互补约束规划问题的一个广义梯度投影算法

本文研究了一类均衡约束最优化问题.利用广义梯度投影法,结合罚函数思想,得到了一个初始点可以任意的广义梯度投影算法.在较弱的条件下,证明了算法的全局收敛性.     

互补约束均衡优化的一个共轭梯度投影法

讨论均衡约束最优化问题,利用一个互补函数和扰动技术将原问题转换为非线性等式约束最优化问题,然后利用共轭梯度投影算法的思想,给出了问题的一个求解算法,在适当的条件下,证明了算法的全局收敛性.     

非线性约束优化一个强收敛的广义投影强次可行方向法

讨论带非线性不等式和等式约束的最优化问题,借助强次可行方向法和半罚函数的思想,给出了问题的一个新的广义投影强次可行方向法.该算法的一个重要特性是有限次迭代后,迭代点落入半罚问题的可行域.在适当的条件下证明了算法的全局收敛性和强收敛性.数值实验表明算法是有效的.     

一个改进的序列二次规划可行下降算法及其全局收敛性

利用广义投影校正技术对搜索方向进行某种修正,改进假设条件,采用一种新型的一阶修正方向并结合SQP技术,建立了求解非线性约束最优化问题(p)的一个新的SQP可行下降算法,在较温和的假设条件下证明了算法的全局收敛性.由于新算法仅需较小的存储,从而适合大规模最优化问题的计算.     

不等式约束最优化无严格互补条件下的快速收敛序列线性方程组算法

本文讨论无严格互补性的非线性不等式约束最优化问题,建立了一个新的序列线性方程组算法。算法每次迭代只需解一个线性方程组或计算一次广义梯度投影,并不要求Lagrange函数的近似Hessian阵正定。在较弱的假设下,证明了算法的整体收敛性、强收敛性、超线性收敛性及二次收敛速度。还对算法进行了有效的数值试验。     

一般约束最优化拓广的强次可行方向法

本文讨论非线性等式与不等式最优化问题,引进一个拟罚函数及其相应的只带不等式约束的辅助问题,然后采用广义投影技术和强次可行方向法思想建立原问题的一个全局收敛新算法,该算法具有初点始任意,结构简单,计算量较小等特点。     

超线性与二次收敛序列线性方程组算法

本文,在无严格互补条件下,对非线性不等式约束最优化问题提出了一个新的序列线性方程组(简称SSLE)算法．算法有两个重要特征:首先,每次迭代,只须求解一个线性方程组或一个广义梯度投影阵,且线性方程组可以无解．其次,初始点可以任意选取．在无严格互补条件下,算法仍有全局收敛性、强收敛性、超线性收敛性及二次收敛性．文章的最后,还对算法进行了初步的数值实验．     

非线性最优化的广义梯度投影法

梯度投影法已有许多有效算法,但这些算法还存在三个问题:1)为了保证算法的收敛性,在算法的每一迭代步,需要选取δ-主动约束集,计算量较大.2)在迭代过程中,需要跟踪主动约束集.3)只能处理非线性不等式约束问题.本文讨论非线性等式与不等式约束的优化问题,给出了一个广义梯度投影法,证明了算法的收敛性并且完满地解决了上述三个问题.本文算法结构简单且其处理技巧有普遍意义.     

不等式约束优化一个基于滤子思想的广义梯度投影算法

讨论非线性不等式约束优化问题, 借鉴于滤子算法思想,提出了一个新型广义梯度投影算法.该方法既不使用罚函数又无真正意义下的滤子.每次迭代通过一个简单的显式广义投影法产生搜索方向,步长由目标函数值或者约束违反度函数值充分下降的Armijo型线搜索产生.算法的主要特点是: 不需要迭代序列的有界性假设;不需要传统滤子算法所必需的可行恢复阶段;使用了ε积极约束集减小计算量.在合适的假设条件下算法具有全局收敛性, 最后对算法进行了初步的数值实验.     

线性约束最优化的一个共轭投影梯度法

本结合共轭梯度法及梯度投影法的思想,建立线性等式约束最优化的一个新算法,称之为共轭投影梯度法。分别对二次凸目标函数和一般目标函数分析和论证了算法的重要性质和收敛性。     

互补约束数学规划问题的一个广义梯度投影罚算法

结合罚函数思想和广义梯度投影技术,提出求解非线性互补约束数学规划问题的一个广义梯度投影罚算法.首先,通过扰动技术和广义互补函数,将原问题转化为序列带参数的近似的标准非线性规划;其次,利用广义梯度投影矩阵构造搜索方向的显式表达式.一个特殊的罚函数作为效益函数,而且搜索方向能保证效益函数的下降性.在适当的假设条件下算法具有全局收敛性.     

