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# point w max crossing point q 01223 4 0 9 m u7 8 c

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Unformatted text preview: /MAX CROSSING DUALITY . w 6 M. %w Min Common %&'()*++*'(,*&'-(./ Point w∗ M Min Common Point w∗ %&'()*++*'(,*&'-(./ M % % ! 0 Max Crossing %#0()1*22&'3(,*&'-(4/ Point q ∗ ! 0 7 u Max Crossing %#0()1*22&'3(,*&'-(4/ 7 u Point q ∗ (a "#\$ ) (b "5\$) . w 6 % M Min Common %&'()*++*'(,*&'-(./ Point w∗ Max Crossing Point q ∗ %#0()1*22&'3(,*&'-(4/ 0 ! 9 M% u7 ("8\$ c) • All of duality theory and all of (convex/concave) minimax theory can be developed/explained in terms of this one ﬁgure. • The machinery of convex analysis is needed to ﬂesh out this ﬁgure, and to rule out the exceptional/pathological behavior shown in (c). 11 ABSTRACT/GENERAL DUALITY ANALYSIS Abstract Geometric Framework (Set M ) Min-Common/Max-Crossing Theorems Special choices of M Minimax Duality ( MinMax = MaxMin ) Constrained Optimization Duality 12 Theorems of the Alternative etc EXCEPTIONAL BEHAVIOR • If convex structure is so favorable, what is the source of exceptional/pathological behavior? • Answer: Some common operations on convex sets do not preserve some basic properties. • Example: A linearly transformed closed convex set need not be closed (contrary to compact and polyhedral sets). − Also the vector sum of two closed convex sets need not be closed. x2 � ⇥ C1 = (x1 , x2 ) | x1 > 0, x2 > 0, x1 x2 ≥ 1 � ⇥ C2 = (x1 , x2 ) | x1 = 0 x1 • This is a major reason for the analytical di⌅culties in convex analysis and pathological behavior in convex optimization (and the favorable character of polyhedral sets). 13 MODERN VIEW OF CONVEX OPTIMIZATION • Traditional view: Pre 1990s − LPs are solved by simplex method − NLPs are solved by gradient/Newton methods − Convex programs are special cases of NLPs LP CONVEX Duality Simplex NLP Gradient/Newton • Modern view: Post 1990s − LPs are often solved by nonsimplex/convex methods − Convex problems are often solved by the same methods as LPs − “Key distinction is not Linear-Nonlinear but Convex-Nonconvex” (Rockafellar) LP Simplex CONVEX Duality Cutting plane Interior point Subgradient 14 NLP Gradient/Newton THE RISE OF THE ALGORITHMIC ERA • Convex programs and LPs connect around − Duality − Large-scale piecewise linear problems • Synergy of: − Duality − Algorit...
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## This document was uploaded on 03/19/2014 for the course EECS 6.253 at MIT.

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