arXiv subjects

Jacek Gondzio

Publications and source records attributed to Jacek Gondzio.

At least 19 recordsLinked to original sources

arXiv2026

Block coordinate descent for joint delay-energy optimization in multi-hop D2D networks

In multi-hop device-to-device (D2D) networks, the optimization of network-level metrics is particularly difficult due to the tight coupling between network-layer routing and physical-layer resource allocation. Departing from traditional average-performance metrics, this paper addresses the joint optimization of routing paths, transmission power, and bandwidth allocation. We formulate a generalized cost function to minimize the maximum transmission time (i.e., the bottleneck delay) alongside the total energy consumption. To tackle the resulting highly non-convex formulation, we propose a novel block coordinate descent (BCD) framework. At the network layer, we develop two adaptive routing algorithms: a matrix-free Frank-Wolfe (MF-FW) algorithm for fast execution in dense topologies, and a low-rank primal-dual interior-point method (LR-PDIPM) that bypasses dense matrix inversions via the Sherman-Morrison formula for high-precision solutions. At the physical layer, we design a parallel dual ascent algorithm leveraging a time-domain perspective transformation to solve the resource allocation subproblem to global optimality. The proposed BCD framework is proven to converge to an ε-neighborhood of a stationary point. Through comprehensive experiments, the proposed BCD framework establishes its superiority in achieving the optimal delay-energy trade-off. Specifically, the LR-PDIPM variant achieves a maximum 9.14-fold reduction in total energy consumption and up to an order of magnitude improvement in energy efficiency, while maintaining a bounded maximum delay gap (up to 3.78-fold) relative to the best baseline. Meanwhile, the warm-start MF-FW variant identifies near-optimal solutions in mere seconds, serving as a highly practical engineering approach.

math.OC↗

arXiv2026

A column generation approach to exact experimental design

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.

math.OC↗

arXiv2025

A factorisation-based regularised interior point method using the augmented system

This paper describes the implementation of a new interior point solver for linear programming for the open-source optimization library HiGHS. The solver uses a direct factorisation to solve the Newton systems, choosing the best approach between the normal equations and augmented system. Details of the multifrontal factorisation routine are given, with attention to the features that allow to achieve high performance, like storage formats, use of efficient dense linear algebra subroutines and parallelism. The paper also describes the use of pivoting and regularisation strategies to ensure that a stable factorisation is obtained, despite the ill-conditioning of the matrices. Results on three different collections of problems are presented which highlight the improved performance of the solver compared to the existing HiGHS interior point method.

math.OC↗

arXiv2024

An efficient active-set method with applications to sparse approximations and risk minimization

In this paper we present an efficient active-set method for the solution of convex quadratic programming problems with general piecewise-linear terms in the objective, with applications to sparse approximations and risk-minimization. The algorithm is derived by combining a proximal method of multipliers (PMM) with a standard semismooth Newton method (SSN), and is shown to be globally convergent under minimal assumptions. Further local linear (and potentially superlinear) convergence is shown under standard additional conditions. The major computational bottleneck of the proposed approach arises from the solution of the associated SSN linear systems. These are solved using a Krylov-subspace method, accelerated by certain novel general-purpose preconditioners which are shown to be optimal with respect to the proximal penalty parameters. The preconditioners are easy to store and invert, since they exploit the structure of the nonsmooth terms appearing in the problem's objective to significantly reduce their memory requirements. We showcase the efficiency, robustness, and scalability of the proposed solver on a variety of problems arising in risk-averse portfolio selection, $L^1$-regularized partial differential equation constrained optimization, quantile regression, and binary classification via linear support vector machines. We provide computational evidence, on real-world datasets, to demonstrate the ability of the solver to efficiently and competitively handle a diverse set of medium- and large-scale optimization instances.

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arXiv2023

Inner product regularized multi-energy X-ray tomography for material decomposition

Multi-energy X-ray tomography is studied for decomposing three materials using three X-ray energies and a classical energy-integrating detector. A novel regularization term comprises inner products between the material distribution functions, penalizing any overlap of different materials. The method is tested on real data measured of a phantom embedded with Na$_2$SeO$_3$, Na$_2$SeO$_4$, and elemental selenium. It is found that the two-dimensional distributions of selenium in different oxidation states can be mapped and distinguished from each other with the new algorithm. The results have applications in material science, chemistry, biology and medicine.

cond-mat.mtrl-sci↗

arXiv2023

A regularized Interior Point Method for sparse Optimal Transport on Graphs

In this work, the authors address the Optimal Transport (OT) problem on graphs using a proximal stabilized Interior Point Method (IPM). In particular, strongly leveraging on the induced primal-dual regularization, the authors propose to solve large scale OT problems on sparse graphs using a bespoke IPM algorithm able to suitably exploit primal-dual regularization in order to enforce scalability. Indeed, the authors prove that the introduction of the regularization allows to use sparsified versions of the normal Newton equations to inexpensively generate IPM search directions. A detailed theoretical analysis is carried out showing the polynomial convergence of the inner algorithm in the proposed computational framework. Moreover, the presented numerical results showcase the efficiency and robustness of the proposed approach when compared to network simplex solvers.