Generalized mildly nonlinear complementarity problems for fuzzy mappings

In this paper, we introduce and study a new class of generalized mildly nonlinear complementarity problems for fuzzy mappings. We use the change of variables technique to establish the equivalence between the generalized mildly nonlinear complementarity problems and the Wiener-Hopf equations. This equivalence is used to suggest and analyze a number of iterative algorithm for solving the generalized mildly nonlinear complementarity problems.     

A New Step-Size Skill for Solving a Class of Nonlinear Projection Equations

1.IntroductionIn[11],aniterativeprojectionandcontraction(PC)methodforlinearcomplemen-taxityproblemswasproposed.Inpractice,thealgorithmbehaveseffectively,butintheorythesteP-sizecannotbeprovedtobeboundedawayfromzero.Sonostatementcanbemadeabouttherateofconvergence.AlthoughavariantoftheprimePCal-gorithmwithconstantstepsizeforlineaJrprogrammminghasalinearconvergence['1,itconvergesmuchslowerinparctice.In[1o],HeproposedanewsteVsizerulefortheprimePCalgorithmforthelinearprogrammingsuchthattheresul…     

Finding solutions of implicit complementarity problems by isotonicity of the metric projection

Isac and Németh [G. Isac and A. B. Németh, Projection method, isotone projection cones and the complementarity problem, J. Math. Anal. App., 153, 258-275(1990)] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this paper, the notion of *-isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on *-isotone projection cones. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by *-isotone projection cones. The question of finding nonzero solutions of these problems is also studied.     

Implicit complementarity problems on isotone projection cones

Isac and Németh [G. Isac and A. B. Németh, Projection methods, isotone projection cones and the complementarity problem, J. Math. Anal. Appl. 153 (1990), pp. 258–275] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this article an iterative algorithm is studied in connection with an implicit complementarity problem. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones, extending the results of Németh [S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350 (2009), pp. 340–370]. Some existing concepts from the latter paper are extended to solve the problem of finding nonzero solutions of the implicit complementarity problem.     

Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs

In this paper, we present an algorithm for solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs. The path cost function considered is comprised of two attributes, travel time and toll, that are combined into a nonlinear generalized cost. Travel demand is determined endogenously according to a travel disutility function. Travelers choose routes with the minimum overall generalized costs. The algorithm involves two components: a bicriteria shortest path routine to implicitly generate the set of non-dominated paths and a projection and contraction method to solve the nonlinear complementarity problem (NCP) describing the traffic equilibrium problem. Numerical experiments are conducted to demonstrate the feasibility of the algorithm to this class of traffic equilibrium problems.     

A General Algorithm and Sensitivity Analysis for Variational Inequalities

The fixed point technique is used to prove the existence of a solution for a class of variational inequalities related with odd order boundary value problems and to suggest a general algorithm. We also make the sensitivity analysis for these variational inequalities and complementarity problems using the projection technique. Several special cases are discussed, which can be obtained from our results.     

用投影方法求耗散广义Hamilton约束系统的李群积分

针对耗散广义Hamilton约束系统,通过引入拉格朗日乘子和采用投影技术,给出了一种保持动力系统内在结构和约束不变性的李群积分法．首先将带约束条件的耗散Hamilton系统化为无约束广义Hamilton系统, 进而讨论了无约束广义Hamilton系统的李群积分法,最后给出了广义Hamilton约束系统李群积分的投影方法．采用投影技术保证了约束的不变性,引入拉格朗日乘子后,在向约束流形投影时不会破坏原动力系统的李群结构．讨论的内容仅限于完整约束系统, 通过数值例题说明了方法的有效性．     

A New Superlinearly Convergent Strongly Subfeasible Sequential Quadratic Programming Algorithm for Inequality-Constrained Optimization

Combining the ideas of generalized projection and the strongly subfeasible sequential quadratic programming (SQP) method, we present a new strongly subfeasible SQP algorithm for nonlinearly inequality-constrained optimization problems. The algorithm, in which a new unified step-length search of Armijo type is introduced, starting from an arbitrary initial point, produces a feasible point after a finite number of iterations and from then on becomes a feasible descent SQP algorithm. At each iteration, only one quadratic program needs to be solved, and two correctional directions are obtained simply by explicit formulas that contain the same inverse matrix. Furthermore, the global and superlinear convergence results are proved under mild assumptions without strict complementarity conditions. Finally, some preliminary numerical results show that the proposed algorithm is stable and promising.     

A Superlinearly Convergent SSLE Algorithm for Optimization Problems with Linear Complementarity Constraints

In this paper we study a special kind of optimization problems with linear complementarity constraints. First, by a generalized complementarity function and perturbed technique, the discussed problem is transformed into a family of general nonlinear optimization problems containing parameters. And then, using a special penalty function as a merit function, we establish a sequential systems of linear equations (SSLE) algorithm. Three systems of equations solved at each iteration have the same coefficients. Under some suitable conditions, the algorithm is proved to possess not only global convergence, but also strong and superlinear convergence. At the end of the paper, some preliminary numerical experiments are reported.