Interior-point solver for convex separable block-angular problems

Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the precondi- tioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using l1 and l2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to l1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.Preprin

Improving an interior-point algorithm for multicommodity flows by quadratic regularizations

One of the best approaches for some classes of multicommodity flow problems is a specialized interior-point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work the authors improved this algorithm (i.e., reduced the spectral radius) for general block-angular problems by adding a quadratic regularization to the logarithmic barrier. This barrier was shown to be self-concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work we focus on linear multicommodity problems, a particular case of primal block-angular ones. General results are tailored for multicommodity flows, allowing a local sensitivity analysis on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior-point algorithm is more efficient than the nonregularized one. From this work it can be concluded that, if interior-point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones.Preprin

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.Peer ReviewedPostprint (author’s final draft

Improving an interior-point approach for large block-angular problems by hybrid preconditioners

The computational time required by interior-point methods is often domi- nated by the solution of linear systems of equations. An efficient spec ialized interior-point algorithm for primal block-angular proble ms has been used to solve these systems by combining Cholesky factorizations for the block con- straints and a conjugate gradient based on a power series precon ditioner for the linking constraints. In some problems this power series prec onditioner re- sulted to be inefficient on the last interior-point iterations, wh en the systems became ill-conditioned. In this work this approach is combi ned with a split- ting preconditioner based on LU factorization, which is main ly appropriate for the last interior-point iterations. Computational result s are provided for three classes of problems: multicommodity flows (oriented and no noriented), minimum-distance controlled tabular adjustment for statistic al data protec- tion, and the minimum congestion problem. The results show that , in most cases, the hybrid preconditioner improves the performance an d robustness of the interior-point solver. In particular, for some block-ang ular problems the solution time is reduced by a factor of 10.Peer ReviewedPreprin

Solving L1L_1-CTA in 3D tables by an interior-point method for primal block-angular problems

The purpose of the field of statistical disclosure control is to avoid that no confidential information can be derived from statistical data released by, mainly, national statistical agencies. Controlled tabular adjustment (CTA) is an emerging technique for the protection of statistical tabular data. Given a table to be protected, CTA looks for the closest safe table. In this work we focus on CTA for three-dimensional tables using the L1 norm for the distance between the original and protected tables. Three L1-CTA models are presented, giving rise to six different primal block-angular structures of the constraint matrices. The resulting linear programming problems are solved by a specialized interior-point algorithm for this constraints structure, which solves the normal equations by a combination of Cholesky factorization and preconditioned conjugate gradients (PCG). In the past this algorithm shown to be one of the most efficient approaches for some classes of block-angular problems. The effect of quadratic regularizations is also analyzed, showing that for three of the six primal block-angular structures the performance of PCG is guaranteed to improve. Computational results are reported for a set of large instances, which provide linear optimization problems of up to 50 millions of variables and 25 millions of constraints. The specialized interior-point algorithm is compared with the state-of-the-art barrier solver of the CPLEX 12.1 package, showing to be a more efficient choice for very large L1-CTA instances.Preprin

A new interior-point approach for large two-stage stochastic problems

Two-stage stochastic models give rise to very large optimization problems. Several approaches havebeen devised for efficiently solving them, including interior-point methods (IPMs). However, usingIPMs, the linking columns associated to first-stage decisions cause excessive fill-in for the solutionof the normal equations. This downside is usually alleviated if variable splitting is applied to first-stage variables. This work presents a specialized IPM that applies variable splitting and exploits thestructure of the deterministic equivalent of the stochastic problem. The specialized IPM combinesCholesky factorizations and preconditioned conjugate gradients for solving the normal equations.This specialized IPM outperforms other approaches when the number of first-stage variables is largeenough. This paper provides computational results for two stochastic problems: (1) a supply chainsystem and (2) capacity expansion in an electric system. Both linear and convex quadratic formu-lations were used, obtaining instances of up to 38 million variables and six million constraints. Thecomputational results show that our procedure is more efficient than alternative state-of-the-art IPMimplementations (e.g., CPLEX) and other specialized solvers for stochastic optimizationPeer ReviewedPreprin

Exploiting total unimodularity for classes of random network problems

Network analysis is of great interest for the study of social , biological and technolog- ical networks, with applications, among others, in busines s, marketing, epidemiology and telecommunications. Researchers are often interested in a ssessing whether an observed fea- ture in some particular network is expected to be found withi n families of networks under some hypothesis (named conditional random networks, i.e., networks satisfying some linear constraints). This work presents procedures to generate ne tworks with specified structural properties which rely on the solution of classes of integer o ptimization problems. We show that, for many of them, the constraints matrices are totally unimodular, allowing the efficient generation of conditional random networks by polynomial ti me interior-point methods. The computational results suggest that the proposed methods ca n represent a general framework for the efficient generation of random networks even beyond the models analyzed in this pa- per. This work also opens the possibility for other applicat ions of mathematical programming in the analysis of complex networks.Preprin