math.OC↗

arXiv2023

An active-set method for sparse approximations. Part I: Separable $\ell_1$ terms

In this paper we present an active-set method for the solution of $\ell_1$-regularized convex quadratic optimization problems. It is derived by combining a proximal method of multipliers (PMM) strategy with a standard semismooth Newton method (SSN). The resulting linear systems are solved using a Krylov-subspace method, accelerated by certain general-purpose preconditioners which are shown to be optimal with respect to the proximal parameters. Practical efficiency is further improved by warm-starting the algorithm using a proximal alternating direction method of multipliers. We show that the outer PMM achieves global convergence under mere feasibility assumptions. Under additional standard assumptions, the PMM scheme achieves global linear and local superlinear convergence. The SSN scheme is locally superlinearly convergent, assuming that its associated linear systems are solved accurately enough, and globally convergent under certain additional regularity assumptions. We provide numerical evidence to demonstrate the effectiveness of the approach by comparing it against OSQP and IP-PMM (an ADMM and a regularized IPM solver, respectively) on several elastic-net linear regression and $L^1$-regularized PDE-constrained optimization problems.

math.OC↗

arXiv2023

An active-set method for sparse approximations. Part II: General piecewise-linear terms

In this paper we present an efficient active-set method for the solution of convex quadratic programming problems with general piecewise-linear terms in the objective, with applications to sparse approximations and risk-minimization. The method exploits the structure of the piecewise-linear terms appearing in the objective in order to significantly reduce its memory requirements, and thus improve its efficiency. We showcase the robustness of the proposed solver on a variety of problems arising in risk-averse portfolio selection, quantile regression, and binary classification via linear support vector machines. We provide computational evidence to demonstrate, on real-world datasets, the ability of the solver of efficiently handling a variety of problems, by comparing it against an efficient general-purpose interior point solver as well as a state-of-the-art alternating direction method of multipliers. This work complements the accompanying paper [``An active-set method for sparse approximations. Part I: Separable $\ell_1$ terms", S. Pougkakiotis, J. Gondzio, D. S. Kalogerias], in which we discuss the case of separable $\ell_1$ terms, analyze the convergence, and propose general-purpose preconditioning strategies for the solution of its associated linear systems.

math.OC↗

arXiv2022

An Interior-Point-Inspired algorithm for Linear Programs arising in Discrete Optimal Transport

Discrete Optimal Transport problems give rise to very large linear programs (LP) 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 usage. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method.

math.OC↗

arXiv2022

A new stopping criterion for Krylov solvers applied in Interior Point Methods

When an iterative method is applied to solve the linear equation system in interior point methods (IPMs), the attention is usually placed on accelerating their convergence by designing appropriate preconditioners, but the linear solver is applied as a black box with a standard termination criterion which asks for a sufficient reduction of the residual in the linear system. Such an approach often leads to an unnecessary 'oversolving' of linear equations. In this paper, an IPM that relies on an inner termination criterion not based on the residual of the linear system is introduced and analyzed. Moreover, new indicators for the early termination of the inner iterations are derived from a deep understanding of IPM needs. The new technique has been adapted to the Conjugate Gradient (CG) and to the Minimum Residual method (MINRES) applied in the IPM context. The new criterion has been tested on a set of quadratic optimization problems including compressed sensing, image processing and instances with partial differential equation constraints, and it has been compared to standard residual tests with variable tolerance. Evidence gathered from these computational experiments shows that the new technique delivers significant improvements in terms of inner (linear) iterations and those translate into significant savings of the IPM solution time.

math.OC↗

arXiv2022

Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming

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 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.

math.OC↗

arXiv2022

Proximal stabilized Interior Point Methods for quadratic programming and low-frequency-updates preconditioning techniques

In this work, in the context of Linear and Quadratic Programming, we interpret Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is strongly supported by theoretical results concerning convergence and the rate of convergence, and can handle degenerate problems. Moreover, in the second part of this work, we analyse the interactions between the regularization parameters and the computational foot-print of the linear algebra routines used to solve the Newton linear systems. In particular, when these systems are solved using an iterative Krylov method, we propose general purpose preconditioners which, exploiting the regularization and a new rearrangement of the Schur complement, remain attractive for a series of subsequent IPM iterations. Therefore they need to be recomputed only in a fraction of the total IPM iterations. The resulting regularized second order methods, for which low-frequency-updates of the preconditioners are allowed, pave the path for an alternative third way in-between first and second order methods.

math.OC↗

arXiv2022

General-purpose preconditioning for regularized interior point methods

In this paper we present general-purpose preconditioners for regularized augmented systems arising from optimization problems, and their corresponding normal equations. We discuss positive definite preconditioners, suitable for CG and MINRES. We consider "sparsifications" which avoid situations in which eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems arising from the application of regularized interior point methods to linear or nonlinear convex programming problems.

math.OC↗

arXiv2021

Sparse Approximations with Interior Point Methods

Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well conditioned problems. In this paper, specialized variants of an interior point-proximal method of multipliers are proposed and analyzed for problems of this class. Computational experience on a variety of problems, namely, multi-period portfolio optimization, classification of data coming from functional Magnetic Resonance Imaging, restoration of images corrupted by Poisson noise, and classification via regularized logistic regression, provides substantial evidence that interior point methods, equipped with suitable linear algebra, can offer a noticeable advantage over first-order approaches.

math.OC↗

arXiv2021

An Interior Point-Proximal Method of Multipliers for Positive Semi-Definite Programming

In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in [An Interior Point-Proximal Method of Multipliers for Convex Quadratic Programming, Computational Optimization and Applications, 78, 307--351 (2021)] for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.

math.OC↗

arXiv2021

Random multi-block ADMM: an ALM based view for the QP case

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 \textit{mitigate} the diverging behaviour of the multi-block extension of ADMM, from the theoretical point of view, it can ensure just the \textit{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, 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 literature which, supported from stronger theoretical guarantees, is able to ensure the convergence of the multi-block generalization of the ADMM method.

math.OC↗

arXiv2021

Material-separating regularizer for multi-energy X-ray tomography

Dual-energy X-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two X-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, Joint Total Variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.

math.NA↗

arXiv2021

A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion

A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems.

math.NA↗