Abderrahim Hantoute’s research while affiliated with University of Alicante and other places

This paper deals with the regularization of the sum of functions defined on a locally convex spaces through their closed-convex hulls in the bidual space. Different conditions guaranteeing that the closed-convex hull of the sum is the sum of the corresponding closed-convex hulls are provided. These conditions are expressed in terms of some epsilon-subdifferential calculus rules for the sum. The case of convex functions is also studied, and exact calculus rules are given under additional continuity/qualifications conditions. As an illustration, a variant of the proof of the classical Rockafellar theorem on convex integration is proposed.

Given a subset $A\times B$ of a locally convex space $X\times Y$ (with A compact) and a function $f:A\times B\rightarrow\overline{\mathbb{R}}$ such that $f(\cdot,y),$ $y\in B,$ are concave and upper semicontinuous, the minimax inequality $\max_{x\in A} \inf_{y\in B} f(x,y) \geq \inf_{y\in B} \sup_{x\in A_{0}} f(x,y)$ is shown to hold provided that $A_{0}$ be the set of $x\in A$ such that $f(x,\cdot)$ is proper, convex and lower semi-contiuous. Moreover, if, in addition, $A\times B\subset f^{-1}(\mathbb{R})$, then we can take as $A_{0}$ the set of $x\in A$ such that $f(x,\cdot)$ is convex. The relation to Moreau's biconjugate representation theorem is discussed, and some applications to convex duality are provided.

This chapter accounts for the most relevant developments of convex analysis in relation to the contents of this book. Specifically, we emphasize the role played by the concept of $\varepsilon$-subgradient of a convex function. Here, X is an lcs and $X^{*}$ is its topological dual space.Unless otherwise stated, we assume that $X^{*}$ (as well as any other involved dual lcs) is endowed with a compatible topology, in particular, the topologies $\sigma (X^{*},X)$ and $\tau (X^{*},X),$ or the dual norm topology when X is a reflexive Banach space. The associated bilinear form is represented by $\langle \cdot ,\cdot \rangle .$

This chapter and the following one offer a crash course in convex analysis, including the fundamental results in the theory of convex functions which are used throughout this book. In the present chapter, we review the Fenchel–Moreau–Rockafellar theory, giving new proofs highlighting the role of separation theorems. These results are then applied to provide dual representations of support functions, which are used in section 4.2 to develop a general duality theory for optimization. In this chapter, we also apply the Fenchel–Moreau–Rockafellar theorem to give slight non-convex extensions of the classical minimax theorems.

In this section, we present some basic concepts and results of functional analysis that will be used throughout the book.

Exercise 1: (i) First, observe that $p_{C}(\theta )=\inf \{\lambda \ge 0:\theta \in \lambda C\}=0$. To check that $p_{C}$ is positively homogeneous on ${\text {dom}}\,p_{C}$, we fix $x\in {\text {dom}}\,p_{C}$ and $\alpha \ge 0.$ If $\alpha =0,$ then $p_{C}(0x)=p_{C}(\theta )=0=0p_{C}(x).$ If $\alpha >0,$ then.

In this chapter, we characterize the subdifferential of pointwise suprema of arbitrarily indexed families of convex functions, defined on a separated locally convex space X.

This last chapter addresses several issues related to the previous chapters. The first part is mainly aimed at deriving optimality conditions for a convex optimization problem, posed in an lcs, with an arbitrary number of constraints. The approach taken is to replace the set of constraints with a unique constraint via the supremum function. Subsequently, we appeal to the properties of the subdifferential of the supremum function that has been exhaustively studied in the previous chapters. With this goal, we extend to infinite convex systems two constraint qualifications that are crucial in linear semi-infinite programming. The first, called the Farkas–Minkowski property, is global in nature, while the other is a local property, called locally Farkas–Minkowski. We obtain two types of Karush–Kuhn–Tucker (KKT, in brief) optimality conditions: asymptotic and non-asymptotic.

This chapter aims to provide formulas for the subdifferential of supremum functions in specific contexts. In these scenarios, an additional structure is available so that the proposed characterizations can adopt particular formats, generally of greater simplicity.

The main objective of this chapter is to establish new formulas for the subdifferential of the sum under conditions that are at an intermediate level of generality between those in Proposition 4.1.16, which hold for lsc proper convex functions, and Proposition 4.1.20, which requires additional continuity assumptions.