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An arc-search interior-point algorithm for nonlinear constrained optimization

COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2025年第3期90卷 969-995页

作者： Yang, Yaguang NASA Goddard Space Flight Ctr 8800 Greenbelt Rd Greenbelt MD 20771 USA

This paper proposes a new arc-search interior-point algorithm for the nonlinear constrained optimization problem. The proposed algorithm uses the second-order derivatives to construct a search arc that approaches the optimizer. Because the arc stays in the interior set longer than any straight line, it is expected that the scheme will generate a better new iterate than a line search method. The computation of the second-order derivatives requires to solve the second linear system of equations, but the coefficient matrix of the second linear system of equations is the same as the first linear system of equations. Therefore, the matrix decomposition obtained while solving the first linear system of equations can be reused. In addition, most elements of the right-hand side vector of the second linear system of equations are already computed when the coefficient matrix is assembled. Therefore, the computation cost for solving the second linear system of equations is insignificant and the benefit of having a better search scheme is well justified. The convergence of the proposed algorithm is established. Some preliminary test results are reported to demonstrate the merit of the proposed algorithm.

关键词： Arc-search interior-point algorithm Nonlinear optimization

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interior-point algorithm for sufficient LCPs based on the technique of algebraically equivalent transformation

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OPTIMIZATION LETTERS 2021年第2期15卷 357-376页

作者： Darvay, Zsolt Illes, Tibor Majoros, Csilla Babes Bolyai Univ Fac Math & Comp Sci 1 Mihail Kogalniceanu St RO-400084 Cluj Napoca Romania Budapest Univ Technol & Econ Dept Differential Equat Budapest Hungary

We present a short-step interior-point algorithm (IPA) for sufficient linear complementarity problems (LCPs) based on a new search direction. An algebraic equivalent transformation (AET) is used on the centrality equation of the central path system and Newton's method is applied on this modified system. This technique was offered by Zsolt Darvay for linear optimization in 2002. Darvay first used the square root function as AET and in 2012 Darvay et al. applied this technique with an other transformation formed by the difference of the identity map and the square root function. We apply the AET technique with the new function to transform the central path equation of the sufficient LCPs. This technique leads to new search directions for the problem. We introduce an IPA with full Newton steps and prove that the iteration bound of the algorithm coincides with the best known one for sufficient LCPs. We present some numerical results to illustrate performance of the proposed IPA on two significantly different sets of test problems and compare it, with related, quite similar variants of IPAs.

关键词： interior-point algorithm Sufficient linear complementarity problem Algebraic equivalent transformation Polynomial complexity

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interior-point algorithm for linear optimization based on a new trigonometric kernel function

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OPERATIONS RESEARCH LETTERS 2015年第5期43卷 471-475页

作者： Li, Xin Zhang, Mingwang China Three Gorges Univ Coll Sci Yi Chang 443002 Peoples R China

In this paper, we present a new primal-dual interior-point algorithm for linear optimization based on a trigonometric kernel function. By simple analysis, we derive the worst case complexity for a large-update primal-dual interior-point method based on this kernel function. This complexity estimate improves a result from El Ghami et al. (2012) and matches the one obtained in Reza Peza Peyghami et al. (2014). (C) 2015 Elsevier B.V. All rights reserved.

关键词： Linear optimization Kernel function interior-point algorithm Large-update Polynomial complexity

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interior-point algorithm FOR SHAKEDOWN ANALYSIS ACCOUNTING FOR LIMITED KINEMATICAL HARDENING

INTERIOR-POINT ALGORITHM FOR SHAKEDOWN ANALYSIS ACCOUNTING F...

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11th International Conference on Computational Plasticity (COMPLAS) - Fundamentals and Applications

作者： Simon, Jaan-W. Weichert, Dieter Rhein Westfal TH Aachen Inst Gen Mech D-52062 Aachen Germany

ISBN: (纸本)9788489925731

We present a numerical method for the computation of shakedown loads of structures under thermo-mechanical loading accounting for limited kinematical hardening. The method is based on the lower bound approach by Melan extended to the hardening using a two-surface model. Both the yield and the bounding surface are defined by the von Mises condition. Melan's shakedown theorem leads to a nonlinear convex optimization problem. This is solved by the interior-point algorithm ipsa recently developed by the authors. In this paper, theoretical and numerical aspects will be presented as well as numerical examples from mechanical engineering.

关键词： Shakedown Analysis Direct Methods interior-point algorithm Limited Kinematical Hardening Nonlinear Programming

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Neuro-Swarm heuristic using interior-point algorithm to solve a third kind of multi-singular nonlinear system

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MATHEMATICAL BIOSCIENCES AND ENGINEERING 2021年第5期18卷 5285-5308页

作者： Sabir, Zulqurnain Raja Zahoor, Muhammad Asif Kamal, Aldawoud Guirao, Juan L. G. Le, Dac-Nhuong Saeed, Tareq Salama, Mohamad Hazara Univ Dept Math & Stat Mansehra Pakistan Natl Yunlin Univ Sci & Technol Future Technol Res Ctr 123 Univ RdSect 3 Touliu 64002 Yunlin Taiwan Mutah Univ Jordan Dept Math & Stat Mutah North Ireland Tech Univ Cartagena Hosp Marina Dept Appl Math & Stat Cartagena 30203 Spain Duy Tan Univ Inst Res & Dev Danang 550000 Vietnam Duy Tan Univ Fac Informat Technol Danang 550000 Vietnam King Abdulaziz Univ Dept Math Fac Sci Nonlinear Anal & Appl Math NAAM Res Grp POB 80203 Jeddah 21589 Saudi Arabia Appl Sci Univ Dept Engn Al Eker Bahrain

The purpose of the present work is to solve a third kind of multi-singular nonlinear system using the neuro-swarm computing solver based on the artificial neural networks (ANNs) optimized with the effectiveness of particle swarm optimization (PSO) maintained by a local search proficiency of interior-point algorithm (IPA), i.e., ANN-PSO-IPA. An objective function is designed using the continuous mapping of ANN for nonlinear multi-singular third order system of Emden-Fowler equations and optimization of fitness function carried out with the integrated strength of PSO-IPA. The motivation to design the ANN-PSO-IPA is to present a feasible, reliable and feasible framework to handle with such complicated nonlinear multi-singular third order system of Emden-Fowler model. The designed ANN-PSO-IPA is tested for three different nonlinear variants of the multi-singular third kind of Emden-Fowler system. The obtained numerical results on single/multiple executions of the designed ANN-PSO-IPA are used to endorse the precision, viability and reliability.

