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A function is convex if its epigraph is convex. This geometrical structure has very strong implications in terms of continuity and differentiability. Separation theorems lead to optimality conditions and duality for convex problems. A function is quasiconvex if its lower level sets are convex. Here again, the geo metrical structure of the level sets implies some continuity and differentiability properties for quasiconvex functions. Optimality conditions and duality can be derived for optimization problems involving such functions as well. Over a period of about fifty years, quasiconvex and other generalized convex functions have been considered in a variety of fields including economies, man agement science, engineering, probability and applied sciences in accordance with the need of particular applications. During the last twenty-five years, an increase of research activities in this field has been witnessed. More recently generalized monotonicity of maps has been studied. It relates to generalized convexity off unctions as monotonicity relates to convexity. Generalized monotonicity plays a role in variational inequality problems, complementarity problems and more generally, in equilibrium prob lems.
Sommario
I Generalized Convexity.- 1 Are Generalized Derivatives Useful for Generalized Convex Functions?.- 2 Stochastic Programs with Chance Constraints: Generalized Convexity and Approximation Issues.- 3 Error Bounds for Convex Inequality Systems.- 4 Applying Generalised Convexity Notions to Jets.- 5 Quasiconvexity via Two Step Functions.- 6 On Limiting Fréchet ?-Subdifferentials.- 7 Convexity Space with Respect to a Given Set.- 8 A Convexity Condition for the Nonexistence of Some Antiproximinal Sets in the Space of Integrable Functions.- 9 Characterizations of ?-Convex Funtions.- II Generalized Monotonicity.- 10 Characterizations of Generalized Convexity and Generalized Monotonicity, A Survey.- 11 Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems.- 12 On the Scalarization of Pseudoconcavity and Pseudomonotonicity Concepts for Vector Valued Functions.- 13 Variational Inequalities and Pseudomonotone Functions: Some Characterizations.- III Optimality Conditions and Duality.- 14 Simplified Global Optimality Conditions in Generalized Conjugation Theory.- 15 Duality in DC Programming.- 16 Recent Developments in Second Order Necessary Optimality Conditions.- 17 Higher Order Invexity and Duality in Mathematical Programming.- 18 Fenchel Duality in Generalized Fractional Programming.- IV Vector Optimization.- 19 The Notion of Invexity in Vector Optimization: Smooth and Nonsmooth Case.- 20 Quasiconcavity of Sets and Connectedness of the Efficient Frontier in Ordered Vector Spaces.- 21 Multiobjective Quadratic Problem: Characterization of the Efficient Points.- 22 Generalized Concavity for Bicriteria Functions.- 23 Generalized Concavity in Multiobjective Programming.
