v2007.09.13 - Convex Optimization

Chapter 4Semidefinite programmingPrior to 1984 , 4.1 linear and nonlinear programming, one a subsetof the other, had evolved for the most part along unconnectedpaths, without even a common terminology. (The use of“programming” to mean “optimization” serves as a persistentreminder of these differences.)−Forsgren, Gill, & Wright (2002) [97]Given some application of convex analysis, it may at first seem puzzling whythe search for its solution ends abruptly with a formalized statement of theproblem itself as a constrained optimization. The explanation is: typicallywe do not seek analytical solution because there are relatively few. (C) If aproblem can be expressed in convex form, rather, then there exist computerprograms providing efficient numerical global solution. [251] [116] [288] [289][290] [296]The goal, then, becomes conversion of a given problem (perhaps anonconvex or combinatorial problem statement) to an equivalent convex formor to an alternation of convex subproblems convergent to a solution of theoriginal problem: A fundamental property of convex optimization problems isthat any locally optimal point is also (globally) optimal. [46,4.2.2] [227,1]Given convex real objective function g and convex feasible set C ⊆domg ,which is the set of all variable values satisfying the problem constraints, we4.1 nascence of interior-point methods of solution [271] [286],2001 Jon Dattorro. CO&EDG version 2007.09.13. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.225