Despite recent interest in multiobjective integer programming, few algorithms exist for solving biobjective mixed integer programs. We present such an algorithm: the Boxed Line Method. For one of its variants, we prove that the number of single-objective integer programs solved is bounded by a linear function of the number of nondominated line segments in the … Read more
In this work, we study the problem of scheduling courses and instructors in the Mathematics Department at the United States Naval Academy (USNA) in a resilient manner. Every semester, the department needs to schedule around 70 instructors and 150-180 course sections into 30 class periods and 30 rooms. We formulate a stochastic integer linear program … Read more
We propose a sigmoidal approximation (SigVaR) for the value-at-risk (VaR) and we use this approximation to tackle nonlinear programming problems (NLPs) with chance constraints. We prove that the approximation is conservative and that the level of conservatism can be made arbitrarily small for limiting parameter values. The SigVar approximation brings computational benefits over exact mixed-integer … Read more
We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould and Toint (2010,2011). To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region … Read more
Given an undirected graph G = (V, E), a vertex k-cut of G is a vertex subset of V the removing of which disconnects the graph in at least k connected components. Given a graph G and an integer k greater than or equal to two, the vertex k-cut problem consists in finding a vertex … Read more
The assortment of products carried by a store has a crucial impact on its success. However, finding the right mix of products to attract a large portion of the customers is a challenging task. Several mathematical models have been proposed to optimize assortments. In particular, rank-based choice models have been acknowledged for representing well high-dimensional … Read more
Recently Dadush, Vegh, and Zambelli (2017) has devised a polynomial submodular function minimization (SFM) algorithm based on their LP algorithm. In the present note we also show a weakly polynomial algorithm for SFM based on the recently developed linear programming feasibility algorithm of Chubanov (2017). Our algorithm is different from Dadush, Vegh, and Zambelli’s but … Read more
This paper discusses the practical use of the saddle variational formulation for the weakly-constrained 4D-VAR method in data assimilation. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationaly coherent variants of the algorithm … Read more
To precondition the normal equation system from the linear programming (LP) interior point method, basis preconditioners choose a basis matrix dependent on column scaling factors. Two criteria for choosing the basis matrix are compared which yield a maximum volume or maximum weight basis. Finding a maximum volume basis requires a combinatorial effort, but it gives … Read more
In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing and machine learning research. The traditional alternating direction method of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of … Read more