Related papers: Convex envelopes for ray-concave functions
The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the…
Deterministic global optimization of nonlinear models is important in many scientific and engineering applications. This framework typically involves repeatedly solving convex relaxations of the nonconvex problem, meaning that the strength…
Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is…
In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their…
We give an extension to a nonconvex setting of the classical radial representation result for lower semicontinuous envelope of a convex function on the boundary of its effective domain. We introduce the concept of radial uniform upper…
We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension.This hinges on a fully nonlinear obstacle formulation [A. M. Oberman, "The convex envelope is the solution of a…
The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their…
The notion of ordinal concavity of utility functions has recently been considered by Hafalir, Kojima, Yenmez, and Yokote in economics while there exist earlier related works in discrete optimization and operations research. In the present…
The motivation of this paper is the development of an optimisation method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios…
In this paper we analyze several new methods for solving nonconvex optimization problems with the objective function formed as a sum of two terms: one is nonconvex and smooth, and another is convex but simple and its structure is known.…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…
In this paper an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Some applications of this method to approximation of distribution…
When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering…
This paper presents the input convex neural network architecture. These are scalar-valued (potentially deep) neural networks with constraints on the network parameters such that the output of the network is a convex function of (some of)…