In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using mainly tools from discrete geometry, we show that some sparsity conditions on the original data matrix allow … Read more
We present new integer linear programming (ILP) models for NP-hard optimisation problems in instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals / Residents problem with Ties (HRT). These models can be used to efficiently solve these optimisation problems when applied to (i) instances derived from … Read more
In this paper, we propose a new approach to solve low-rank tensor completion and robust tensor PCA. Our approach is based on some novel notion of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank. We discuss the connections between these new tensor ranks and the CP-rank and … Read more
A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program … Read more
We consider optimal transport based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded … Read more
In recent years the use of decision diagrams within the context of discrete optimization has proliferated. This paper continues this expansion by proposing the use of decision diagrams for modeling and solving binary optimization problems with quadratic constraints. The model proposes the use of multiple decision diagrams to decompose a quadratic matrix so that each … Read more
We propose a new Mixed Integer Programming formulation for the Asymmetric Traveling Salesman Problem with Pickup and Delivery, along with valid inequalities for the Sarin-Sherali-Bhootra formulation. We study these models in their complete forms, relax complicating constraints of these models, and compare their performance. Finally, we present computational results showing the promise of these formulations … Read more
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proximal linearized augmented Lagrangian algorithm (PLAM) … Read more
The limited memory BFGS (L-BFGS) method is widely used for large-scale unconstrained optimization, but its behavior on nonsmooth problems has received little attention. L-BFGS can be used with or without “scaling”; the use of scaling is normally recommended. A simple special case, when just one BFGS update is stored and used at every iteration, is … Read more
The paper “An algorithmic characterization of P-matricity” (SIAM Journal on Matrix Analysis and Applications, 34:3 (2013) 904–916, by the same authors as here) implicitly assumes that the iterates generated by the Newton-min algorithm for solving a linear complementarity problem of dimension n, which reads 0 ⩽ x ⊥ (M x + q) ⩾ 0, are … Read more