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Convex Optimization Problems and Equivalent Transformations

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  1. CSE203BWeek5 Discuss Session Xinyuan Wang 02/06/2019

  2. Contents • ConvexOptimization Problems(Ref.Chap 4) • Optimization problem in standard form • Convex optimization problem • Linear optimization • Quadratic optimization • Geometric programming

  3. opt. value not achieved unbounded below unique opt. local opt. global op.

  4. Proof: use first-order condition of the convex functions

  5. Practice: prove the optimality criterion for convex opt. problems

  6. example

  7. Equivalent convex problems Two problems are equivalent if the solution of one is readily obtained from the solution of the other, and vice-versa The transformations preserve convexity (Chap 4.2.4) • Eliminating equality constraints • Introducing equality constraints • Introducing slack variables for linear inequalities • Epigraph form • Minimizing over some variables

  8. More examples in Chap 4.3 • Linear fractional program : transform to LP

  9. More examples in Chap 4.4

  10. More examples in Chap 4.5

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