Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as a mixed-integer program with an objective function composed of a convex quadratic loss function and a regularization penalty subject to linear constraints. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions. However, the state-of-the-art optimization solvers are not able to obtain provably optimal solutions to the existing mathematical formulations for medium-size problems within reasonable computational times. To address this difficulty, we tackle the problem from both computational and statistical perspectives. On the one hand, we propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution to the mixed-integer program, and establish the consistency of this approximate solution. On the other hand, we improve the existing formulations by replacing the linear "big-$M$" constraints that represent the relationship between the continuous and binary indicator variables with second-order conic constraints. Our numerical results demonstrate the effectiveness of the proposed approaches

Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion

Mixed-Integer Second-Order Cone Programs (MISOCPs) form a nice class of mixed-inter convex programs, which can be solved very efficiently due to the recent advances in optimization solvers. Our paper bridges the gap between modeling a class of optimization problems and using MISOCP solvers. It is shown how various performance metrics of M/G/1 queues can be molded by different MISOCPs. To motivate our method practically, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service networks. Four different MISOCPs are developed and compared on sets of benchmark test problems. The new formulations efficiently solve large-size test problems, which cannot be solved by the best existing method. Then, the general applicability of our method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

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

SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression

This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, tipically d<5, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem.Comment: 12 pages, 3 figures, 2 table