@MISC{_lecture4:, author = {}, title = {Lecture 4: Convex optimization problems • Linear programming • Convex sets and functions • Semidefinite programming • Duality • Algorithms}, year = {} }

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Abstract

Linear algebra and optimization (L4) Convex optimization problems 1 / 31Linear programming (LP) optimization problem with linear cost function and affine constraints Linear program in a standard form: minimize c ⊤ x subject to Gx ≤ h and Ax = b (LP) c, G, h, A, b are given (problem data) x is an unknown vector of optimization variables Contrary to least-squares and least-norm, (LP) has no analytic solution however, it can be solved very efficiently by iterative methods. Note: recurrent theme — use of quickly convergent iterative methods. Even for LS and LN problems, iterative methods may have advantage. Linear algebra and optimization (L4) Convex optimization problems 2 / 31Geometric interpretation of LP Let a ⊤ i be the ith row of A, and g ⊤ i be the ith row of G a ⊤ i x = bi is a hyperplane, perpendicular to ai (assuming ai = 0) g ⊤ i x ≥ hi is a half space (assuming hi = 0) ai = [ 1 1 gi = [ 1