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In discrete convex analysis, the scaling and proximity properties for the class of L$^\natural$-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of…
A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efficient implementations of the proximity calculation for a useful class of functions; the implementations exploit…
Approximating a convex function by a polyhedral function that has a limited number of facets is a fundamental problem with applications in various fields, from mitigating the curse of dimensionality in optimal control to bi-level…
Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems.…
We define w-invex set, w-preinvex, w-strictly preinvex, w-quasi preinvex, w-strictly quasi preinvex, w-semi-strictly quasi preinvex, and w-pre pseudo-invex functions in this context. And these form a class of real functions, which is the…
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role,…
Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the…
We study the composition of two set-valued functions defined on locally convex topological linear spaces. We assume that these functions map into certain complete lattices of sets that have been used to establish a conjugation theory for…
The optimization problem concerning the determination of the minimizer for the sum of convex functions holds significant importance in the realm of distributed and decentralized optimization. In scenarios where full knowledge of the…
A real-valued set function is (additively) approximately submodular if it satisfies the submodularity conditions with an additive error. Approximate submodularity arises in many settings, especially in machine learning, where the function…
In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the…
Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit…
This paper aims at providing further studies of the notion of quasi-relative interior for convex sets introduced by Borwein and Lewis. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings…
In this paper, we perform sensitivity analysis for the maximal value function which is the optimal value function for a parametric maximization problem. Our aim is to study various subdifferentials for the maximal value function. We obtain…
In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is…
Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation…
Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min-max theorem for a pair of integer-valued M-natural-convex functions generalizes the min-max formulas for polymatroid intersection and…
In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by…
During the study of the topic of limit summability of functions (introduced by the author in 2001), we encountered some types of functions that are related to the mean value theorem. In this paper, we formally define mean value and…
The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…