Invexity and Optimization

The convexity of sets and the convexity or concavity of functions have been the object of many investigations during the past century. This is mainly due to the development of the theory of mathematical programming, both linear and nonlinear, which is closely tied with convex analysis. Optimality conditions, duality and related algorithms were mainly established for classes of problems involving the optimization of convex objective functions over convex feasible regions. Such assumptions were very convenient, due to the basic properties of convex (or concave) functions concerning optimality conditions. However, not all practical problems, when formulated as mathematical problems, fulfill the requirements of convexity (or concavity). Fortunately, such problems were often found to have some characteristics in common with convex problems and these properties could be exploited to establish theoretical results or develop algorithms. In the second half of the past century various generalizations of convex functions have been introduced. We mention here the early work by de Finetti [54], Fenchel [65], Arrow and Enthoven [5], Mangasarian [142], Ponstein [203] and Karamardian [109]. Usually such generalizations were introduced by a particular problem in economics, management science or optimization theory. In 1980 the first International Conference on generalized convexity/concavity and related fields was held in Vancouver (Canada) and since then, similar international symposia have been organized every year. So, at present we dispose of the proceedings of such conferences, published by Academic Press [221], Analytic Publishing [222], Springer Verlag [25,52,80,129,130], and Kluwer Academic Publishers [51]. Moreover, a monograph on generalized convexity was published by Plenum Publishing Corporation in 1988 (see [10]) and Handbook of Generalized Convexity and Generalized Monotonicity was published by Springer in 2005 (see, [81]). A useful survey is provided by Pini and Singh [202]. The Working Group on Generalized Convexity (WGGC) was founded during the 15th International Symposium on Mathematical Programming in Ann Arbor (Michigan, USA), August 1994. It is a working group of researchers who carry on their interests in generalized convexity, generalized monotonicity and related fields.

The relationship between mathematical programming and classical calculus of variation was explored and extended by Hanson [82]. Duality results are obtained for scalar valued variational problems in Mond and Hanson [170] under convexity.

There are few papers dealing with invexity of quadratic forms and functions; we know only the contributions of Smart [224], Mond and Smart [177] and Molho and Schaible [166]. The study of invex quadratic functions can improve optimality and duality results for that important class of problems formed by quadratic programming problems.