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

Pougkakiotis, Spyridon; Gondzio, Jacek

doi:10.1007/s10957-021-01954-4

Cited by 10 publications

(18 citation statements)

References 28 publications

(100 reference statements)

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“…Let us now focus on the case of the regularized saddle point systems (and their respective normal equations) arising from the application of regularized IPMs on convex programming problems. The MATLAB code, which is based on the IP-PMM presented in [40,41], can be found on GitHub. 1 In all the presented experiments a 6-digit accurate solution is requested.…”

Section: Regularized Ipms: Numerical Resultsmentioning

confidence: 99%

“…For the rest of this section, we assume that = ( ) = ( ) . This assumption is based on the developments in [40,41], where a polynomially convergent regularized IPM is derived for convex quadratic and linear positive semi-definite programming problems, respectively. Following [7], we could precondition the matrix M using the following matrix:…”

Section: Let Us Initially Focus On Linear Programming (Lp) Problems O...mentioning

confidence: 99%

“…Then, we test the preconditioners on examples of Partial Differential Equation (PDE) optimal control problems. All preconditioning approaches have been implemented within an Interior Point-Proximal Method of Multipliers (IP-PMM) framework, which is a polynomially convergent primal-dual regularized IPM, based on the developments in [40,41]. A robust implementation is provided.…”

Section: Introductionmentioning

confidence: 99%

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General-purpose preconditioning for regularized interior point methods

2022

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In this paper we present general-purpose preconditioners for regularized augmented systems, and their corresponding normal equations, arising from optimization problems. 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.

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Section: Regularized Ipms: Numerical Resultsmentioning

confidence: 99%

Section: Let Us Initially Focus On Linear Programming (Lp) Problems O...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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General-purpose preconditioning for regularized interior point methods

2022

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“…The optimal PSO presentations are applied as a primary input in the IPA. Recently, IPA is used in the phase-field approach to brittle and ductile fracture 56 , power system observability 57 , multipliers for linear positive semi-definite programming 58 , simulation and optimization of dynamic flux balance analysis models 59 , fourth order singular systems 60 , multistage nonlinear nonconvex programs 61 and monotone weighted linear complementarity problems 62 .…”

Section: Methodsmentioning

confidence: 99%

A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics

Sabir

Saïd

Al‐Mdallal

et al. 2022

Sci Rep

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The current work provides a mathematical second order perturbed singular delay differential model (SO-PSDDM) by using the standard form of the Lane-Emden model. The inclusive structures based on the delay terms, singular-point and perturbation factor and shape forms of the SO-PSDDM are provided. The novel form of the SO-PSDDM is numerically solved by using the procedures of artificial neural networks (ANNs) along with the optimization measures based on the swarming procedures (PSO) and interior-point algorithm (IPA). An error function is optimized through the swarming PSO procedure along with the IPA to solve the SO-PSDDM. The precision, substantiation and validation are observed for three problems of the SO-PSDDM. The exactness of the novel SO-PSDDM is observed by comparing the obtained and exact solutions. The reliability, stability and convergence of the proposed stochastic algorithms are observed for 30 independent trials to solve the novel SO-PSDDM.

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“…This scheme was then utilized for the solution of linear programming problems in [35], and for lassoregularized problems in [34]. A similar primal-dual approach for ℓ 1 -regularized convex quadratic programming problems was developed and analyzed in our accompanying paper [39] and was shown to be especially efficient for the solution of elastic-net linear regression and L 1 -regularized partial differential equation constrained optimization problems. In fact, the proposed active set method developed in this work is a direct extension of the method given in [39], altered in a specific way so that it can efficiently handle most piecewise-linear terms that appear in practice, via restricting its memory requirements.Indeed, we showcase that each of the nonsmooth terms in the objective of (P) can be utilized for reducing the memory requirements of the proposed method.…”

mentioning

confidence: 99%

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

Pougkakiotis¹,

Gondzio²,

Kalogerias³

2023

Preprint

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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 ℓ1 terms", S. Pougkakiotis, J. Gondzio, D. S. Kalogerias], in which we discuss the case of separable ℓ1 terms, analyze the convergence, and propose general-purpose preconditioning strategies for the solution of its associated linear systems.Finally, it is important to note that model (P) allows for multiple piecewise-linear terms of the form max{Cx + d,since we can always adjust l to account for more than one terms. Hence, one can observe that (P) is quite general and can be used to model a plethora of very important problems that arise in practice.In light of the discussion in Remark 1, it is easily observed that problem (P) can model a plethora of very important problems arising in several application domains spanning, among others, operational research, machine learning, data science, and engineering. More specifically, various lasso and fussed lasso instances (with applications to sparse approximations for classification and regression [19,54], portfolio allocation [1], or medical diagnosis [23], among many others) can be readily modeled by (P). Additionally, various risk-minimization problems with linear random cost functions can be modeled by (P) (e.g. see [33,44,48]). Furthermore, even risk-minimization problems with nonlinear random cost functions, which are typically solved via Gauss-Newton schemes (e.g. see [10]), often require the solution of sub-problems like (P). Finally, continuous relaxations of integer programming problems with applications to operational research (e.g. [32]) often take the form of (P). Given the multitude of problems requiring easy access to (usually accurate) solutions of (P), the derivation of efficient, robust, and scalable solution methods is of paramount importance.Problem (P) can be solved by various first-or second-order methods. In particular, using a standard reformulation, by introducing several auxiliary variables, (P) can be written as a convex quadratic programming (QP) one and efficiently solved by, among others, an interior point method (IPM; e.g. [19,38]),...

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

Cited by 10 publications

References 28 publications

General-purpose preconditioning for regularized interior point methods

General-purpose preconditioning for regularized interior point methods

A neuro swarm procedure to solve the novel second order perturbed delay Lane-Emden model arising in astrophysics

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

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