Pseudolinear Functions and Optimization: 1st Edition (Hardback) book cover

Pseudolinear Functions and Optimization

1st Edition

By Shashi Kant Mishra, Balendu Bhooshan Upadhyay

Chapman and Hall/CRC

510 pages | 6 B/W Illus.

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pub: 2014-12-18
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Description

Pseudolinear Functions and Optimization is the first book to focus exclusively on pseudolinear functions, a class of generalized convex functions. It discusses the properties, characterizations, and applications of pseudolinear functions in nonlinear optimization problems.

The book describes the characterizations of solution sets of various optimization problems. It examines multiobjective pseudolinear, multiobjective fractional pseudolinear, static minmax pseudolinear, and static minmax fractional pseudolinear optimization problems and their results. The authors extend these results to locally Lipschitz functions using Clarke subdifferentials. They also present optimality and duality results for h-pseudolinear and semi-infinite pseudolinear optimization problems.

The authors go on to explore the relationships between vector variational inequalities and vector optimization problems involving pseudolinear functions. They present characterizations of solution sets of pseudolinear optimization problems on Riemannian manifolds as well as results on pseudolinearity of quadratic fractional functions. The book also extends n-pseudolinear functions to pseudolinear and n-pseudolinear fuzzy mappings and characterizations of solution sets of pseudolinear fuzzy optimization problems and n-pseudolinear fuzzy optimization problems. The text concludes with some applications of pseudolinear optimization problems to hospital management and economics.

This book encompasses nearly all the published literature on the subject along with new results on semi-infinite nonlinear programming problems. It will be useful to readers from mathematical programming, industrial engineering, and operations management.

Reviews

"This is a study of the mathematics of pseudolinear functions and their applications. … [The book includes] some applications of pseudolinear optimization problems to hospital management and economics."

Zentralblatt MATH 1326

"There is a real need for this book in the market. … There are numerous books on generalized convexity with applications, but there are only a few books concerned with pseudolinear functions. There are many books on nonconvex optimization with applications, but they do not contain pseudolinear functions. There are also several books on vector optimization, but they do not contain pseudolinear functions."

—Nan-Jing Huang, Sichuan University, Chengdu, People’s Republic of China

Table of Contents

Introduction

Convex Sets

Convex Functions and Properties

Alternative and Separation Theorems

Unconstrained and Constrained Optimization Problems

Nonlinear Multiobjective Optimization and Pareto Optimality

Subdifferential Calculus

Optimality Criteria and Duality

Pseudolinear Functions: Characterizations and Properties

Introduction

Differentiable Pseudolinear Functions

Local Characterizations of Pseudolinear Functions

Second Order Characterizations of Pseudolinear Functions

Semilocally Pseudolinear Functions

Dini Derivatives and Pseudolinear Functions

Characterizations of Locally Lipschitz Pseudolinear Functions using Clarke Subdifferential

Weakly Pseudolinear Functions

Properties of the Functional Class of Differentiable at a Point Functions

Constrained Pseudolinear Optimization Problems: Characterizations of Solution Sets

Introduction

Characterizations of Solution Sets of Differentiable Pseudolinear Optimization Problems

Solution Sets of Linear Fractional Programs

Characterizations of Solution Sets of Semilocally Pseudolinear Optimization Problems

Characterizations of Solution Sets of Pseudolinear Optimization Problems in Terms of Dini Derivatives

Characterizations of Solution Sets of h-Pseudolinear Optimization Problems

Characterizations of Solution Sets of Nonsmooth Pseudolinear Optimization Problems

Constrained Pseudolinear Optimization Problems: Characterizations of Solutions Sets in Terms of Lagrange Multipliers

Introduction

Definitions and Preliminaries

Pseudolinear Optimization Problems

Linear Fractional Programming Problem

Vector Linear Fractional Programming Problem

Nonsmooth Pseudolinear Optimization Problems with Linear Inequality Constraints

Nonsmooth Vector Pseudolinear Optimization Problems with Linear Inequality Constraints

Pseudolinearity in terms of Bifunction

Necessary Optimality Conditions for a Mathematical Program in terms of Bifunctions

