Continuous Optimization and Variational Inequalities

✍ Scribed by Anurag Jayswal (editor), Tadeusz Antczak (editor)

• Publisher: Chapman and Hall/CRC
• Year: 2022
• Tongue: English
• Leaves: 379
• Edition: 1
• Category: Library

✦ Synopsis

The proposed book provides a comprehensive coverage of theory and methods in the areas of continuous optimization and variational inequality. It describes theory and solution methods for optimization with smooth and non-smooth functions, for variational inequalities with single-valued and multivalued mappings, and for related classes such as mixed variational inequalities, complementarity problems, and general equilibrium problems. The emphasis is made on revealing generic properties of these problems that allow creation of efficient solution methods.

Salient Features

•The book presents a deep, wide-ranging introduction to the theory of the optimal control of processes governed by optimization techniques and variational inequality.

•Several solution methods are provided which will help the reader to develop various optimization tools for real-life problems which can be modeled by optimization techniques involving linear and nonlinear functions.

•The book focuses on most recent contributions in the nonlinear phenomena, which can appear in various areas of human activities.

•This book also presents relevant mathematics clearly and simply to help solve real life problems in diverse fields such as mechanical engineering, management, control behavior, traffic signal, industry, etc.

This book is aimed primarily at advanced undergraduates and graduate students pursuing computer engineering and electrical engineering courses. Researchers, academicians and industry people will also find this book useful.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Editors
Preface
Contributors
Symbols
1. Mixed Type Duality and Saddle Point Criteria for Multiobjective Programming Problems with Nonsmooth Generalized Invexity
1.1. Introduction
1.2. Preliminaries
1.3. Optimality Conditions
1.4. Mixed Type Duality
1.5. Mixed Lagrange Function and Saddle Point Criteria
1.6. Conclusions
Bibliography
2. Optimality and Duality for Nonsmooth Optimization Problems with Switching Constraints
2.1. Introduction
2.2. Preliminaries
2.3. Stationary Conditions for Nonsmooth Optimization Problems with Switching Constraint
2.4. Constraint Qualification for (NOPSC)
2.5. Sufficient Optimality Conditions of Nonsmooth Optimization Problems with Switching Constraints
2.6. Duality
2.7. Conclusions
Bibliography
3. Optimality Conditions for Approximate Solutions of Nonsmooth Semi-Infinite Vector Optimization Problems
3.1. Introduction
3.2. Preliminaries
3.3. Main Results
3.3.1. Fuzzy Necessary Optimality Condition for (Weak) ε-Pareto Solutions
3.3.2. Exact Necessary Optimality Condition for (Weak) ε-Pareto Solutions
3.4. Conclusions
Bibliography
4. Density Aspects in Semi-Infinite Vector Optimization
4.1. Introduction
4.2. Preliminaries
4.3. Continuity of Constraint Set Mapping
4.4. Density Results
4.5. Conclusions
Bibliography
5. Optimality Conditions and Duality for Nondifferentiable Multiobjective Fractional Programming Problems
5.1. Introduction
5.2. Preliminaries
5.3. Necessary and Sufficient Conditions
5.4. Proper Efficiency
5.5. Generalized Mond–Weir Type Duality
5.6. Converse Duality
5.7. Conclusions
Bibliography
6. Nonconvex Nonsmooth Minimax Fractional Programming Involving Generalized Semidifferentiable Preinvex Functions with Different Directions
6.1. Introduction
6.2. Preliminaries
6.3. Why It is Worth and Interesting to Consider (ηi)i-Semidifferentiability instead of η-Semidifferentiability
6.4. Necessary Optimality Conditions
6.5. Sufficient Optimality Conditions
6.6. The First Dual Model – Wolfe Type Dual
6.7. The Second Dual Model – Mond–Weir Type Dual
6.8. The Third Dual Model
6.9. Conclusions
Bibliography
7. Optimality Conditions and Wolfe Duality Results for LU-Efficiency in Differentiable Vector Optimization Problems with Multiple Interval-Valued Objective Function and Vanishing Constraints
7.1. Introduction
7.2. Preliminaries
7.3. Optimality Conditions
7.4. Wolfe Duality
7.5. Conclusions
Bibliography
8. Interval-Valued Multi-Time Control Problem with Applications
8.1. Introduction
8.2. Preliminaries
8.3. Exact l1 Penalty Function Method
8.4. Lagrange Functional for (IVMCOP)
8.5. Conclusions
Bibliography
9. Generalized Minty and Stampacchia Vector Variational-Like Inequalities and Interval-Valued Vector Optimization Problems
9.1. Introduction
9.2. Preliminaries
9.3. Relationship Between Generalized Vector Variational-Like inequalities and Nonsmooth Interval-Valued Vector Optimization Problems
9.4. Conclusions
Bibliography
10. Sufficient Optimality Conditions and Duality for a Nonsmooth Interval-Valued Optimization Problem with Generalized Convexity via gH-Clarke Subgradients
10.1. Introduction
10.2. Preliminaries
10.3. gH-Clarke Subdifferential and gH-Generalized Convex IVF
10.4. Optimality Conditions for a Nonsmooth Optimization Problem
10.5. Duality Theory for NIOP
10.6. Conclusions
Bibliography
11. Forward-Backward Extragradient Methods for Quasimonotone Variational Inequalities
11.1. Introduction
11.2. Preliminaries
11.3. Main Results
11.4. Numerical Illustration
11.5. Conclusions
Bibliography
12. Multidimensional Split Variational Inequality in Traffic Analysis
12.1. Introduction
12.2. Preliminaries
12.3. An Equivalence Relation
12.4. Existence of Equilibria
12.5. A Motivational Example
12.6. Conclusions
Bibliography
13. Controlled Nonlinear Dynamics for Constrained Optimization Problems Involving Second-Order Partial Derivatives
13.1. Introduction
13.2. Second-Order PDE Constrained Controlled Optimization Problem
13.3. Isoperimetric Constrained Controlled Optimization Problem
13.3.1. Isoperimetric Constraints Defined by Controlled Curvilinear Integral Functionals
13.4. Conclusions
Bibliography
14. A Non-Parametric Dual Control Algorithms of Multidimensional Static Systems with Delay
14.1. Introduction
14.2. The Statement of the Problem
14.3. Non-parametric Dual Control Algorithm
14.4. Numerical Experiments
14.5. Conclusions
Bibliography
15. Well-Posedness of Set Optimization Using Nonlinear Scalarization Function
15.1. Introduction
15.2. Preliminaries
15.3. A Scalarization Function based on Gerstewitz Function
15.4. Well-Posedness of (P)
15.5. Scalarization Results
15.6. Conclusions
Bibliography
Abbreviations
Index