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Generalized Convexity and Generalized Monotonicity, Jean-Pierre Crouzeix
Convexity and monotonicity are very useful in optimization problems and variational problems. They lead to important results concerning the existence of solutions and their characterizations, and the structure of the set of solutions. They allow to design efficient algorithms to compute the solutions of the problems. The epigraph of a convex function is convex, this geometrical structure induces many properties. The close connection between algebraic and topological interiors of convex sets, the separation theorems for convex sets and so many other properties of convex sets are transposed to convex functions. Differentiability is a major concern in convexity. Although a convex function is not necessarily differentiable, it is subdifferentiable at any point in the interior of its domain. Sensibility results in convex programming are obtained thanks to the continuity properties of the subgradient.
Besides these continuity properties, the subgradient of a convex function enjoys another major property, the property of monotonicity. Convex (constrained and unconstrained) optimization problems can be viewed as variational inequality problems. Although a monotone variational inequality problem does not necessarily derive from a convex optimization problem, monotonicity and convexity are closely related.
It appears that, in many situations, convexity is two restrictive. For instance, in economics, it is unrealistic to assume the convexity of utility functions. But, the convexity of the level sets of utility functions can be retained. A function is said to be quasiconvex if its level sets are convex.
Another useful relaxation of convexity is pseudoconvexity which involves the differentiability of the function. Generalized monotonicity of maps is related to generalized convexity of functions as monotonicity is related to convexity.
The aim of these lecture notes is to introduce basic properties of generalized convexity and generalized monotonicity.