Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

Ek, David; Forsgren, Anders

doi:10.1007/s10589-021-00265-8

Cited by 3 publications

(4 citation statements)

References 32 publications

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“…Lemma 2. There exists a starting point (x 0 , y 0 , s 0 ) for which, when solving an LP or a QP, the exact direction satisfies (14) with M = O(n 2 ).…”

Section: Algorithm Ipm-i Interior Point Methods With Early Stopping O...mentioning

confidence: 99%

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

Zanetti¹,

Gondzio²

2021

Preprint

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A surprising result is presented in this paper with possible far reaching consequences for any optimization technique which relies on Krylov subspace methods employed to solve the underlying linear equation systems. In this paper the advantages of the new technique are illustrated in the context of Interior Point Methods (IPMs). When an iterative method is applied to solve the linear equation system in IPMs, the attention is usually placed on accelerating their convergence by designing appropriate preconditioners, but the linear solver is applied as a black box solver 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 a new specialized termination criterion for Krylov methods used in IPMs is designed. It is derived from a deep understanding of IPM needs and is demonstrated to preserve the polynomial worst-case complexity of these methods. The new criterion 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 linear and quadratic optimization problems including compressed sensing, image processing and instances with partial differential equation constraints. 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.

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“…Lemma 2. There exists a starting point (x 0 , y 0 , s 0 ) for which, when solving an LP or a QP, the exact direction satisfies (14) with M = O(n 2 ).…”

Section: Algorithm Ipm-i Interior Point Methods With Early Stopping O...mentioning

confidence: 99%

“…[13,29,30]) and in particular on the effect that an inexact linear solver has on the convergence properties of IPMs ( [6,11,24,28]). Many other improvements have been made regarding predictor-correctors strategies [12,21], regularization strategies [3,19,36] or the use of quasi-Newton approaches [14,15,26].…”

Section: Introductionmentioning

confidence: 99%

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

Zanetti¹,

Gondzio²

2021

Preprint

View full text Add to dashboard Cite

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“…It was first used to solve unconstrained optimization problems and then gradually applied to optimization problems with constraints. It is an important research direction in the field of optimization in recent years [17]. The traditional line search method often fails in solving ill-posed problems because the selected step size is too large.…”

Section: Trust Region Algorithmmentioning

confidence: 99%

“…Solve an approximate problem of the original objective function in the trust region and obtain the solution as a trial step. This approximate problem is often called a subproblem [18]. Generally, the trust region subproblem of unconstrained optimization problem can be…”

Section: Trust Region Algorithmmentioning

confidence: 99%

New Solutions of Nonlinear Equations Constrained by Bounded Variables under Edge Computation

Wang

2022

Wireless Communications and Mobile Computing

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Aiming at the problem of low global convergence and local convergence rate of trust region interior points of bounded variable constrained nonlinear equations, a trust region interior point algorithm for bounded variable constrained nonlinear equations under edge calculation is designed. By constructing the basic function form of nonlinear equations constrained by bounded variables, the boundary of nonlinear equations is determined by Gauss Newton iterative process to ensure the global convergence of changes; solve the original objective function, analyze the trust region subproblem of the unconstrained optimization problem, and generate an acceptable region. The region is generated through two-dimensional example interpretation. The interior points in the acceptable region are determined by the primal dual interior point method, and the interior points in the acceptable region are optimized. With the help of edge calculation, the trust region interior point programming model of bounded variable constrained nonlinear equations is designed to realize the algorithm design. The experimental results show that the designed algorithm can improve the trust region interior point global convergence and local convergence rate of nonlinear equations with bounded variable constraints.

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A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

Forsgren

2023

Comput Optim Appl

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The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions. We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

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Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

Cited by 3 publications

References 32 publications

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

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

New Solutions of Nonlinear Equations Constrained by Bounded Variables under Edge Computation

A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

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