v2010.10.26 - Convex Optimization

226 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSwhere |x| = α ⋆ + β ⋆ because of complementarity α ⋆T β ⋆ = 0 at optimality.(508) (510) (511) represent linear programs, (509) is a semidefinite program.Over some convex set C given vector constant y or matrix constant Yarg infx∈C ‖x − y‖ 2 = arg infx∈C ‖x − y‖2 2 (512)arg infX∈ C ‖X − Y ‖ F = arg infX∈ C ‖X − Y ‖2 F (513)are unconstrained convex quadratic problems. Equality does not hold for asum of norms. (5.4.2.3.2) Optimal solution is norm dependent: [61, p.297]minimizex∈R n ‖x‖ 1subject to x ∈ C≡minimize 1 T tx∈R n , t∈R nsubject to −t ≼ x ≼ tx ∈ C(514)minimizex∈R n ‖x‖ 2subject to x ∈ C≡minimizex∈R n , t∈Rsubject tot[ tI xx T tx ∈ C]≽S n+1+0(515)minimize ‖x‖ ∞x∈R nsubject to x ∈ C≡minimize tx∈R n , t∈Rsubject to −t1 ≼ x ≼ t1x ∈ C(516)In R n these norms represent: ‖x‖ 1 is length measured along a grid(taxicab distance), ‖x‖ 2 is Euclidean length, ‖x‖ ∞ is maximum |coordinate|.minimizex∈R n ‖x‖ 1subject to x ∈ C≡minimizeα∈R n , β∈R n 1 T (α + β)subject to α,β ≽ 0x = α − βx ∈ C(517)These foregoing problems (508)-(517) are convex whenever set C is. Theirequivalence transformations make objectives smooth.