Perturbation Analysis of Optimization Problems

The main subject of this book is perturbation analysis of continuous optimization problems. In the last two decades considerable progress has been made in that area, and it seems that it is time now to present a synthetic view of many important results that apply to various classes of problems. The model problem that is considered throughout the book is of the form (P) Min/(x) subjectto G(x) E K. xeX Here X and Y are Banach spaces, K is a closed convex subset of Y, and / : X -+ IR and G : X -+ Y are called the objective function and the constraint mapping, respectively. We also consider a parameteriZed version (P ) of the above u problem, where the objective function / (x, u) and the constraint mapping G(x, u) are parameterized by a vector u varying in a Banach space U. Our aim is to study continuity and differentiability properties of the optimal value v(u) and the set S(u) of optimal solutions of (P ) viewed as functions of the parameter vector u.

This is a research-level book in optimization. The results herein can be applied to various fields, from physics to economics. A nice feature is its great number of examples, which help the reader to understand the meaning of various statements and assumptions. It will be very useful not only to research professionals in the field of optimization, nonlinear programming and optimal control, but also to users of optimization in mechanics, physics, statistics, finance and economics.

Basic notation.- Introduction.- Background material.- Optimality conditions.- Basic perturbation theory.- Second order analysis of the optimal value and optimal solutions.- Optimal Control.- References.