A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities with Applications to Bilevel Optimization and Gas Transport

We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate … Read more

Superiorization as a novel strategy for linearly constrained inverse radiotherapy treatment planning

Objective: We apply the superiorization methodology to the intensity-modulated radiation therapy (IMRT) treatment planning problem. In superiorization, linear voxel dose inequality constraints are the fundamental modeling tool within which a feasibility-seeking projection algorithm will seek a feasible point. This algorithm is then perturbed with gradient descent steps to reduce a nonlinear objective function. Approach: Within … Read more

Cutting-plane algorithm for sparse estimation of the Cox proportional-hazards model

Survival analysis is a family of statistical methods for analyzing event occurrence times. In this paper, we address the mixed-integer optimization approach to sparse estimation of the Cox proportional-hazards model for survival analysis. Specifically, we propose a high-performance cutting-plane algorithm based on reformulation of bilevel optimization for sparse estimation. This algorithm solves the upper-level problem … Read more

Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity, which is both of theoretical as well as practical interest. This area of … Read more

Blessing of Nonconvexity in Deep Linear Models: Depth Flattens the Optimization Landscape Around the True Solution

This work characterizes the effect of depth on the optimization landscape of linear regression, showing that, despite their nonconvexity, deeper models have more desirable optimization landscape. We consider a robust and over-parameterized setting, where a subset of measurements are grossly corrupted with noise and the true linear model is captured via an $N$-layer linear neural … Read more

The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning

In this paper we study the split minimization problem that consists of two constrained minimization problems in two separate spaces that are connected via a linear operator that maps one space into the other. To handle the data of such a problem we develop a superiorization approach that can reach a feasible point with reduced … Read more

A combined model for chain expansion including the possibility of locating a new facility and modification and/or closing of existing facilities

The problem of an expanding chain (it already has some facilities) in a given area is considered. It may locate a new facility, or vary (up or down) the quality of its existing facilities, or close some of them, or a combination of all those possibilities, whatever it is the best to maximize its profit, … Read more

Dual solutions in convex stochastic optimization

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental … Read more

Solving large-scale unit-commitment problems using dual dynamic programming and open-source solvers

The astonishing dimensions and complexity of power systems render them impossible to be managed without the help of cutting-edge software. Due to a lack of scalable, reliable and well documented free and open-source solutions, system operators, regulators, and government agencies often rely on proprietary software to provide them information that ultimately will be used to … Read more

Metrizing Fairness

We study supervised learning problems for predicting properties of individuals who belong to one of two demographic groups, and we seek predictors that are fair according to statistical parity. This means that the distributions of the predictions within the two groups should be close with respect to the Kolmogorov distance, and fairness is achieved by … Read more