Related papers: Functions with uniform sublevel sets and scalariza…

Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in…

We consider supervised learning problems in which set predictions provide explicit uncertainty estimates. Using Choquet integrals (a.k.a. Lov{\'a}sz extensions), we propose a convex loss function for nondecreasing subset-valued functions…

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance…

Many computer vision tasks address the problem of scene understanding and are naturally interrelated e.g. object classification, detection, scene segmentation, depth estimation, etc. We show that we can leverage the inherent relationships…

This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces…

A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary…

Soft sets, as a mathematical tool for dealing with uncertainty, have recently gained considerable attention, including some successful applications in information processing, decision, demand analysis, and forecasting. To construct new soft…

In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results…

Computing explicitly the {\epsilon}-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class…

Submodularity is an important concept in combinatorial optimization, and it is often regarded as a discrete analog of convexity. It is a fundamental fact that the set of minimizers of any submodular function forms a distributive lattice.…

Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of Gerstewitz function [1], a vectorizing function is defined to replace a given set-valued…

The essential variables in a finite function $f$ are defined as variables which occur in $f$ and weigh with the values of that function. The number of essential variables is an important measure of complexity for discrete functions. When…

In this paper, we study some optimization problems in uniformly convex and uniformly smooth Bochner spaces. We consider four cases of the underlying subsets: closed and convex subsets, closed and convex cones, closed subspaces and closed…

Function (linear) spaces on which an arbitrary function operates (i.e. the space is stable w.r.t. the pointwise unary operation defined by the function) were investigated, for continuous real or complex operations, by deLeeuw-Katznelson,…

Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not…

An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and their subclasses have attracted attention in signal processing and machine learning. However,…

Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic…

This paper introduces and studies the convergence properties of a new class of explicit $\epsilon$-subgradient methods for the task of minimizing a convex function over the set of minimizers of another convex minimization problem. The…

We consider the class of smooth convex functions defined over an open convex set. We show that this class is essentially different than the class of smooth convex functions defined over the entire linear space by exhibiting a function that…

In many learning tasks, the structure of the target space of a function holds rich information about the relationships between evaluations of functions on different data points. Existing approaches attempt to exploit this relationship…