Necessary and Sufficient Optimality Conditions in DC Semi-infinite Programming

This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section.The first author was partially supported by ANID Chile under grant Fondecyt Regular 1190110. The second author is supported by Research Project PGC2018-097960-B-C21 from MICINN Spain, and Australian ARC–Discovery Projects DP 180100602. The third author was partially supported by ANID Chile under grants Fondecyt regular 1190110, Fondecyt regular 1200283, and Programa Regional Mathamsud 20-Math-08 CODE: MATH190003

Optimal pricing for optimal transport

Suppose that $c(x,y)$ is the cost of transporting a unit of mass from $x\in X$ to $y\in Y$ and suppose that a mass distribution $\mu$ on $X$ is transported optimally (so that the total cost of transportation is minimal) to the mass distribution $\nu$ on $Y$. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price $f(x)$ for a unit of mass sold (say by the producer to the distributor) at $x$ and a price $g(y)$ for a unit of mass sold (say by the distributor to the end consumer) at $y$ such that for any $x\in X$ and $y\in Y$, the price difference $g(y)-f(x)$ is not greater than the cost of transportation $c(x,y)$ and such that there is equality $g(y)-f(x)=c(x,y)$ if indeed a nonzero mass was transported (via the optimal transportation plan) from $x$ to $y$. We consider the following optimal pricing problem: suppose that a new pricing policy is to be determined while keeping a part of the optimal transportation plan fixed and, in addition, some prices at the sources of this part are also kept fixed. From the producers' side, what would then be the highest compatible pricing policy possible? From the consumers' side, what would then be the lowest compatible pricing policy possible? In the framework of $c$-convexity theory, we have recently introduced and studied optimal $c$-convex $c$-antiderivatives and explicit constructions of these optimizers were presented. In the present paper we employ optimal $c$-convex $c$-antiderivatives and conclude that these are natural solutions to the optimal pricing problems mentioned above. This type of problems drew attention in the past and existence results were previously established in the case where $X=Y=R^n$ under various specifications. We solve the above problem for general spaces $X,Y$ and real-valued, lower semicontinuous cost functions $c$

Subdifferential Calculus Rules in Convex Analysis: A Unifying Approach Via Pointwise Supremum Functions

We provide a rule to calculate the subdifferential set of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space. Our formula is given exclusively in terms of the data functions and does not require any assumption either on the index set on which the supremum is taken or on the involved functions. Some other calculus rules, namely chain rule formulas of standard type, are obtained from our main result via new and direct proofs.Research supported by grants MTM2005-08572-C03 (01) from MEC (Spain) and FEDER (E.U.), ACOMP06/117 and ACOMP/2007/247-292 from Generalitat Valenciana (Spain), and ID-PCE-379 (Romania)

Abstract convex approximations of nonsmooth functions

In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.Comment: This is a slightly edited version of Accepted Manuscript of an article published by Taylor & Francis in Optimization on 13/01/2014, available online: https://doi.org/10.1080/02331934.2013.86981