Mathematical Program with h-Convex Objective and j h-Pseudolinear Constraints

Mathematical Program with h-Pseudolinear Objective and j h-Pseudolinear Constraints

Pseudolinear Multiobjective Optimization

Introduction

Necessary Optimality Conditions

Sufficient Optimality Conditions

Duality in Multiobjective Programming

Multiobjective Fractional Programming Problems

Necessary Optimality Conditions

Sufficient Optimality Conditions

Duality in Multiobjective Fractional Programming

Nonsmooth Pseudolinear Multiobjective Optimization

Introduction

Necessary Optimality Conditions

Sufficient Optimality Conditions

Duality Theorems

Nonsmooth Multiobjective Fractional Programming

Necessary Optimality Conditions

Sufficient Optimality Conditions

Duality Theorems

Static Minimax Programming and Pseudolinear Functions

Minimax Programming Problems

Necessary Optimality Conditions

Sufficient Optimality Conditions

Duality for Minimax Programming Problem

Minimax Fractional Programming Problem

Necessary Optimality Conditions

Sufficient Optimality Conditions

Duality for Minimax Fractional Programming

Nonsmooth Static Minimax Programming and Pseudolinear Functions

Introduction

Definitions and Preliminaries

Necessary and Sufficient Optimality Conditions

Dual Model I

Dual Model II

Nonsmooth Multiobjective Pseudolinear Programming: Optimality and Duality in Terms of Bifunctions

Introduction

Definitions and Preliminaries

Necessary and Sufficient Optimality Conditions

Duality Theorem

Numerical Examples

Pseudolinear Multiobjective Semi-Infinite Programming Problems

Introduction

Necessary and Sufficient Optimality Conditions

Duality Theorems

Nonsmooth Semi-infinite Programming Problems

Necessary and Sufficient Optimality Conditions

Duality Theorems

Numerical Examples

Vector Variational Inequality and Vector Pseudolinear Optimization Problems

Introduction

Vector Variational Inequality Problems

Necessary and Sufficient Optimality Conditions

Nonsmooth Vector Variational Inequality Problems

Necessary and Sufficient Optimality Conditions

Extension of Pseudolinear Functions and Variational Inequality Problems

Introduction

PPM maps and their Characterizations

Affine PPM Maps

Characterizations of the Solution Sets of a PPM Variational Inequality Problem

Generalizations

Ƞ Pseudolinear Functions: Characterizations and Properties of Solution Set

Introduction

Characterizations of Differentiable Ƞ-Pseudolinear Functions

Characterizations of Solution set of Differentiable Ƞ-Pseudolinear Program

Ƞ-Pseudolinear Functions and Vector Optimization

Characterizations of Locally Lipschitz Ƞ-Pseudolinear Functions

Characterizations of Solution set of Nonsmooth Ƞ-Pseudolinear Program

Locally Lipschitz Ƞ-Pseudolinear Functions and Nonsmooth Vector Optimization

Smooth Pseudolinear Functions and Riemannian Manifolds

Introduction

Definitions and Preliminaries

Explicit Formulation of Gradient of Pseudolinear Functions

Construction of Pseudolinear Functions

Generalizations

Pseudolinear Quadratic Fractional Functions

Quadratic Fractional Functions

Pseudolinearity of Quadratic Fractional Functions and their Characterization

Gradient of Pseudolinear Quadratic Fractional Functions

Computational Aspect

Pseudolinear Fuzzy Mapping

Introduction

Definitions and Preliminaries

Characterizations of Pseudolinear Fuzzy Mappings

Characterizations of Solution Sets

Numerical Examples

Pseudolinear Function and Its Applications

Pseudolinear Functions and Hospital Management

Pseudolinear Functions and Economics

Pseudolinear Functions and Simplex Algorithms

Subject Categories

BISAC Subject Codes/Headings:
BUS049000
BUSINESS & ECONOMICS / Operations Research
MAT003000
MATHEMATICS / Applied
TEC029000
TECHNOLOGY & ENGINEERING / Operations Research