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.     

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.AMS Subject Classification: 90C, 49MThis work was supported by the National Natural Science Foundation (10261001) and the Guangxi Province Science Foundation (0236001, 0249003) of China.     

A New Superlinearly Convergent SQP Algorithm for Nonlinear Minimax Problems

In this paper, the nonlinear minimax problems are discussed. By means of the Sequential Quadratic Programming （SQP）, a new descent algorithm for solving the problems is presented. At each iteration of the proposed algorithm, a main search direction is obtained by solving a Quadratic Programming （QP） which always has a solution. In order to avoid the Maratos effect, a correction direction is obtained by updating the main direction with a simple explicit formula. Under mild conditions without the strict complementarity, the global and superlinear convergence of the algorithm can be obtained. Finally, some numerical experiments are reported.     

A Globally and Superlinearly Convergent SQP Algorithm for Nonlinear Constrained Optimization

Based on a continuously differentiable exact penalty function and a regularization technique for dealing with the inconsistency of subproblems in the SQP method, we present a new SQP algorithm for nonlinear constrained optimization problems. The proposed algorithm incorporates automatic adjustment rules for the choice of the parameters and makes use of an approximate directional derivative of the merit function to avoid the need to evaluate second order derivatives of the problem functions. Under mild assumptions the algorithm is proved to be globally convergent, and in particular the superlinear convergence rate is established without assuming that the strict complementarity condition at the solution holds. Numerical results reported show that the proposed algorithm is promising.     

一个线性约束优化问题的广义梯度投影法

本文对线性约束优化问题提出了一个新的广义梯度投影法，该算法采用了非精确线性搜索，并在每次迭代运算中结合了广义投影矩阵和变尺度方法的思想确定其搜索方向．在通常的假设条件下，证明了该算法的整体收敛性和超线性收敛速度．     

不等式约束最优化超线性与二次收敛的强次可行SQP算法

利用ＳＱＰ方法、广义投影技术和强次可行方（向）法思想,建立不等式约束优化一个新的初始点任意的快速收敛算法． 算法每次迭代仅需解一个总存在可行解的二次子规划,或用广义投影计算“一阶”强次可行下降辅助搜索方向;采用曲线搜索与直线搜索相结合的方法产生步长． 在较温和的条件下,算法具有全局收敛性、强收敛性、超线性与二次收敛性． 给出了算法有效的数值试验．     

非线性互补约束问题一个全局收敛的SQP算法

本文研究非线性互补约束优化问题,利用Fischer-Burmeister函数将非线性互补问题转化为非光滑方程,提出一个求解非线性互补约束问题的SQP算法,并在适当的假设下证明这个算法是全局收敛的.     

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 Robust SQP Method for Mathematical Programs with Linear Complementarity Constraints

The relationship between the mathematical program with linear complementarity constraints (MPLCC) and its inequality relaxation is studied. Based on this relationship, a new sequential quadratic programming (SQP) method is presented for solving the MPLCC. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived without assuming the linear independence constraint qualification for MPEC, the nondegeneracy condition, and any feasibility condition of the quadratic programming subproblems. Preliminary numerical results are reported. Research is partially supported by Singapore-MIT Alliance and School of Business, National University of Singapore.     

线性均衡约束最优化的一个广义投影强次可行方向法

本文讨论带线性均衡约束最优化问题,首先利用摄动技术和一个互补函数将问题等价转化为一般约束最优化问题,然后结合广义投影技术和强次可行方向法思想,建立了问题的一个新算法.算法在迭代过程中保证搜索方向不为零,从而使得每次迭代只需计算一次广义投影.在适当的条件下,证明了算法的全局收敛性,并对算法进行了初步的数值试验.     

A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints

This paper presents a sequential quadratic programming algorithm for computing a stationary point of a mathematical program with linear complementarity constraints. The algorithm is based on a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Burmeister functional, combined with a classical penalty function method for solving constrained optimization problems. Global convergence of the algorithm is established under appropriate assumptions. Some preliminary computational results are reported.     

LC1约束规划问题的超线性收敛ODE型信赖域算法

本文对带线性等式约束的LC^1优化问题提出了一个新的ODE型信赖域算法，它在每一次迭代时，不必求解带信赖域界的子问题，仅解一线性方程组而求得试验步。从而可以降低计算的复杂性，提高计算效率，在一定的条件下，文中还证明了该算法是超线性收敛的。     

不等式约束优化的一个超线性收敛FSSLE算法

本文针对不等式约束优化问题,提出了一个可行序列线性方程组(FSSLE)算法.该算法每次迭代只需求解四个具有相同系数矩阵的线性方程组,因而计算量较小.在没有假设算法产生的聚点是孤立点和近似乘子列有界的条件下,证明了算法具有全局收敛性.在一般条件下,证明了算法具有超线性收敛性.     

非线性不等式约束最优化一个超线性与二次收敛的强次可行方法

本文讨论非线性不等式约束最优化问题，借助于序列线性方程组技术和强次可行方法思想，建立了问题的一个初始点任意的快速收敛新算法．在每次迭代中，算法只需解一个结构简单的线性方程组．算法的初始迭代点不仅可以是任意的，而且不使用罚函数和罚参数，在迭代过程中，迭代点列的可行性单调不减．在相对弱的假设下，算法具有较好的收敛性和收敛速度，即具有整体与强收敛性，超线性与二次收敛性．文中最后给出一些数值试验结果．     

A Linearly Convergent Derivative-Free Descent Method for Strongly Monotone Complementarity Problems

We establish the first rate of convergence result for the class of derivative-free descent methods for solving complementarity problems. The algorithm considered here is based on the implicit Lagrangian reformulation [26, 35] of the nonlinear complementarity problem, and makes use of the descent direction proposed in [42], but employs a different Armijo-type linesearch rule. We show that in the strongly monotone case, the iterates generated by the method converge globally at a linear rate to the solution of the problem.     

不等式约束优化一个超线性收敛的可行内点型算法

本文针对非线性不等式约束优化问题,提出了-个可行内点型算法.在每次迭代中,基于积极约束集策略,该算法只需求解三个线性方程组,因而其计算工作量较小.在-般的条件下,证明了算法具有全局收敛及超线性收敛性.     

隐互补约束优化问题的磨光SQP算法

本文提出了一类隐互补约束优化问题的磨光SQP算法.首先,我们给出了这类优化问题的最优性和约束规范性条件.然后,在适当假设条件下,我们证明了算法具有全局收敛性.     

非线性互补约束均衡问题的一个SQP算法

提出了一个求解非线性互补约束均衡问题(MPCC)的逐步逼近光滑SQP算法.通过一系列光滑优化来逼近MPCC.引入l<,1>精确罚函数,线搜索保证算法具有全局收敛性.进而,在严格互补及二阶充分条件下,算法是超线性收敛的.此外,当算法有限步终止,当前迭代点即为MPEC的一个精确稳定点.     

Globally and Superlinearly Convergent QP-Free Algorithm for Nonlinear Constrained Optimization

A new, infeasible QP-free algorithm for nonlinear constrained optimization problems is proposed. The algorithm is based on a continuously differentiable exact penalty function and on active-set strategy. After a finite number of iterations, the algorithm requires only the solution of two linear systems at each iteration. We prove that the algorithm is globally convergent toward the KKT points and that, if the second-order sufficiency condition and the strict complementarity condition hold, then the rate of convergence is superlinear or even quadratic. Moreover, we incorporate two automatic adjustment rules for the choice of the penalty parameter and make use of an approximated direction as derivative of the merit function so that only first-order derivatives of the objective and constraint functions are used.     

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．