Related papers: Convex envelopes for ray-concave functions
Chapter 1 deals with the problem of the existence of an upper/lower envelope from a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space…
A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low…
In recent years, the success of deep learning has inspired many researchers to study the optimization of general smooth non-convex functions. However, recent works have established pessimistic worst-case complexities for this class…
Recent advances in the efficiency and robustness of algorithms solving convex quadratically constrained quadratic programming (QCQP) problems motivate developing techniques for creating convex quadratic relaxations that, although more…
This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional…
Solutions to network optimization problems have greatly benefited from developments in nonlinear analysis, and, in particular, from developments in convex optimization. A key concept that has made convex and nonconvex analysis an important…
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common…
We give a relation between verbatim generating functions of what we call Pythagorean languages and matrix convexity. Namely, several multivariate matrix convex functions occurring in the existing matrix analysis literature arise naturally…
In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth…
Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions…
In this paper, we focus on the decentralized composite optimization for convex functions. Because of advantages such as robust to the network and no communication bottle-neck in the central server, the decentralized optimization has…
Traditionally, most complex intelligence architectures are extremely non-convex, which could not be well performed by convex optimization. However, this paper decomposes complex structures into three types of nodes: operators, algorithms…
We consider the problem of packing congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…
We introduce and study various algorithms for solving nonconvex minimization with inequality constraints, based on the construction of convex surrogate envelopes that majorize the objective and the constraints. In the case where the…
Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In…
Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify…
We present a distributed solution to optimizing a convex function composed of several non-convex functions. Each non-convex function is privately stored with an agent while the agents communicate with neighbors to form a network. We show…