初始点任意的一个非线性优化的广义梯度投影法

广义投影算法的优点是避免转轴运算。它成功地给出了线性约束问题、初始点任意的只带非线性不等式约束问题，以及利用辅助规划来处理带等式与不等式约束问题的算法．后者完满地解决了投影算法对于非线性等式约束问题的处理，但要求满足不等式约束的初始点．本文据此利用广义投影与罚函数技巧给出了一个初始点任意的等式与不等式约束问题的算法，省去了求初始解的计算，并保持了上述方法的优点，证明了算法的全局收敛性     

用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

非线性约束最优化一族超线性收敛的可行方法

本文建立求解非线性不等式约束最优化一族含参数的可行方法．算法每次迭代仅需解一个规模较小的二次规划．在一定的假设条件下,证明了算法族的全局收敛性和超线性收敛性．     

无罚函数和滤子的QP-free非可行域方法

提出了求解光滑不等式约束最优化问题的无罚函数和无滤子QP-free非可行域方法. 通过乘子和非线性互补函数, 构造一个等价于原约束问题一阶KKT条件的非光滑方程组. 在此基础上, 通过牛顿-拟牛顿迭代得到满足KKT最优性条件的解, 在迭代中采用了无罚函数和无滤子线搜索方法, 并证明该算法是可实现,具有全局收敛性. 另外, 在较弱条件下可以证明该方法具有超线性收敛性.     

新的滤子方法(英)

本文定义了一种新的滤子方法,并提出了求解光滑不等式约束最优化问题的滤子QP-free非可行域方法. 通过乘子和分片线性非线性互补函数,构造一个等价于原约束问题一阶KKT条件的非光滑方程组.在此基础上, 通过牛顿-拟牛顿迭代得到满足KKT最优条件的解,在迭代中采用了滤子线搜索方法,证明了该算法是可实现,并具有全局收敛性. 另外,在较弱条件下可以证明该方法具有超线性收敛性.     

新的滤子方法

本文定义了一种新的滤子方法,并提出了求解光滑不等式约束最优化问题的滤子QP-free非可行域方法.通过乘子和分片线性非线性互补函数,构造一个等价于原约束问题一阶KKT条件的非光滑方程组.在此基础上,通过牛顿-拟牛顿迭代得到满足KKT最优条件的解,在迭代中采用了滤子线搜索方法,证明了该算法是可实现,并具有全局收敛性.另外,在较弱条件下可以证明该方法具有超线性收敛性.     

解约束优化的分段线性有理NCP函数

本文给出新的NCP函数,这些函数是分段线性有理正则伪光滑的,且具有良好的性质.把这些NCP函数应用到解非线性优化问题的方法中.例如,把求解非线性约束优化问题的KKT点问题分别用QP-free方法,乘子法转化为解半光滑方程组或无约束优化问题.然后再考虑用非精确牛顿法或者拟牛顿法来解决该半光滑方程组或无约束优化问题.这个方法是可实现的,且具有全局收敛性.可以证明在一定假设条件下,该算法具有局部超线性收敛性.     

求非线性规划的非可行滤子无二次子规划方法

本文研究了不等式约束的非线性规划问题.利用带滤子的无二次子规划(QP-free)非可行域方法,构造一个等价于原约束问题的一阶KKT条件的非光滑方程组,给出解这个方程组的迭代算法,并获得算法的全局收敛性.     

带新NCP函数的Lagrangian乘子方法

在经营管理、工程设计、科学研究、军事指挥等方面普遍存在着最优化问题,而实际问题中出现的绝大多数问题都被归纳为非线性规划问题之中。作为带等式、不等式约束的复杂事例,最优化问题的求解向来较为繁琐、困难。适当条件下,非线性互补函数(NCP)可以与约束优化问题相结合,其中NCP函数的无约束极小解对应原约束问题的解及其乘子。本文提出了一类新的NCP函数用于解决等式和不等式约束非线性规划问题,结合新的NCP函数构造了增广Lagrangian函数。在适当假设条件下,证明了增广Lagrangian函数与原问题的解之间的一一对应关系。同时构造了相应算法,并证明了该算法的收敛性和有效性。     

一个改进的SQP型算法

本文建立非线性等式和不等式约束规划问题的一个序列二次规划(SQP)型算法.算法的每次迭代只需解一个确实可解的二次规划，然后对其解进行简单的显式校正，便可产生关于罚函数是下降的搜索方向，克服Maratos效应.在适当的假设条件下，还论证了算法的全局收敛性和超级收敛性.     

GLOBAL CONVERGENCE AND IMPLEMENTATION OF NGTN METHOD FOR SOWING LARGE-SCALE SMRSE NONLINEAR PROGRAMMING PROBLEMS

1. IntroductionConsider the following NLP problemwhere the function f: Re --+ RI and gi: Re - R', j E J are twice continuously dtherentiable.In particular, we discuss the cajse, where the nUmber of variables and the nUmber of constraintsin (1.1) are large and second derivatives in (1.1) are sparse.There are some methods whiCh can solve largesscale problems, e.g. Lancelot in [2] andTDSQPLM in [9]. But they can not take adVantage of sparse structtire of the problem. A newefficient meth…     

求解非线性规划的原始对偶内点法(英文)

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.     

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.

     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

A projection-filter method for solving nonlinear complementarity problems

The Josephy-Newton method attacks nonlinear complementarity problems which consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. Utilizing the filter method, we presented a new globalization strategy for this Newton method applied to nonlinear complementarity problem without any merit function. The strategy is based on the projection-proximal point and filter methodology. Our linesearch procedure uses the regularized Newton direction to force global convergence by means of a projection step which reduces the distance to the solution of the problem. The resulting algorithm is globally convergent to a solution. Under natural assumptions, locally superlinear rate of convergence was established.     

Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems.In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver.A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of up to 3 improvement over branch-and-bound is observed.     

Convergence of a projected gradient method with trust region for nonlinear constrained optimization†

In this paper we describe a projected gradient algorithm with trust region, introducing a nondifferentiable merit function for solving nonlinear constrained optimization problems. We show that this method is globally convergent even if conditions are weak. It is also proved that, when the strict complementarity condition holds, the proposed algorithm can be solved by an equality constrained problem, allowing locally rate of superlinear convergence.     

解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

A nonmonotone derivative-free algorithm for nonlinear complementarity problems based on the new generalized penalized Fischer–Burmeister merit function

In this paper, we propose a new generalized penalized Fischer–Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455–471, 2009), which confirm the superior behaviour of the new merit function.     

Probabilistic version of the method of feasible directions

A probabilistic version of the method of feasible directions (MFD) for solving nonlinear programming (NLP) problems of the type min{f(x): f_j(x)0,j=1,2,…,m} is presented. Randomization is applied to modify the algorithm and a global convergence Theorem is used in the analysis of convergence. Some numerical experiments on problems with known solutions serve to compare this method with the traditional deterministic versions.