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