v2009.01.01 - Convex Optimization

Chapter 4
Semidefinite programming
Prior to 1984, linear and nonlinear programming, 4.1 one a subset
of the other, had evolved for the most part along unconnected
paths, without even a common terminology. (The use of
“programming” to mean “optimization” serves as a persistent
reminder of these differences.)
−Forsgren, Gill, & Wright (2002) [121]
Given some application of convex analysis, it may at first seem puzzling why
the search for its solution ends abruptly with a formalized statement of the
problem itself as a constrained optimization. The explanation is: typically
we do not seek analytical solution because there are relatively few. (C,
3.1.7.2) If a problem can be expressed in convex form, rather, then there
exist computer programs providing efficient numerical global solution. [141]
[335] [342] The goal, then, becomes conversion of a given problem (perhaps
a nonconvex or combinatorial problem statement) to an equivalent convex
form or to an alternation of convex subproblems convergent to a solution of
the original problem:
A fundamental property of convex optimization problems is that any
locally optimal point is also (globally) optimal. [53,4.2.2] [265,1] Given
convex real objective function g and convex feasible set C ⊆dom g , which
is the set of all variable values satisfying the problem constraints, we pose a
4.1 nascence of interior-point methods of solution [316] [333].
Linear programming ⊂ (convex ∩ nonlinear) programming.
2001 Jon Dattorro. CO&EDG version 2009.01.01. All rights reserved.
Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,
Meboo Publishing USA, 2005.
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