Robust support vector machines via conic optimization

We consider the problem of learning support vector machines robust to uncertainty. It has been established in the literature that typical loss functions, including the hinge loss, are sensible to data perturbations and outliers, thus performing poorly in the setting considered. In contrast, using the 0-1 loss or a suitable non-convex approximation results in robust … Read more

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of non-lower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance … Read more

Continuous Selections of Solutions to Parametric Variational Inequalities

This paper studies the existence of a (Lipschitz) continuous (single-valued) solution function of parametric variational inequalities under functional and constraint perturbations. At the most elementary level, this issue can be explained from classical parametric linear programming and its resolution by the parametric simplex method, which computes a solution trajectory of the problem when the objective … Read more

On the Number of Pivots of Dantzig’s Simplex Methods for Linear and Convex Quadratic Programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegative constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency … Read more

On polynomial time solvability of combinatorial Markov random fields

The problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled as a mixed-integer program. This motivates us to study a general class of convex submodular optimization problems with indicator variables, which we show to be polynomially solvable in this paper. The key insight is that, possibly after … Read more

Some Strongly Polynomially Solvable Convex Quadratic Programs with Bounded Variables

This paper begins with a class of convex quadratic programs (QPs) with bounded variables solvable by the parametric principal pivoting algorithm with $\mbox{O}(n^3)$ strongly polynomial complexity, where $n$ is the number of variables of the problem. Extension of the Hessian class is also discussed. Our research is motivated by a recent reference [7] wherein the … Read more

Compact extended formulations for low-rank functions with indicator variables

We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions Extended formulations describing the convex hull of such … Read more

Comparing Solution Paths of Sparse Quadratic Minimization with a Stieltjes Matrix

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “L0-path” where the discontinuous L0-function provides the exact sparsity count; the “L1-path” where the L1-function provides a convex surrogate of sparsity count; … Read more

Single-neuron convexifications for binarized neural networks

Binarized neural networks are an important class of neural network in deep learning due to their computational efficiency. This paper contributes towards a better understanding of the structure of binarized neural networks, specifically, ideal convex representations of the activation functions used. We describe the convex hull of the graph of the signum activation function associated … Read more

2×2-convexifications for convex quadratic optimization with indicator variables

In this paper, we study the convex quadratic optimization problem with indicator variables. For the bivariate case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the bivariate case as a building block, we derive … Read more