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Kernel-Based Full-Newton Step Feasible interior-point algorithm for P∗(κ)-Weighted Linear Complementarity Problem

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期202卷 108-132页

作者： Chi, Xiaoni Wang, Guoqiang Lesaja, Goran Guilin Univ Elect Technol Guangxi Coll Sch Math & Comp Sci Guilin 541004 Guangxi Peoples R China Guilin Univ Elect Technol Univ Key Lab Data Anal & Computat Guilin 541004 Guangxi Peoples R China Shanghai Univ Engn Sci Sch Math Phys & Stat Shanghai 201620 Peoples R China Georgia Southern Univ Dept Math Sci Statesboro GA 30458 USA

In this paper, we consider a kernel-based full-Newton step feasible interior-point method (IPM) for P & lowast;(K)-Weighted Linear Complementarity Problem (WLCP). The specific eligible kernel function is used to define an equivalent form of the central path, the proximity measure, and to obtain search directions. Full-Newton steps are adopted to avoid the line search at each iteration. It is shown that with appropriate choices of the parameters, and a certain condition on the starting point, the iterations always lie in the defined neighborhood of the central path. Assuming strict feasibility of P & lowast;(K)-WLCP, it is shown that the IPM converges to the epsilon-approximate solution of P & lowast;(K)-WLCP in a polynomial number of iterations. Few numerical results are provided to indicate the computational performance of the algorithm.

关键词： P*(kappa)-weighted linear complementarity problem interior-point algorithm Full-Newton step Polynomial complexity

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

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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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A path-following interior-point algorithm for monotone LCP based on a modified Newton search direction

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RAIRO-OPERATIONS RESEARCH 2023年第3期57卷 1059-1073页

作者： Grimes, Welid Achache, Mohamed Univ Ferhat Abbas Setif 1 Fac Sci Dept Math Lab Fundamental & Numer Math Setif 19000 Algeria

In this paper, we propose a short-step feasible full-Newton step path-following interior-point algorithm (IPA) for monotone linear complementarity problems (LCPs). The proposed IPA uses the technique of algebraic equivalent transformation (AET) induced by an univariate function to transform the centering equations which defines the central-path. By applying Newton's method to the modified system of the central-path of LCP, a new Newton search direction is obtained. Under new appropriate defaults of the threshold t which defines the size of the neighborhood of the central-path and of ? which determines the decrease in the barrier parameter, we prove that the IPA is well-defined and converges locally quadratically to a solution of the monotone LCPs. Moreover, we derive its iteration bound, namely, O(vnlogn/c) which coincides with the best-known iteration bound for such algorithms. Finally, some numerical results are presented to show its efficiency.

关键词： Monotone linear complementarity problems interior-point algorithm Algebraic equivalent transformation Full-Newton step Polynomial complexity

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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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A Full-Newton Step Feasible interior-point algorithm for the Special Weighted Linear Complementarity Problems Based on a Kernel Function

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Wuhan University Journal of Natural Sciences 2024年第1期29卷 29-37页

作者： GENG Jie ZHANG Mingwang ZHU Dechun Mathematics Department Anhui Institute of Information TechnologyWuhu 241000AnhuiChina College of Science China Three Gorges UniversityYichang 443002HubeiChina

In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and *** algorithm employs a kernel function with a linear growth term to derive the search direction,and by introducing new technical results and selecting suitable parameters,we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point ***,numerical results illustrate the efficiency of the proposed method.

关键词： interior-point algorithm weighted linear complementarity problem full-Newton step kernel function iteration complexity

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Patch loading resistance prediction of steel plate girders using a deep artificial neural network and an interior-point algorithm

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STEEL AND COMPOSITE STRUCTURES 2022年第2期45卷 159-173页

作者： Mai, Sy Hung Tran, Viet-Linh Nguyen, Duy-Duan Nguyen, Viet Tiep Thai, Duc-Kien Hanoi Univ Civil Engn HUCE Dept Hydraul Engn & Construct Hanoi Vietnam Vinh Univ Dept Civil Engn Vinh 461010 Vietnam Sejong Univ Dept Civil & Environm Engn 98 Gunja Dong Seoul South Korea

