Complexity of optimizing over the integers

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like “input”, “size” and “complexity” in the context of general mathematical optimization, avoiding context dependent definitions which is one of the … Read more

Complexity of cutting planes and branch-and-bound in mixed-integer optimization

We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In particular, we study the relative efficiency of BB and CP, when both are based on the same family of disjunctions. We extend a result of Dash to the nonlinear setting which shows that for convex 0/1 problems, CP … Read more

Admissibility of solution estimators for stochastic optimization

We look at stochastic optimization problems through the lens of statistical decision theory. In particular, we address admissibility, in the statistical decision theory sense, of the natural sample average estimator for a stochastic optimization problem (which is also known as the empirical risk minimization (ERM) rule in learning literature). It is well known that for … Read more

Admissibility of solution estimators for stochastic optimization

We look at stochastic optimization problems through the lens of statistical decision theory. In particular, we address admissibility, in the statistical decision theory sense, of the natural sample average estimator for a stochastic optimization problem (which is also known as the empirical risk minimization (ERM) rule in learning literature). It is well known that for … Read more

Mixed-integer bilevel representability

We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these … Read more

Can cut generating functions be good and efficient?

Making cut generating functions (CGFs) computationally viable is a central question in modern integer programming research. One would like to nd CGFs that are simultaneously good, i.e., there are good guarantees for the cutting planes they generate, and ecient, meaning that the values of the CGFs can be computed cheaply (with procedures that have some … Read more

Approximation of Minimal Functions by Extreme Functions

In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The n-dimensional version of this result was left as an open question. In the present paper, we settle this query in the affirmative: for any … Read more

Optimal cutting planes from the group relaxations

We study quantitative criteria for evaluating the strength of valid inequalities for Gomory and Johnson’s finite and infinite group models and we describe the valid inequalities that are optimal for these criteria. We justify and focus on the criterion of maximizing the volume of the nonnegative orthant cut off by a valid inequality. For the … Read more

Understanding Deep Neural Networks with Rectified Linear Units

In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give the first-ever polynomial time (in the size of data) algorithm to train a ReLU DNN with one hidden layer to {\em global optimality}. This follows from our complete characterization of the ReLU DNN … Read more