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

The Quadratic Cycle Cover Problem: special cases and efficient bounds

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic

In this paper, we present a multistage time consistent Expected Conditional Risk Measure for minimizing a linear combination of the expected mean and the expected variance, so-called Expected Mean-Variance. The model is formulated as a multistage stochastic mixed-integer quadratic programming problem combining risk-sensitive cost and scenario analysis approaches. The proposed problem is solved by a matheuristic based on the Branch-and-Fix Coordination method. The multistage scenario cluster primal decomposition framework is extended to deal with large-scale quadratic optimization by means of stage-wise reformulation techniques. A specific case study in risk-sensitive production planning is used to illustrate that a remarkable decrease in the expected variance (risk cost) is obtained. A competitive behavior on the part of our methodology in terms of solution quality and computation time is shown when comparing with plain use of CPLEX in 150 benchmark instances, ranging up to 711,845 constraints and 193,000 binary variables.project MTM2015-65317-P (MINECO/FEDER/EU); BERC 2014-2017; IT-928-16; and by the University of the Basque Country UPV/EHU; BCAM Severo Ochoa excellence accreditation Grant SEV-2013-0323; BERC 2014-201

Quadratic Binary Programming Models in Computational Biology

In this paper we formulate four problems in computational molecular biology as 0-1 quadratic programs. These problems are all NP-hard and the current solution methods used in practice consist of heuristics or approximation algorithms tailored to each problem. Using test problems from scientific databases, we address the question, “Can a general-purpose solver obtain good answers in reasonable time?” In addition, we use the latest heuristics as incumbent solutions to address the question, “Can a general-purpose solver confirm optimality or find an improved solution in reasonable time?” Our computational experiments compare four different reformulation methods: three forms of linearization and one form of quadratic convexification

Reformulation Techniques and Solution Approaches for Fractional 0-1 Programs and Applications

Fractional binary programs (FPs) form a broad class of nonlinear integer optimization problems, where the objective is to optimize the sum of ratios of (linear) binary functions. FPs arise naturally in a number of important real-life applications such as scheduling, retail assortment, facility location, stochastic service systems, and machine learning, among others. This dissertation studies methods that improve the performance of solution approaches for fractional binary programs in their general structure. In particular, we first explore the links between equivalent mixed-integer linear programming (MILP) and conic quadratic programming reformulations of FPs. Thereby, we show that integrating the ideas behind these two types of reformulations of FPs allows us to push further the limits of the current state-of-the-art results and tackle larger-size problems. In practice, the parameters of an optimization problem are often subject to uncertainty. To deal with uncertainties in FPs, we extend the robust methodology to fractional binary programming. In particular, we study robust fractional binary programs (RFPs) under a wide-range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator, and the latter accounts for different forms of inter-relatedness between them. We demonstrate that, unlike the deterministic case, single-ratio RFP is NP-hard under general polyhedral uncertainty sets. However, if the uncertainty sets are imbued with a certain structure - variants of the well-known budgeted uncertainty - the disjoint and joint single-ratio RFPs are polynomially-solvable when the deterministic counterpart is. We also propose MILP formulations for multiple-ratio RFPs and evaluate their performances by using real and synthetic data sets. One interesting application of FPs arises in feature selection which is an essential preprocessing step for many machine learning and pattern recognition systems and involves identification of the most characterizing features from the data. Notably, correlation-based and mutual-information-based feature selection problems can be reformulated as single-ratio FPs. We study approaches that ensure globally optimal solutions for medium- and reasonably large-sized instances of the aforementioned problems, where the existing MILPs in the literature fail. We perform computational experiments with diverse classes of real data sets and report encouraging results

Mixed Integer Linear Programming Formulation Techniques

A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques. However, in some cases the resulting MIP can be either too weak or too large to be effectively solved by state of the art solvers. In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems