In preparing this lecture note a special effort has been made to obtain a selfcontained treatment of the subjects; so we hope that this may be a suitable source or a beginner in this fast growing area of research, a semester graduate course in nonlinear programing, and a good reference book. This book may be useful to theoretical economists, engineers, and applied researchers involved in this area of active research. The lecture note is divided into eight chapters:
- Chapter 1 briefly deals with the notion of nonlinear programing problems withbasic notations and preliminaries.
- Chapter 2 deals with various concepts of convex sets, convex functions, invex set,invex functions, quasiinvex functions, pseudoinvex functions, type I and generalizedtype I functions, V-invex functions, and univex functions.
- Chapter 3 covers some new type of generalized convex functions, such asType I univex functions, generalized type I univex functions, nondifferentiabled-type I, nondifferentiable pseudo-d-type I, nondifferentiable quasi d-type I andrelated functions, and similar concepts for continuous-time case, for nonsmoothcontinuous-time case, and for n-set functions are introduced.
- Chapter 4 deals with the optimality conditions for multiobjective programingproblems, nondifferentiable programing problems, minimax fractional programingproblems, mathematical programing problems in Banach spaces, in complex spaces,continuous-time programing problems, nonsmooth continuous-time programing problems, and multiobjective fractional subset programing problems under the assumptions of some generalized convexity given in Chap. 3.
- In Chap. 5 we give Mond–Weir type and General Mond–Weir type duality results for primal problems given in Chap. 4. Moreover, duality results for nonsmooth programing problems and control problems are also given in Chap. 5.
- Chapter 6 deals with second and higher order duality results for minimax programing problems, nondifferentiable minimax programing problems, nondifferentiable mathematical programing problems under assumptions generalized convexity conditions.
- Chapter 7 is about symmetric duality results for mathematical programing problems, mixed symmetric duality results for nondifferentiablemultiobjective programing problems, minimax mixed integer programing problems, and symmetric duality results for nondifferentiablemultiobjective fractional variational problems.
- Chapter 8 is about relationships between vector variational-like inequality problems and vector optimization problems under various assumptions of generalized convexity. Such relationships are also studied for nonsmooth vector optimization problems as well. Some characterization of generalized univex functions using generalized monotonicity are also given in this chapter.
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