This paper proposes a hybrid machine-learning model, which is called DANN-IP, that combines a deep artificial neural network (DANN) and an interior-point (IP) algorithm in order to improve the prediction capacity on the patch loading resistance of steel plate girders. For this purpose, 394 steel plate girders that were subjected to patch loading were tested in order to construct the DANN-IP model. Firstly, several DANN models were developed in order to establish the relationship between the patch loading resistance and the web panel length, the web height, the web thickness, the flange width, the flange thickness, the applied load length, the web yield strength, and the flange yield strength of steel plate girders. Accordingly, the best DANN model was chosen based on three performance indices, which included the R(boolean AND)2, RMSE, and a20-index. The IP algorithm was then adopted to optimize the weights and biases of the DANN model in order to establish the hybrid DANN-IP model. The results obtained from the proposed DANN-IP model were compared with of the results from the DANN model and the existing empirical formulas. The comparison showed that the proposed DANN-IP model achieved the best accuracy with an R(boolean AND)2 of 0.996, an RMSE of 23.260 kN, and an a20-index of 0.891. Finally, a Graphical User Interface (GUI) tool was developed in order to effectively use the proposed DANN-IP model for practical applications.

关键词： artificial neural network interior-point algorithm machine learning patch loading resistance steel plate girder

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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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Second-order cone programming for frictional contact mechanics using interior point algorithm

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OPTIMIZATION METHODS & SOFTWARE 2024年第3期39卷 634-663页

作者： Acary, Vincent Armand, Paul Nguyen, Hoang Minh Shpakovych, Maksym Univ Grenoble Alpes Inst Engn Inria CNRSGrenoble INP Grenoble France Univ Limoges Lab XLIM Limoges France Lab XLIM 123 Ave Albert Thomas F-87060 Limoges France

We report experiments of an implementation of a primal-dual interior point algorithm for solving mechanical models of one-sided contact problems with Coulomb friction. The objective is to recover an optimal solution with high accuracy and as quickly as possible. These developments are part of the design of Siconos, an open-source software for modelling and simulating non-smooth dynamical systems. Currently, Siconos uses mainly first-order methods for the numerical solution of these systems. These methods are very robust, but suffer from a linear rate of convergence and are therefore too slow to find accurate solutions in a reasonable time. Our main objective is to apply second-order methods to speed up convergence. In this paper, we focus on solving a relaxed formulation of the initial mechanical model, corresponding to the optimality conditions of a convex quadratic minimization problem with second-order cone constraints. We will present in detail a primal-dual interior point algorithm for solving this type of problem. The main difficulty in implementing this algorithm arises from the fact that at each iteration of the algorithm, a change of variable, called a scaling, has to be performed in order to guarantee the non-singularity of the linear system to be solved, as well as to recover a symmetric system. Although this scaling strategy is very nice from a theoretical point of view, it leads to an enormous deterioration of the conditioning of the linear system when approaching the optimal solution and therefore to all the numerical difficulties that result from it. We will describe in detail the numerical algebra that we have developed in our implementation, in order to overcome these problems of numerical instability. We will also present the solution of the models resulting from the problems of rolling friction, for which the cone of constraints is no longer self-dual like the Lorentz cone.

关键词： Second-order cone optimization interior-point algorithm contact mechanics Coulomb friction

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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 wide neighborhood interior-point algorithm with arc-search for P*(κ) linear complementarity problem

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APPLIED NUMERICAL MATHEMATICS 2019年 136卷 293-304页

作者： Yuan, Beibei Zhang, Mingwang Huang, Zhengwei China Three Gorges Univ Coll Sci Yi Chang 443002 Peoples R China China Three Gorges Univ Coll Econ & Management Yi Chang 443002 Peoples R China

We propose a wide neighborhood interior-point algorithm with arc-search for P-*(kappa) linear complementarity problem (LCP). Along the ellipsoidal approximation of the central path, the algorithm searches optimizers in every iteration. Assuming a strictly starting point is available, we show that the algorithm has O((1 + 2 kappa)(1 + 18 kappa)(2)n log (x(0))(T)s(0)/epsilon) iteration complexity. It matches the currently best known iteration bound for P-*(K) LCP. Some preliminary numerical results show that the proposed algorithm is efficient and reliable. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Arc-search interior-point algorithm P-*(kappa) linear complementarity problem Polynomial complexity

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