Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory … Read more
This paper describes a new primal relaxation for nonlinear integer programming problems with linear constraints. This relaxation, contrary to the standard Lagrangean relaxation, can be solved efficiently. It requires the solution of a nonlinear penalized problem whose linear constraint set is known only implicitly, but whose solution is made possible by the use of a … Read more
A finite test set for an integer maximization problem enables us to verify whether a feasible point attains the global maximum. We establish in this paper several general results that apply to integer maximization problems with nonlinear objective functions. Citation Operations Research Letters 36 (2008) 439–443 Article Download View On Test Sets for Nonlinear Integer … Read more
A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These … Read more
We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based on only the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation. Citation Appeared in Operations Research … Read more
In the context of a variation of the standard UFL (Uncapacitated Facility Location) problem, but with an objective function that is a separable convex quadratic function of the transportation costs, we present some techniques for improving relaxations of MINLP formulations. We use a disaggregation principle and a strategy of developing model-specific valid inequalities (some nonlinear), … Read more
This paper develops a linear programming based branch-and-bound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other linear programming based branch-and-bound algorithms for mixed integer nonlinear programs in that, it … Read more
We present experimental results on portfolio optimization problems with return errors under the robust optimization framework. We use several a histogram-like model for return deviations, and a model that allows correlation among errors, together with a cutting-plane algorithm which proves effective for large, real-life data sets. Citation Columbia Center for Financial Engineering Report 2007-01 Columbia … Read more
In this paper, we study extensions of the classical Markowitz mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint which imposes that the expected return of the constructed portfolio must exceed a prescribed return threshold with a high confidence level. We study the deterministic equivalents … Read more
We describe improvements to Smith’s branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in R^d. Nodes in the B&B tree correspond to full Steiner topologies associated with a subset of the terminal nodes, and branching is accomplished by “merging” a new terminal node with each edge in the current Steiner tree. For a given topology … Read more