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 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

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

Research Report UPC-DEIO DR 2018-01. November 2018The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows problems in bipartite networks, for convex objective functions with diagonal Hessians (i.e., either linear, quadratic or separable nonlinear objectives). After analyzing the theoretical properties of the interior-point method for this kind of problems, we provide extensive computational experiments with linear and quadratic instances of up to one billion arcs and 200 and five million nodes in each subset of the node partition. For linear and quadratic instances our approach is compared with the barriers algorithms of CPLEX (both standard path-following and homogeneous-self-dual); for linear instances it is also compared with the different algorithms of the state-of-the-art network flow solver LEMON (namely: network simplex, capacity scaling, cost scaling and cycle canceling). The specialized interior-point approach significantly outperformed the other approaches in most of the linear and quadratic transportation instances tested. In particular, it always provided a solution within the time limit and it never exhausted the 192 Gigabytes of memory of the server used for the runs. For assignment problems the network algorithms in LEMON were the most efficient option.Peer ReviewedPreprin

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

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On geometrical properties of preconditioners in IPMs for classes of block-angular problems

J. Castro, S. Nasini, On geometrical properties of preconditioners in IPMs for classes of block-angular problems, Research Report DR 2016/03, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, 2016.One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraints structure. In particular, it is seen that the principal angles between the subspaces generated by the diagonal blocks and the linking constraints can be used to estimate ex-ante the efficiency of the preconditioner. Numerical validation is provided with some generated optimization problems. An application to the solution of multicommodity network flow problems with nodal capacities and equal flows of up to 127 million of variables and up to 7.5 million of constraints is also presentedPreprin

Video-on-demand optimization using an interior-point algorithm

A Content Delivery Network (CDN) aims to provide efficient movement of massive digital content (multimedia and files) across the Internet. This is achieved by putting the content in servers closer to the costumer. Video-On-Demand service is an application of CDN where videos have to be located strategically to avoid network congestion and servers saturation. Therefore, the problem of optimal placement of videos arises. This problem has a block diagonal structure with linking constraints on links and servers capacities. In this project, we solve huge instances of a video placement problem over three real network topologies with a specialized interior point solver named BlockIP. The evaluated instances range from 7 to 300 millions of variables and the difficulty of the instances depends on the size of servers, links bandwidth and network topology. Our results: 1) verified characteristics of BlockIP like regularization and the intensive computation in the last iterations and 2) showed that BlockIP found optimal solution in all the evaluated instances with a good optimality gap. On the contrary, state-of-art CPLEX cannot reach an optimal, feasible solution in some difficult instances and needs almost twice the memory of BlockIP. However, CPLEX solved most of feasible instances at least twice faster than BlockI

New Preconditioners for KKT Systems of Network Flow Problems

We propose a new set of preconditioners for the iterative solution, via a Preconditioned Conjugate Gradient (PCG) method, of the KKT systems that must be solved at each iteration of an Interior Point (IP) algorithm for the solution of linear Min Cost Flow (MCF) problems. These preconditioners are based on the idea of extracting a proper triangulated subgraph of the original graph which strictly contains a spanning tree. We define a new class of triangulated graphs, called Brother-Connected Trees (BCT), and discuss some fast heuristics for finding BCTs of "large" weight. Computational experience shows that the new preconditioners can complement tree preconditioners, outperforming them both in iterations count and in running time on some classes of graphs