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1. Convex Functions

Convex and concave functions have many important properties that are useful in Economics and Optimization. In this Chapter the basic properties of convex and concave functions are explained, including some fundamental results involving these functions. In particular, the role of convexity and concavity in Optimization is stressed. Since a function ƒ is concave if and only if − ƒ is convex, any result related to a convex function can easily be translated for a concave function. For this reason only the proofs related to convex functions are presented. For the sake of completeness, the corresponding results for the concave case are summarized in Appendix B.

2. Non-Differentiable Generalized Convex Functions

In several economic models convexity appears to be a restrictive condition. For instance, classical assumptions in Economics include the convexity of the production set in producer theory and the convexity of the upper level sets of the utility function in consumer theory.

3. Differentiable Generalized Convex Functions

In this chapter we shall consider, under the differentiability assumption, the classes of generalized convex functions introduced in the previous chapter. Furthermore, a new class is defined: that of pseudoconvex functions, which is perhaps the most important of all.

4. Optimality and Generalized Convexity

In this chapter, the role of generalized convexity in Optimization is stressed. After presenting the Fritz John and Karush-Kuhn-Tucker necessary optimality conditions, which are proven by means of separation theorems, some constraint qualifications involving generalized convexity are illustrated.

5. Generalized Convexity and Generalized Monotonicity

As convexity plays an important role in solving mathematical programming problems, so, too, does monotonicity in solving variational inequality and nonlinear complementarity problems. Pioneering work was done by Cottle, Dantzig, Karamardian, Stampacchia, and many others (see for instance [71, 74, 134, 154, 155]).

6. Generalized Convexity of Quadratic Functions

Generalized convexity of quadratic functions has been widely studied; the main historical references are Martos [209, 210, 211], Ferland [108], Cottle and Ferland [73], Schaible [236, 243, 242, 248].

7. Generalized Convexity of Some Classes of Fractional Functions

Economic applications are often characterized by maximizing the efficiency of an economic system. This leads to optimization problems whose objective function is a ratio. Examples include maximization of productivity, maximization of return on investment, maximization of return/risk, minimization of cost/time. Linear fractional and generalized fractional problems may be found in different fields such as data envelopment analysis, tax programming, risk and portfolio theory, logistics and location theory (see for instance [14, 15, 66, 67, 166, 214]). The interest in studying fractional problems is confirmed in the extensive survey (with twelve hundred entries) which appeared in [256]; another updated survey may be found in [114].

8. Sequential Methods for Generalized Convex Fractional Programs

In spite of the relevance of the role played by generalized convexity in mathematical programming, research in finding efficient numerical methods for solving generalized convex optimization problems has not yet been sufficiently developed. The only text-book in which solution methods for pseudolinear functions and generalized convex quadratic functions are proposed is Martos’ [211].

9. Solutions

Without Abstract


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