Jordi Castro

A Specialized Interior-Point Algorithm for Multicommodity Network Flows

Despite the efficiency shown by interior-point methods in large-scale linear programming, they usually perform poorly when applied to multicommodity flow problems. The new specialized interior-point algorithm presented here overcomes this drawback. This specialization uses both a preconditioned conjugate gradient solver and a sparse Cholesky factorization to solve a linear system of equations at each iteration of the algorithm. The ad hoc preconditioner developed by exploiting the structure of the problem is instrumental in ensuring the efficiency of the method. An implementation of the algorithm is compared to state-of-the-art packages for multicommodity flows. The computational experiments were carried out using an extensive set of test problems, with sizes of up to 700,000 variables and 150,000 constraints. The results show the effectiveness of the algorithm.

Minimum-distance controlled perturbation methods for large-scale tabular data protection

Abstract National Statistical Agencies routinely release large amounts of tabular information. Prior to dissemination, tabular data needs to be processed to avoid the disclosure of individual confidential information. One widely used class of methods is based on the modification of the table cells values. However, previous approaches were not able to preserve the values of the marginal cells and the additivity relations for a general table of any dimension, size and structure. This void was recently filled by the controlled tabular adjustment and one of its variants, the quadratic minimum-distance controlled perturbation method. Although independently developed, both approaches rely on the same strategy: given a set of tables to be protected, they find the minimum-distance values to the original cells that make the released information safe. Controlled tabular adjustment uses the L 1 distance; the quadratic minimum-distance variant considers L 2 . This work presents both approaches within an unified framework, and includes a new variant based on L ∞ . Among other benefits, the unified framework permits the simple comparison of the three distances, and a single general result about their disclosure risk. The three distances are evaluated with the unique standard library for tabular data protection currently available. Some of the complex instances were contributed by National Statistical Agencies, and, therefore, are good representatives of theirs real needs. Unlike alternative methods, the three distances were able to solve all the instances, requiring only few seconds for each of them on a personal computer using a general purpose solver. The results show that this class of methods are an effective and promising tool for the protection of large volumes of tabular data. All the linear and quadratic problems solved in the paper are delivered to the optimization community in MPS format.

A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.

An interior-point approach for primal block-angular problems

Abstract Multicommodity flows belong to the class of primal block-angular problems. An efficient interior-point method has already been developed for linear and quadratic network optimization problems. It solved normal equations, using sparse Cholesky factorizations for diagonal blocks, and a preconditioned conjugate gradient for linking constraints. In this work we extend this procedure, showing that the preconditioner initially developed for multicommodity flows applies to any primal block-angular problem, although its efficiency depends on each particular linking constraints structure. We discuss the conditions under which the preconditioner is effective. The procedure is implemented in a user-friendly package in the MATLAB environment. Computational results are reported for four primal block-angular problems: multicommodity flows, nonoriented multicommodity flows, minimum-distance controlled tabular adjustment for statistical data protection, and the minimum congestion problem. The results show that this procedure holds great potential for solving large primal-block angular problems efficiently.

Recent advances in optimization techniques for statistical tabular data protection

One of the main services of National Statistical Agencies (NSAs) for the current Information Society is the dissemination of large amounts of tabular data, which is obtained from microdata by crossing one or more categorical variables. NSAs must guarantee that no confidential individual information can be obtained from the released tabular data. Several statistical disclosure control methods are available for this purpose. These methods result in large linear, mixed integer linear, or quadratic mixed integer linear optimization problems. This paper reviews some of the existing approaches, with an emphasis on two of them: cell suppression problem (CSP) and controlled tabular adjustment (CTA). CSP and CTA have concentrated most of the recent research in the tabular data protection field. The particular focus of this work is on methods and results of practical interest for end-users (mostly, NSAs). Therefore, in addition to the resulting optimization models and solution approaches, computational results comparing the main optimization techniques – both optimal and heuristic – using real-world instances are also presented.

Quadratic regularizations in an interior-point method for primal block-angular problems

One of the most efficient interior-point methods for some classes of primal block-angular problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. Its efficiency depends on the spectral radius—in [0,1)— of a certain matrix in the definition of the preconditioner. Spectral radius close to 1 degrade the performance of the approach. The purpose of this work is twofold. First, to show that a separable quadratic regularization term in the objective reduces the spectral radius, significantly improving the overall performance in some classes of instances. Second, to consider a regularization term which decreases with the barrier function, thus with no need for an extra parameter. Computational experience with some primal block-angular problems confirms the efficiency of the regularized approach. In particular, for some difficult problems, the solution time is reduced by a factor of two to ten by the regularization term, outperforming state-of-the-art commercial solvers.

A Shortest-Paths Heuristic for Statistical Data Protection in Positive Tables

National statistical agencies (NSAs) routinely release large amounts of tabular information. Prior to dissemination, tabular data need to be processed to avoid disclosure of individual confidential information. Cell suppression is one of the most widely used techniques by NSAs. Optimal procedures for cell suppression are computationally expensive with large real-world data sets, so heuristic procedures are used. Most heuristics for positive tables (i.e., cell values are nonnegative) rely on the solution of minimum-cost network-flows subproblems. A very efficient heuristic based on shortest paths already exists, but it is only appropriate for general tables (i.e., cell values can be either positive or negative), whereas in practice most tables are positive. We present a method that sensibly combines and improves previous approaches, overcoming some of their drawbacks: it is designed for positive tables and requires only the solution of shortest-path subproblems---therefore being much more efficient than other network-flows heuristics. We report extensive computational experience in the solution of randomly generated and real-world instances, comparing the heuristic with alternative procedures. The results show that the method, currently included in a software package for statistical data protection, fits NSA needs: it is extremely efficient and provides good solutions.

Solving Difficult Multicommodity Problems with a Specialized Interior-Point Algorithm

Due to recent advances in the development of linear programming solvers, some of the formerly considered difficult multicommodity problems can today be solved in few minutes, even faster than with specialized methods. However, for other kind of multicommodity instances, general linear solvers can still be quite inefficient. In this paper we will give an overview of the current state-of-the-art in solving large-scale multicommodity problems, comparing an specialized interior-point algorithm with CPLEX 6.5 in the solution of difficult multicommodity problems of up to 1 million of variables and 300,000 constraints.

Using the analytic center in the feasibility pump

Abstract The feasibility pump (FP) has proved to be a successful heuristic for finding feasible solutions of mixed integer linear problems. Briefly, FP alternates between two sequences of points: one of feasible solutions for the relaxed problem, and another of integer points. This short paper extends FP, such that the integer point is obtained by rounding a point on the (feasible) segment between the computed feasible point and the analytic center for the relaxed linear problem.

Quadratic interior-point methods in statistical disclosure control

Abstract.The safe dissemination of statistical tabular data is one of the main concerns of National Statistical Institutes (NSIs). Although each cell of the tables is made up of the aggregated information of several individuals, the statistical confidentiality can be violated. NSIs must guarantee that no individual information can be derived from the released tables. One widely used type of methods to reduce the disclosure risk is based on the perturbation of the cell values. We consider a new controlled perturbation method which, given a set of tables to be protected, finds the closest safe ones - thus reducing the information loss while preserving confidentiality. This approach means solving a quadratic optimization problem with a much larger number of variables than constraints. Real instances can provide problems with millions of variables. We show that interior-point methods are an effective choice for that model, and, also, that specialized algorithms which exploit the problem structure can be faster than state-of-the art general solvers. Computational results are presented for instances of up to 1000000 variables.