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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 ) MinCommon/MaxCrossing
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 LinearNonlinear but
ConvexNonconvex” (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
− Largescale 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.
 Spring '12
 DimitrisBertsekas
 Optimization

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