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Lagrange-Type Functions in Constrained Non-Convex Optimization

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ISBN-10: 1402076274

ISBN-13: 9781402076275

Edition: 2003

Authors: Aleksandr Moiseevich Rubinov, Xiaoqi Yang

List price: $149.99
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Description:

This volume provides a systematic examination of Lagrange-type functions and augmented Lagrangians. Weak duality, zero duality gap property and the existence of an exact penalty parameter are examined. Weak duality allows one to estimate a global minimum. The zero duality gap property allows one to reduce the constrained optimization problem to a sequence of unconstrained problems, and the existence of an exact penalty parameter allows one to solve only one unconstrained problem. By applying Lagrange-type functions, a zero duality gap property for nonconvex constrained optimization problems is established under a coercive condition. It is shown that the zero duality gap property is…    
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Book details

List price: $149.99
Copyright year: 2003
Publisher: Springer
Publication date: 11/30/2003
Binding: Hardcover
Pages: 286
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.430
Language: English

Preface
Acknowledgments
Introduction
Introduction and motivation
Duality
Mathematical tools
Notation
Abstract Convexity
Abstract convexity
Definitions and preliminary results
Fenchel-Moreau conjugacy and subdifferential
Abstract convex at a point functions
Subdifferential
Abstract convex sets
Increasing positively homogeneous (IPH) functions
IPH functions: definitions and examples
IPH functions defined on R[superscript 2 subscript ++] and R[superscript 2 subscript +]
Associated functions
Strictly IPH functions
Multiplicative inf-convolution
Lagrange-Type Functions
Conditions for minimum in terms of separation functions
Problem P(f,g) and its image space
Optimality conditions through the intersection of two sets
Optimality conditions via separation functions: linear separation
Optimality conditions via separation functions: general situation
Perturbation function
Lower semicontinuity of perturbation function
Lagrange-type functions and duality
Convolution functions
Lagrange-type functions
Lagrange-type functions with multipliers
Linear outer convolution function
Penalty-type functions
Auxiliary functions for methods of centers
Augmented Lagrangians
Duality: a list of the main problems
Weak duality
Problems with a positive objective function
Giannessi scheme and RWS functions
Zero duality gap
Zero duality gap property
Special convolution functions
Alternative approach
Zero duality gap property and perturbation function
Saddle points
Weak duality
Saddle points
Saddle points and separation
Saddle points, exactness and strong exactness
Penalty-Type Functions
Problems with a single constraint
Reformulation of optimization problems
Transition to problems with a single constraint
Optimal value of the transformed problem with a single constraint
Penalization of problems with a single constraint based on IPH convolution functions
Preliminaries
Class P
Modified perturbation functions
Weak duality
Associated function of the dual function
Zero duality gap property
Zero duality gap property (continuation)
Exact penalty parameters
The existence of exact penalty parameters
Exact penalization (continuation)
The least exact penalty parameter
Some auxiliary results. Class B[subscript X]
The least exact penalty parameter (continuation)
Exact penalty parameters for function s[subscript k]
The least exact penalty parameter for function s[subscript k]
Comparison of the least exact penalty parameters for penalty functions generated by s[subscript k]
Lipschitz programming and penalization with a small exact penalty parameter
Strong exactness
The least exact penalty parameters via different convolution functions
Comparison of exact penalty parameters
Equivalence of penalization
Generalized Lagrange functions for problems with a single constraint
Generalized Lagrange and penalty-type functions
Exact Lagrange parameters: class P[subscript *]
Zero duality gap property for generalized Lagrange functions
Existence of Lagrange multipliers and exact penalty parameters for convolution functions s[subscript k]
Augmented Lagrangians
Convex augmented Lagrangians
Augmented Lagrangians
Convex augmenting functions
Abstract augmented Lagrangians
Definition of abstract Lagrangian
Zero duality gap property and exact parameters
Abstract augmented Lagrangians
Augmented Lagrangians for problem P(f, g)
Zero duality gap property for a class of Lagrange-type functions
Level-bounded augmented Lagrangians
Zero duality gap property
Equivalence of zero duality gap properties
Exact penalty representation
Sharp augmented Lagrangians
Geometric interpretation
Sharp augmented Lagrangian for problems with a single constraint
Dual functions for sharp Lagrangians
An approach to construction of nonlinear Lagrangians
Links between augmented Lagrangians for problems with equality and inequality constraints
Supergradients of the dual function
Optimality Conditions
Mathematical preliminaries
Penalty-type functions
Differentiable penalty-type functions
Nondifferentiable penalty-type functions
Augmented Lagrangian functions
Proximal Lagrangian functions
Augmented Lagrangian functions
Approximate optimization problems
Approximate optimal values
Approximate optimal solutions
Appendix: Numerical Experiments
Numerical methods
Results of numerical experiments
Index