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利用光滑对称扰动Fischer-Burmeister函数将广义非线性互补问题转化为非线性方程组,提出新的光滑化拟牛顿法求解该方程组.然后证明该算法是全局收敛的,且在一定条件下证明该算法具有局部超线性(二次)收敛性.最后用数值实验验证了该算法的有效性. 相似文献
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给出求解圆锥规划问题的一种新光滑牛顿方法.基于圆锥互补函数的一个新光滑函数,将圆锥规划问题转化成一个非线性方程组,然后用光滑牛顿方法求解该方程组.该算法可从任意初始点开始,且不要求中间迭代点是内点.运用欧几里得代数理论,证明算法具有全局收敛性和局部超线性收敛速度.数值算例表明算法的有效性. 相似文献
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投影信赖域策略结合非单调线搜索算法解有界约束非线性半光滑方程组.基于简单有界约束的非线性优化问题构建信赖域子问题,半光滑类牛顿步在可行域投影得到投影牛顿的试探步,获得新的搜索方向,结合非单调线搜索技术得到回代步,获得新的步长.在合理的条件下,证明算法不仅具有整体收敛性且保持超线性收敛速率.引入非单调技术能克服高度非线性的病态问题,加速收敛性进程,得到超线性收敛速率. 相似文献
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研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性. 相似文献
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《高校应用数学学报(A辑)》2006,21(3):377-378
面对成比例交易成本的情况下外汇期权的定价;带常利率的Erlang(2)风险模型的罚金折现期望函数;具有反馈、不可靠服务台和二次多选择服务的M/G/1重试排队系统;解非线性优化问题的锥模型信赖域方法;基于非拟牛顿方法无约束最优化问题的全局收敛性;具有分片线性NCP函数的QP-free可行域方法。 相似文献
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《数学的实践与认识》2016,(23)
由于退化解会导致再生方程的奇异性,非线性互补问题的求解通常采用基于半光滑技术的广义牛顿法.基于2-正则性的概念,提出了一类利用光滑互补函数求解互补问题的光滑牛顿算法.算法采用积极集技术,能在解的附近估计出退化指标,并把原问题降阶为一个非奇异方程组,从而保证了迭代效率.算法具有整体收敛性和局部超线性收敛性,数值实验显示算法是有效的. 相似文献
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本文研究了非线性互补的光滑化问题.利用一个新的光滑NCP函数将非线性互补问题转化为等价的光滑方程组,并在此基础上建立了求解P0-函数非线性互补问题的一个完全光滑化牛顿法,获得了算法的全局收敛性和局部二次收敛性的结果.并给出数值实验验证了理论分析的正确性. 相似文献
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In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given. 相似文献
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Liqun Qi 《Journal of Global Optimization》2006,35(2):343-366
The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints
can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a
sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the
NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function
is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function,
which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth
NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent
to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem,
an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers
only.
This work is supported by the Research Grant Council of Hong Kong
This paper is dedicated to Alex Rubinov on the occasion of his 65th Birthday 相似文献
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We study convergence of a semismooth Newton method for generalized semi-infinite programming problems with convex lower level
problems where, using NCP functions, the upper and lower level Karush-Kuhn-Tucker conditions of the optimization problem are
reformulated as a semismooth system of equations. Nonsmoothness is caused by a possible violation of strict complementarity
slackness. We show that the standard regularity condition for convergence of the semismooth Newton method is satisfied under
natural assumptions for semi-infinite programs. In fact, under the Reduction Ansatz in the lower level and strong stability
in the reduced upper level problem this regularity condition is satisfied. In particular, we do not have to assume strict
complementary slackness in the upper level. Numerical examples from, among others, design centering and robust optimization
illustrate the performance of the method.
相似文献
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Aurél Galántai 《Computational Optimization and Applications》2012,52(3):805-824
The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions. 相似文献
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Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P
0-Functions 总被引:1,自引:0,他引:1
By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth
equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust
region algorithm for solving the NCP with P
0 functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under
a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict
complementarity condition.
This work was partially supported by the Research Grant Council of Hong Kong and the National Natural Science Foundation of
China (Grant 10171030). 相似文献
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Xiaojiao Tong Felix F. Wu Liqun Qi 《Numerical Functional Analysis & Optimization》2013,34(3-4):467-485
This paper investigates the convergence of decoupled optimal power flow (DOPF) methods used in power systems. In order to make the analysis tractable, a rigorous mathematical reformation of DOPF is presented first to capture the essence of conventional heuristic decompositions. By using a nonlinear complementary problem (NCP) function, the Karush–Kuhn–Tucker (KKT) systems of OPF and its subproblems of DOPF are reformulated as a set of semismooth equations, respectively. The equivalent systems show that the sequence generated by DOPF methods is identical to the sequence generated by Gauss–Seidel methods with respect to nonsmooth equations. This observation motivates us to extend the classical Gauss–Seidel method to semismooth equations. Consequently, a so-called semismooth Gauss–Seidel method is presented, and its related topics such as algorithm and convergence are studied. Based on the new theory, a sufficient convergence condition for DOPF methods is derived. Numerical examples of well-known IEEE test systems are also presented to test and verify the convergence theorem. 相似文献
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Jacobian smoothing Brown’s method for nonlinear complementarity problems (NCP) is studied in this paper. This method is a generalization of classical Brown’s method. It belongs to the class of Jacobian smoothing methods for solving semismooth equations. Local convergence of the proposed method is proved in the case of a strictly complementary solution of NCP. Furthermore, a locally convergent hybrid method for general NCP is introduced. Some numerical experiments are also presented. 相似文献
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Hou-duo Qi 《计算数学(英文版)》2000,(3)
1. IntroductionConsider the nonlinear complementarity problem (NCP for short), which is to findan x E M" such thatwhere F: Wu - ac and the inequalities are taken componentwise. This problem havemany important applications in various fields. [13, 7, 5].Due to the less storage in computation, derivative--free descent method, which meansthe search direction used does not involye the Jacobian matrix of F, is popular infinding solutions of nonlinear complementarity Problems. We briefly view som… 相似文献
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Huo-Duo Qi & Yu-Zhong Zhang 《计算数学(英文版)》2000,18(3):251-264
Based on a class of functions, which generalize the squared Fischer-Burmeister NCP function and have many desirable properties as the latter function has, we reformulate nonlinear complementarity problem (NCP for short) as an equivalent unconstrained optimization problem, for which we propose a derivative-free descent method in monotone case. We show its global convergence under some mild conditions. If $F$, the function involved in NCP, is $R_0$-function, the optimization problems has bounded level sets. A local property of the merit function is discussed. Finally,we report some numerical results. 相似文献
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In this paper, a new hybrid method is proposed for solving nonlinear complementarity problems (NCP) with P
0 function. In the new method, we combine a smoothing nonmonotone trust region method based on a conic model and line search
techniques. We reformulate the NCP as a system of semismooth equations using the Fischer-Burmeister function. Using Kanzow’s
smooth approximation function to construct the smooth operator, we propose a smoothing nonmonotone trust region algorithm
of a conic model for solving the NCP with P
0 functions. This is different from the classical trust region methods, in that when a trial step is not accepted, the method
does not resolve the trust region subproblem but generates an iterative point whose steplength is defined by a line search.
We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity
condition, the superlinear convergence of the algorithm is established without a strict complementarity condition. 相似文献