Jacek Gondzio

In this work, we address the exact D-optimal experimental design problem when the number of design vectors is large. First, we propose a customized column generation algorithm to solve the continuous relaxation of the problem. In the approach, each restricted master problem is constructed carefully so that the number of variables stays small and therefore the subproblem can be solved efficiently by a Primal-Dual Interior-Point-based Semidefinite Programming solver. The support of this solution provides a subset of design points to be used in a local search algorithm for the solution of the integer problem. We prove that a local search algorithm restricted to points of this subset provides an exact design that is provably close to the exact D-optimal design. Our numerical experiments show that, for large-scale instances in which the number of regression points exceeds by far the number of experiments, our approach achieves superior performance compared to existing branch-and-bound-based algorithms in both computational efficiency and solution quality.

In this work, we address the exact D-optimal experimental design problem by proposing an efficient algorithm that rapidly identifies the support of its continuous relaxation. Our method leverages a column generation framework to solve such a continuous relaxation, where each restricted master problem is tackled using a Primal-Dual Interior-Point-based Semidefinite Programming solver. This enables fast and reliable detection of the design's support. The identified support is subsequently used to construct a feasible exact design that is provably close to optimal. We show that, for large-scale instances in which the number of regression points exceeds by far the number of experiments, our approach achieves superior performance compared to existing branch-and-bound-based algorithms in both computational efficiency and solution quality.

A regularized version of the primal-dual Interior Point Method (IPM) for the solution of Semidefinite Programming Problems (SDPs) is presented in this paper. Leveraging on the proximal point method, a novel Proximal Stabilized Interior Point Method for SDP (PS-SDP-IPM) is introduced. The method is strongly supported by theoretical results concerning its convergence: the worst-case complexity result is established for the inner regularized infeasible inexact IPM solver. The new method demonstrates an increased robustness when dealing with problems characterized by ill-conditioning or linear dependence of the constraints without requiring any kind of pre-processing. Extensive numerical experience is reported to illustrate advantages of the proposed method when compared to the state-of-the-art solver.

New Exact Approaches Tailored for Kidney Exchange Programs with Hierarchical Objectives Kidney exchange programs increase the rate of living donor kidney transplantation. Whereas effective integer programming models aimed at maximizing the total number of transplants have been proposed in the literature, these cannot always be extended to handle a hierarchy of objectives, which is often a requirement in practice. In “New Algorithms for Hierarchical Optimization in Kidney Exchange Programs,” Delorme, García, Gondzio, Kalcsics, Manlove, and Pettersson introduce a new integer programming framework to solve large-size instances of kidney exchange programs. The authors use an ad hoc preprocessing and a reduced-cost variable fixing algorithm to dramatically decrease the size of the models, and they devise a diving algorithm that exploits the hierarchical structure of the problem to significantly reduce the number of integer programs that need to be solved. They also show that it is possible to transition between models as different layers are traversed in the hierarchy, allowing each layer to be solved with the most effective model. Experiments on three different European kidney exchange programs show that running times can be reduced by up to three orders of magnitude.

Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context, quasi-Newton algorithms compute low-rank updates of the matrix associated with the Newton systems, instead of computing it from scratch at every iteration. In this work, we show that a simplified quasi-Newton primal-dual interior point algorithm for linear programming, which alternates between Newton and quasi-Newton iterations, enjoys polynomial worst-case iteration complexity. Feasible and infeasible cases of the algorithm are considered and the most common neighborhoods of the central path are analyzed. To the best of our knowledge, this is the first attempt to deliver polynomial worst-case iteration complexity bounds for these methods. Unsurprisingly, the worst-case complexity results obtained when quasi-Newton directions are used are worse than their counterparts when Newton directions are employed. However, quasi-Newton updates are very attractive for large-scale optimization problems where the cost of factorizing the matrices is much higher than the cost of solving linear systems.

Embedding randomization procedures in the Alternating Direction Method of Multipliers (ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in practice, the introduction of such techniques could mitigate the diverging behaviour of the multi-block extension of ADMM, from the theoretical point of view, it can ensure just the convergence in expectation, which may not be a good indicator of its robustness and efficiency. In this work, analysing the strongly convex quadratic programming case from a linear algebra perspective, we interpret the block Gauss–Seidel sweep performed by the multi-block ADMM in the context of the inexact Augmented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative technique to those present in the literature which, supported from stronger theoretical guarantees, is able to ensure the convergence of the multi-block generalization of the ADMM method.

Discrete optimal transport problems give rise to very large linear programs (LPs) with a particular structure of the constraint matrix. In this paper, we present a hybrid algorithm that mixes an interior point method (IPM) and column generation, specialized for the LP originating from the Kantorovich optimal transport problem. Knowing that optimal solutions of such problems display a high degree of sparsity, we propose a column generation–like technique to force all intermediate iterates to be as sparse as possible. The algorithm is implemented nearly matrix-free. Indeed, most of the computations avoid forming the huge matrices involved and solve the Newton system using only a much smaller Schur complement of the normal equations. We prove theoretical results about the sparsity pattern of the optimal solution, exploiting the graph structure of the underlying problem. We use these results to mix iterative and direct linear solvers efficiently in a way that avoids producing preconditioners or factorizations with excessive fill-in and at the same time guaranteeing a low number of conjugate gradient iterations. We compare the proposed method with two state-of-the-art solvers and show that it can compete with the best network optimization tools in terms of computational time and memory use. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: F. Zanetti received funding from the University of Edinburgh, in the form of a PhD scholarship. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0184 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0184 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .