Related papers: Most Convex Functions Have Unique Minimizers

We identity the optimal non-infinitesimal direction of descent for a convex function. An algorithm is developed that can theoretically minimize a subset of (non-convex) functions.

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical…

This survey reviews a clustering method based on solving a convex optimization problem. Despite the plethora of existing clustering methods, convex clustering has several uncommon features that distinguish it from prior art. The…

The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in…

We prove that if the given compact set $K$ is convex then a minimizer of the functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty, $$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset B_R\}$…

This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under…

Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning or batch Bayesian optimization, the objective function…

We provide a unified framework for a systematic analysis of the existence of solutions to general nonconvex problems, relying on asymptotic and retractive cones for functions and sets. Using this framework we develop new necessary and…

A procedure for generating non-differentiable, continuously differentiable, and twice continuously differentiable classes of test functions for multiextremal multidimensional box-constrained global optimization and a corresponding package…

Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods…

Detecting hidden convexity is one of the tools to address nonconvex minimization problems. After giving a formal definition of hidden convexity, we introduce the notion of conditional infimum, as it will prove instrumental in detecting…

We present a general procedure to construct examples of convex scalar variational problems which admit a minimizers with large singular sets. The dimension of the set of singularities is maximal and the minimizer has no higher integrability…

We consider the general problem of minimizing an objective function which is the sum of a convex function (not strictly convex) and absolute values of a subset of variables (or equivalently the l1-norm of the variables). This problem…

The aim of this article is to establish new two-functions minimax inequalities extending classical results such as Simons' minimax theorem. Our results will be proved in a non-compact setting. We also prove, under general conditions, that…

We answer two questions from {\it V.Bykov, On Baire class one functions on a product space, Topol. Appl. {199} (2016) 55--62,} and prove that every Baire one function on a subspace of a countable perfectly normal product is the pointwise…

The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the…

The problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below. A system of sufficient and necessary first-order optimality…

Making the gradients small is a fundamental optimization problem that has eluded unifying and simple convergence arguments in first-order optimization, so far primarily reserved for other convergence criteria, such as reducing the…

Let $\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $\Delta_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost…

Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…