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A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of K. Adaricheva and M. Bolat (2016) and the Polymath REU 2020 team, continues to investigate representations of convex geometries…
Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the…
Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this…
We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…
We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. Our aim is a scalable algorithm for application on supercomputers, which can only be achieved by mesh-independent convergence. The impact of…
We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are…
Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These…
In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility…
We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs…
In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map $c(\cdot)$. A number of non-convex problems ranging from…
A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh as a unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems.…
Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables.…
This paper presents a novel method for reformulating non-differentiable collision avoidance constraints into smooth nonlinear constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid…
A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential…
A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added…
We describe a convex programming approach to the calculation of lower bounds on the minimum cost of constrained decentralized control problems with nonclassical information structures. The class of problems we consider entail the…
We consider the multilinear polytope defined as the convex hull of the feasible region of a linearized binary polynomial optimization problem. We define a relaxation in an extended space for this polytope, which we refer to as the complete…
We propose a new shape-based modeling technique for applications in imaging problems. Given a collection of shape priors (a shape dictionary), we define our problem as choosing the right dictionary elements and geometrically composing them…
Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…