Convex Analysis and Minimization Algorithms I: FundamentalsConvex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics making up the title of the book, reflect the two origins of the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books. |
Contents
Putting the Mechanism in Perspective | 24 |
Conjugacy in Convex Analysis | 35 |
Introduction to Optimization Algorithms | 47 |
LineSearches | 70 |
Convex Sets | 87 |
Approximate Subdifferentials of Convex Functions | 91 |
Abstract Duality for Practitioners | 137 |
Convex Functions of Several Variables | 143 |
Subdifferentials of Finite Convex Functions | 237 |
The Implementable Algorithm | 248 |
First Examples | 258 |
Numerical Illustrations | 263 |
Further Examples | 275 |
A Variety of Stabilized Algorithms | 285 |
Bibliographical Comments | 331 |
References | 337 |
Illustrations | 161 |
Classical Dual Algorithms | 170 |
Local and Global Behaviour of a Convex Function | 173 |
Putting the Method in Perspective | 178 |
First and SecondOrder Differentiation | 183 |
Sublinearity and Support Functions | 195 |
Computing the Direction | 233 |
Index | 345 |
Notations | 385 |
Bibliographical Comments | 401 |
| 407 | |
| 415 | |
Other editions - View all
Convex Analysis and Minimization Algorithms I: Fundamentals Jean-Baptiste Hiriart-Urruty,Claude Lemarechal No preview available - 2012 |
Convex Analysis and Minimization Algorithms I: Fundamentals Jean-Baptiste Hiriart-Urruty,Claude Lemarechal No preview available - 2011 |
Common terms and phrases
affine function affine hull affine manifold algorithm arbitrary C₁ calculus rule closed convex function closed convex set compact set compute consider constraints Conv convergence convex analysis convex combination convex cone convex function convex hull defined denoted derivative df(x differentiable epi f epigraph equivalent example exists f xk f xo f₁ finite fj(x function f gradient half-spaces holds hyperplane implies inequality infimal convolution int dom f iteration Lemma Let f line-search linear minimization problem nonnegative norm normal cone notation obtain operator optimal positive definite positive homogeneity PROOF Proposition quadratic relative interior Remark resp result S₁ saddle-point satisfying scalar product semi-continuous solution solve steepest-descent direction stepsize subdifferential subgradient sublevel-set sublinear functions subspace support function symmetric t₁ TC(x Theorem vector x₁ xk+1