This paper focuses on algorithms for multi-stage stochastic linear programming (MSLP). We propose an ensemble method named the “compromise policy”, which not only reduces the variance of the function approximation but also reduces the bias of the estimated optimal value. It provides a tight lower bound estimate with a confidence interval. By exploiting parallel computing, … Read more
Sparse Principal Component Analysis (sPCA) is a cardinal technique for obtaining combinations of features, or principal components (PCs), that explain the variance of high-dimensional datasets in an interpretable manner. This involves solving a sparsity and orthogonality-constrained convex maximization problem, which is extremely computationally challenging. Most existing works address sparse PCA via methods—such as iteratively computing … Read more
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades. On the one hand, we prove that when the rank of the positive semidefinite matrix in the decomposition … Read more
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
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
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
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
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
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
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