Optimizing the Path Towards Plastic-Free Oceans

Increasing ocean plastic pollution is irreversibly harming ecosystems and human economic activities. We partner with a non-profit organization and use optimization to help clean up oceans from plastic faster. Specifically, we optimize the route of their plastic collection system in the ocean to maximize the quantity of plastic collected over time. We formulate the problem … Read more

Optimal Low-Rank Matrix Completion: Semidefinite Relaxations and Eigenvector Disjunctions

Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly scalable and often identifying high-quality solutions, do not possess any optimality guarantees. We reexamine matrix completion with an optimality-oriented eye. We reformulate … Read more

A Stochastic Benders Decomposition Scheme for Large-Scale Stochastic Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a stochastic version where operational costs are uncertain due to fluctuating demand and estimated as a sample average from historical data. This problem is computationally challenging, and instances with as few as  100 … Read more

Sparse PCA With Multiple Components

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

Minkowski Centers via Robust Optimization: Computation and Applications

Centers of convex sets are geometric 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 be formulated as … 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