Sparse PCA With Multiple Components

Sparse Principal Component Analysis 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. At its heart, this involves solving a sparsity and orthogonality constrained convex maximization problem, which is extremely computationally challenging. Most existing work address sparse PCA via heuristics … Read more

Minkowski Centers via Robust Optimization: Computation and Applications

Centers of convex sets are geometrical objects that have received extensive attention in the mathematical and optimization literature, both from a theoretical and practical standpoint. For instance, they serve as initialization points for many algorithms such as interior-point, hit-and-run, or cutting-planes methods. First, we observe that computing a Minkowski center of a convex set can … Read more

A new perspective on low-rank optimization

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function — the matrix analog of the perspective function — and characterize explicitly … Read more

Hospital-wide Inpatient Flow Optimization

To improve quality and delivery of care, operations need to be coordinated and optimized across all services in real-time. We propose a multi-stage adaptive robust optimization approach combined with machine learning techniques to achieve this goal. Informed by data and predictions, our framework unifies the bed assignment process across the entire hospital and accounts for … Read more

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2 = Y$, the matrix analog of binary variables that satisfy $z^2 = z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization … Read more

Robust Convex Optimization: A New Perspective That Unifies And Extends

Robust convex constraints are difficult to handle, since finding the worst-case scenario is equivalent to maximizing a convex function. In this paper, we propose a new approach to deal with such constraints that unifies approaches known in the literature and extends them in a significant way. The extension is either obtaining better solutions than the … Read more

Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality

Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than $p=100s$ of variables. By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a … Read more

Probabilistic guarantees in Robust Optimization

We develop a general methodology to derive probabilistic guarantees for solutions of robust optimization problems. Our analysis applies broadly to any convex compact uncertainty set and to any constraint affected by uncertainty in a concave manner, under minimal assumptions on the underlying stochastic process. Namely, we assume that the coordinates of the noise vector are … Read more

A Unified Approach to Mixed-Integer Optimization Problems With Logical Constraints

We propose a unified framework to address a family of classical mixed-integer optimization problems with logically constrained decision variables, including network design, facility location, unit commitment, sparse portfolio selection, binary quadratic optimization, sparse principal component analysis and sparse learning problems. These problems exhibit logical relationships between continuous and discrete variables, which are usually reformulated linearly … Read more