Convex Optimization Theory

by Dimitri P. Bertsekas

An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory. Convexity theory is first developed in a simple accessible manner, using easily visualized proofs. Then the focus shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex sets and functions in terms of points and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality, and game theory to develop the sharpest possible duality results within a highly visual geometric framework. The book may be used as a text for a theoretical convex optimization course; the author has taught several variants of such a course at MIT and elsewhere over the last ten years. It may also be used as a supplementary source for nonlinear programming classes, and as a theoretical foundation for classes focused on convex optimization models (rather than theory). It is an ideal companion to the books Convex Optimization Algorithms, and Nonlinear Programming by the same author.

Genres: Mathematics

246 pages, Hardcover

First published June 30, 2009

Reviews

[Samuel Wang · Rating 5 out of 5]

This book covers basic grounds of convex analysis (on R^n), similar to the classic Rockafeller's style, with a focus on duality theory.
It starts with the basic properties and definitions for convex sets, functions (taking possibly values on the extended real line), and moves on to establishing a unifying geometric framework (the max-crossing min-common (MC/MC) framework) for various optimization duality results (such as duality for linear/convex optimization problems (including slater's conditions, complementary slackness), properties of subgradients, as well minimax problems).
The book is rich with illustrative, graphical examples and counterexamples, proofs are often written with substantial details, though at parts it could be simplified. The use of the MC/MC framework (which is possibly a unique feature of this book) and epigraph correspondence is also a special feature of this book, giving a unified geometric interpretation of the various duality results that one might encounter in the convex optimization world.