关键词： nonlinear singular system artificial neural networks multi-singular interior-point algorithm statistical analysis hybrid approach

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An adaptive updating full-Newton step interior-point algorithm with modified Newton direction

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APPLIED NUMERICAL MATHEMATICS 2015年 91卷 98-106页

作者： Xu, Yinghong Zhang, Lipu Zhang, Jing Luo, Hua Zhejiang Sci Tech Univ Dept Math Hangzhou 310018 Zhejiang Peoples R China Zhejiang A&F Univ Dept Math Hangzhou 311300 Zhejiang Peoples R China

We adopt the self-adaptive strategy to update the barrier parameter of a feasible primal-dual interior-point algorithm. We obtain two adaptive updating methods, namely, cheap updates and sharp updates. We compare the effectiveness of the short updates with the adaptive update methods on some benchmark problems. The numerical results show that the sharp updates method is superior to short updates and cheap updates methods. (C) 2015 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Modified Newton-direction interior-point algorithm Adaptive update

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A primal-dual interior-point algorithm with arc-search for semidefinite programming

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OPTIMIZATION LETTERS 2019年第5期13卷 1157-1175页

作者： Zhang, Mingwang Yuan, Beibei Zhou, Yiyuan Luo, Xiaoyu Huang, Zhengwei China Three Gorges Univ Coll Sci Yichang 443002 Peoples R China China Three Gorges Univ Coll Econ & Management Yichang 443002 Peoples R China

We propose a wide neighborhood primal-dual interior-point algorithm with arc-search for semidefinite programming. In every iteration, the algorithm constructs an ellipse and searches an epsilon-approximate solution of the problem along the ellipsoidal approximation of the central path. Assuming a strictly feasible starting point is available, we show that the algorithm has the iteration complexity bound O (n(3/4) log X-0.S-0/epsilon) for the Nesterov-Todd direction, which is similar to that of the corresponding algorithm for linear programming. The numerical results show that our algorithm is efficient and promising.

关键词： Semidefinite programming Arc-search interior-point algorithm Polynomial complexity

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A full-step interior-point algorithm for second-order cone optimization based on a simple locally kernel function

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OPTIMIZATION METHODS & SOFTWARE 2013年第3期28卷 619-639页

作者： Zhang, Lipu Bai, Yanqin Xu, Yinghong Zhejiang A&F Univ Dept Math Hangzhou 311300 Zhejiang Peoples R China Shanghai Univ Dept Math Shanghai 200444 Peoples R China Zhejiang Sci Tech Univ Dept Math Hangzhou 310018 Zhejiang Peoples R China

In this paper, we investigate the properties of a simple locally kernel function. As an application, we present a full-step interior-point algorithm for second-order cone optimization (SOCO). The algorithm uses the simple locally kernel function to determine the search direction and define the neighbourhood of central path. The full step used in the algorithm has local quadratic convergence property according to the proximity function which is constructed by the simple locally kernel function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bound for SOCO.

关键词： second-order cone optimization interior-point algorithm complexity analysis locally kernel function 17C99 90C25 90C51

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An interior-point algorithm for the minimization arising from 3D contact problems with friction

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OPTIMIZATION METHODS & SOFTWARE 2013年第6期28卷 1195-1217页

作者： Kucera, R. Machalova, J. Netuka, H. Zencak, P. VSB TU Ostrava Ctr Excellence IT4I Ostrava 70833 Czech Republic Palacky Univ Fac Sci Dept Math Anal & Applicat Math Olomouc 77146 Czech Republic

The paper deals with a variant of the interior-point method for the minimization of strictly quadratic objective function subject to simple bounds and separable quadratic inequality constraints. Such minimizations arise from the finite element approximation of contact problems of linear elasticity with friction in three space dimensions. The main goal of the paper is the convergence analysis of the algorithm and its implementation. The optimal preconditioners for solving ill-conditioned inner linear systems are proposed. Numerical experiments illustrate the computational efficiency for large-scale problems.

关键词： interior-point algorithm convergence preconditioners contact problems friction 65K05 90C51 74M10

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A note on the local convergence of a predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem based on the Alizadeh-Haeberly-Overton search direction

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SIAM JOURNAL ON OPTIMIZATION 2005年第4期15卷 1147-1154页

作者： Lu, ZS Monteiro, RDC Georgia Tech Sch ISyE Atlanta GA 30332 USA

This note points out an error in the local quadratic convergence proof of the predictor-corrector interior-point algorithm for solving the semidefinite linear complementarity problem based on the Alizadeh-Haeberly-Overton search direction presented in [M. Kojima, M. Shida, and S. Shindoh, SIAM J. Optim., 9 (1999), pp. 444-465]. Their algorithm is slightly modified and the local quadratic convergence of the resulting method is established.

关键词： semidefinite linear complementarity problem semidefinite programming interior-point algorithm predictor-corrector algorithm local quadratic convergence

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