Optimization Techniques for Solving Nonlinear Interval Programming Problems using Generalized Hukuhara Difference

Abstract

The objective of the thesis, entitled Optimization Techniques for Solving NonlinearInterval Programming Problems Using Generalized Hukuhara Difference is tostudy and analyze nonlinear interval optimization problems and develop some techniquesto solve such problems under the framework of Generalized Hukuhara-basedinterval calculus. The primary goal of this study is to develop some effective solutiontechniques for nonlinear interval programming problems using a hybrid approach involvingstochastic programming and interval analysis. The strategies created havebeen employed to obtain an optimal scheduling scheme for domestic multi-energysystems in smart homes. newline newlineOver the last few decades, interval optimization techniques have primarily evolvedin the domain of optimization under uncertainty as an alternative to traditionalstochastic and fuzzy optimization. Different methods have been proposed by the researchersfor addressing the uncertainty factor that occurs in mathematical modeling,using random variables and suitable probability distributions, membership functions,min-max criteria, etc., but still, there are certain issues regarding convergence andtime-space complexity in the developed algorithms. An effort has been made througha hybrid approach consisting of the concepts of stochastic programming and intervalanalysis, which eliminates the selection of the weight functions by the decision-makerintuitively or using experience. Also, the available solution techniques provide resultsin the form of intervals, whereas the proposed methodologies provide specific resultsthat are more realistic and implementable. newline newlineThe proposed work in this thesis is described as: newline newlineIn Chapter 1, the relevance of the theme, a detailed literature survey, and theframework of the thesis has been introduced. The fundamental concepts of intervalarithmetic, interval analysis, and interval calculus, along with the notion of generalized Hukuhara difference, are included. newline newlineIn Chapter 2, optimality conditions by the traditional Karush-Kuhn-Tucker condi