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We introduce a novel approach addressing global analysis of a difficult class of nonconvex-nonsmooth optimization problems within the important framework of Lagrangian-based methods. This genuine nonlinear class captures many problems in…
Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…
The task of inferring logical formulas from examples has garnered significant attention as a means to assist engineers in creating formal specifications used in the design, synthesis, and verification of computing systems. Among various…
We present algorithms for efficiently learning regularizers that improve generalization. Our approach is based on the insight that regularizers can be viewed as upper bounds on the generalization gap, and that reducing the slack in the…
To optimize efficiently over discrete data and with only few available target observations is a challenge in Bayesian optimization. We propose a continuous relaxation of the objective function and show that inference and optimization can be…
The aim of this paper is to solve linear semidefinite programs arising from higher-order Lasserre relaxations of unconstrained binary quadratic optimization problems. For this we use an interior point method with a preconditioned conjugate…
Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and…
This paper presents a novel robust trajectory optimization method for constrained nonlinear dynamical systems subject to unknown bounded disturbances. In particular, we seek optimal control policies that remain robustly feasible with…
We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of…
An optimal transport problem on finite spaces is a linear program. Recently, a relaxation of the optimal transport problem via strictly convex functions, especially via the Kullback--Leibler divergence, sheds new light on data sciences.…
We consider convex optimization problems formulated using dynamic programming equations. Such problems can be solved using the Dual Dynamic Programming algorithm combined with the Level 1 cut selection strategy or the Territory algorithm to…
This paper proposes tight semidefinite relaxations for polynomial optimization. The optimality conditions are investigated. We show that generally Lagrange multipliers can be expressed as polynomial functions in decision variables over the…
The relaxed maximum entropy problem is concerned with finding a probability distribution on a finite set that minimizes the relative entropy to a given prior distribution, while satisfying relaxed max-norm constraints with respect to a…
This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the…
This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented…
We consider the problem of learning the structure of undirected graphical models with bounded treewidth, within the maximum likelihood framework. This is an NP-hard problem and most approaches consider local search techniques. In this…
Binary Decision Diagram (BDD) based set bounds propagation is a powerful approach to solving set-constraint satisfaction problems. However, prior BDD based techniques in- cur the significant overhead of constructing and manipulating graphs…
This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming. The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable…
We are interested in optimally driving a dynamical system that can be influenced by exogenous noises. This is generally called a Stochastic Optimal Control (SOC) problem and the Dynamic Programming (DP) principle is the natural way of…
We consider Lagrangian duality based approaches to design and analyze algorithms for online energy-efficient scheduling. First, we present a primal-dual framework. Our approach makes use of the Lagrangian weak duality and convexity to…