Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

A Scalable Algorithm For Sparse Portfolio Selection

The sparse portfolio selection problem is one of the most famous and frequently-studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal expected return and minimum variance, subject to an upper bound on the number of positions, linear inequalities and minimum investment constraints. Existing certifiably optimal approaches to this problem do not converge within a practical amount of time at real world problem sizes with more than 400 securities. In this paper, we propose a more scalable approach. By imposing a ridge regularization term, we reformulate the problem as a convex binary optimization problem, which is solvable via an efficient outer-approximation procedure. We propose various techniques for improving the performance of the procedure, including a heuristic which supplies high-quality warm-starts, a preprocessing technique for decreasing the gap at the root node, and an analytic technique for strengthening our cuts. We also study the problem's Boolean relaxation, establish that it is second-order-cone representable, and supply a sufficient condition for its tightness. In numerical experiments, we establish that the outer-approximation procedure gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Computational Methods for Discrete Conic Optimization Problems

This thesis addresses computational aspects of discrete conic optimization. Westudy two well-known classes of optimization problems closely related to mixedinteger linear optimization problems. The case of mixed integer second-ordercone optimization problems (MISOCP) is a generalization in which therequirement that solutions be in the non-negative orthant is replaced by arequirement that they be in a second-order cone. Inverse MILP, on the otherhand, is the problem of determining the objective function that makes a givensolution to a given MILP optimal.Although these classes seem unrelated on the surface, the proposedsolution methodology for both classes involves outer approximation of a conicfeasible region by linear inequalities. In both cases, an iterative algorithmin which a separation problem is solved to generate the approximation isemployed. From a complexity standpoint, both MISOCP and inverse MILP areNP--hard. As in the case of MILPs, the usual decision version ofMISOCP is NP-complete, whereas in contrast to MILP, we provide the firstproof that a certain decision version of inverse MILP is rathercoNP-complete.With respect to MISOCP, we first introduce a basic outer approximationalgorithm to solve SOCPs based on a cutting-plane approach. As expected, theperformance of our implementation of such an algorithm is shown to lag behindthe well-known interior point method. Despite this, such a cutting-planeapproach does have promise as a method of producing bounds when embedded withina state-of-the-art branch-and-cut implementation due to its superior ability towarm-start the bound computation after imposing branching constraints. Ourouter-approximation-based branch-and-cut algorithm relaxes both integrality andconic constraints to obtain a linear relaxation. This linear relaxation isstrengthened by the addition of valid inequalities obtained by separatinginfeasible points. Valid inequalities may be obtained by separation from theconvex hull of integer solution lying within the relaxed feasible region or byseparation from the feasible region described by the (relaxed) conicconstraints. Solutions are stored when both integer and conic feasibility isachieved. We review the literature on cutting-plane procedures for MISOCP andmixed integer convex optimization problems.With respect to inverse MILP, we formulate this problem as a conicproblem and derive a cutting-plane algorithm for it. The separation problem inthis algorithm is a modified version of the original MILP. We show that thereis a close relationship between this algorithm and a similar iterativealgorithm for separating infeasible points from the convex hull of solutions tothe original MILP that forms part of the basis for the well-known result ofGrotschel-Lovasz-Schrijver that demonstrates the complexity-wiseequivalence of separation and optimization.In order to test our ideas, we implement a number of software librariesthat together constitute DisCO, a full-featured solver for MISOCP. Thefirst of the supporting libraries is OsiConic, an abstract base classin C++ for interfacing to SOCP solvers. We provide interfaces using thislibrary for widely used commercial and open source SOCP/nonlinear problemsolvers. We also introduce CglConic, a library that implements cuttingprocedures for MISOCP feasible set. We perform extensive computationalexperiments with DisCO comparing a wide range of variants of our proposedalgorithm, as well as other approaches. As DisCO is built on top of a libraryfor distributed parallel tree search algorithms, we also perform experimentsshowing that our algorithm is effective and scalable when parallelized