Further development of multiple centrality correctors for interior point methods

Colombo, Marco; Gondzio, Jacek

doi:10.1007/s10589-007-9106-0

Cited by 58 publications

(71 citation statements)

References 17 publications

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“…We observe that the bounds on the eigenvalues with the largest possible absolute value λ −n and λ m given by inequalities (24) and (27), respectively, are similar to those obtained for the original system (2) and given by (10) and (13), respectively. On the other hand, the bounds on small eigenvalues have been changed significantly by the presence of primal and dual regularization.…”

Section: Regularized Kkt Systemsupporting

confidence: 71%

“…The matrix-free interior point method has been implemented in HOPDM [2,10]. We use an infeasible primal-dual path following algorithm in which the Newton search directions are computed from the regularized system (19).…”

Section: Implementation and Numerical Resultsmentioning

confidence: 99%

“…The primal-dual Newton direction is computed using multiple centrality correctors [10]. The use of centrality correctors is not an obvious choice in the implementation which relies on iterative methods to solve the linear equations because the effort to compute the preconditioner is relatively small compared to the one needed for every extra backsolve.…”

Section: End-whilementioning

confidence: 99%

“…Their practical performance is much better than that guaranteed by the worst-case complexity analysis. Indeed, they usually converge in a number of iterations which is proportional to log n [3,10]. Even with such a small number of iterations these methods can occasionally be computationally expensive.…”

Section: Introductionmentioning

confidence: 99%

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Matrix-free interior point method

Gondzio

2010

Comput Optim Appl

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In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.

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Section: Regularized Kkt Systemsupporting

confidence: 71%

Section: Implementation and Numerical Resultsmentioning

confidence: 99%

Section: End-whilementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Matrix-free interior point method

Gondzio

2010

Comput Optim Appl

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“…As τ ranges over (0, ∞), the unique (if it exists) 2 solution (x, y, s) to this system traces out the primal-dual "central path". Newton-type steps for system (3), which are well defined, e.g., when X and S are positive definite and A has full rank, are obtained by solving one or more linear systems of the form…”

Section: Introductionmentioning

confidence: 99%

A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm

et al. 2011

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Consider linear programs in dual standard form with n constraints and m variables. When typical interior-point algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires O(nm 2 ) operations per iteration. When n m it is often the case that at each iteration most of the constraints are not very relevant for the construction of a good update and could be ignored to achieve computational savings. This idea was considered in the 1990s by Dantzig and Ye, Tone, Kaliski and Ye, den Hertog et al. and others. More recently, Tits et al. proposed a simple "constraint-reduction" scheme and proved global and local quadratic convergence for a dual-feasible primal-dual affine-scaling method modified according to that scheme. In the present work, similar convergence results are proved for a dual-feasible constraint-reduced variant of Mehrotra's predictor-corrector algorithm, under less restrictive nondegeneracy assumptions. These stronger results extend to primal-dual affine scaling as a limiting case. Promising numerical results are reported. As a special case, our analysis applies to standard (unreduced) primal-dual affine scaling. While we do not prove polynomial complexity, our algorithm allows for much larger steps than in previous convergence analyses of such algorithms.

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A Novel Distributed Dynamic Economic Dispatch Based on Dual ADMM and IPM

Yang

et al. 2020

IEEJ Transactions Elec Engng

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This paper presents a novel distributed method for dynamic economic dispatch (DED) problems with cubic fuel cost functions based on the dual alternating direction method of multipliers (D-ADMM) and interior point method (IPM). This scheme enables us to combine the excellent parallelism of ADMM with the superior convergence properties of the IPM. The idea adopted by the proposed algorithm is that the outer-loop uses ADMM decoupling, and the inner-loop uses IPM to solve the cubic polynomial optimization subproblems. When the unit set is divided into multiple partitions, D-ADMM also has the nice central processing unit (CPU) runtime as the well as the number of iterations. In addition, we use a Decoupling-Decomposition-Backtracking and parallel improved multiple centrality corrections decoupling interior point method (DDB-PIMCCD) to solve the aforementioned larger subproblems more efficiently. Finally, the numerical results, in both serial and parallel modes, on a set of 22 DED cases with the range of units from 8 to 1000 units show that the proposed method is very promising for large scale distributed DED problems. Because it can keep the unit information about each subset in secret to suit the market environment and can obtain high-quality solutions with reasonable time.

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Further development of multiple centrality correctors for interior point methods

Cited by 58 publications

References 17 publications

Matrix-free interior point method

Matrix-free interior point method

A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm

A Novel Distributed Dynamic Economic Dispatch Based on Dual ADMM and IPM

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