Two Classes of Smooth Objective Penalty Functions for Constrained Problems

This research aims at solving constrained problems by providing two classes of objective penalty functions which respectively approach to a class of exact penalty functions smoothly. Meanwhile, the authors present two algorithms based on the two objective penalty functions, and give the conclusion that all of the cluster points of the two sequences generated by the two algorithms are the optimal points of the original problem. Furthermore, this article discusses that both of the two penalty functions are of well-condition. Finally, the authors report numerical results to show the applicability of the two presented algorithms.     

Minimax Theorems Involving Two Functions and Strictly Monotone Transformations of Their Values

Two two-function minimax theorems are proved.The concavity-convexity conditions of the twofunctions involve strictly monotone transformations and mixing of the values of the two functions,and aredescribed by the inequalities as upward and weakly downward conditions.     

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.     

化不等式约束问题为无约束的广义可微限域精确罚函数方法

本文给出了广义可微精确罚函数的概念及一类所谓广义限域可微精确罚函数.本文预先选定罚因子,将不等式约束问题化为单一的无约束问题,并给出了具全局收敛性的算法.本文的罚函数构造简单,假设条件少而且算法的构造与收敛性结果是独特的.     

一类新的罚函数与罚算法(英)

在本文中,我们提出了带不等式约束的非线性规划问题的一类新的罚函数,它的一个子类可以光滑逼近$l_1$罚函数. 基于此类新的罚函数我们给出了一种罚算法,这个算法的特点是每次迭代求出罚函数的全局精确解或非精确解. 在很弱的条件下算法总是可行的. 我们在不需要任何约束规范的情况下,证明了算法的全局收敛性. 最后给出了数值实验.     

一种新的求解带约束的有限极大极小问题的精确罚函数

提出了一种新的精确光滑罚函数求解带约束的极大极小问题．仅仅添加一个额外的变量,利用这个精确光滑罚函数,将带约束的极大极小问题转化为无约束优化问题． 证明了在合理的假设条件下,当罚参数充分大,罚问题的极小值点就是原问题的极小值点．进一步,研究了局部精确性质．数值结果表明这种罚函数算法是求解带约束有限极大极小问题的一种有效算法．     

非线性整规划中的精确光滑罚函数

本文提出了几个非线性整规划 的全局精确光滑罚函数，每个罚函数有两个参数，并且给出了每个罚函数的精确罚参数的估计值，最后，我们举例说明了所提出的罚方法在具有整系数多项式目标函数以约束函数的整数规划中的应用。     

一类单调函数的数学期望

对一类单调可微的有界函数 ,利用相对变化率的概念 ,定义了一种由该函数生成的概率密度函数 ,讨论了有关数学期望的计算和性质 ,并给出了在函数上升或下降速度比较、药动学模型识别中的应用 .     

存零约束优化问题的序列二次方法北大核心CSCD

存零约束优化(MPSC)问题是近年来提出的一类新的优化问题,因存零约束的存在,使得常用的约束规范不满足,以至于现有算法的收敛性结果大多不能直接应用于该问题.应用序列二次规划(SQP)方法求解该问题,并证明在存零约束的线性独立约束规范下,子问题解序列的聚点为原问题的Karush-Kuhn-Tucker点.同时为了完善各稳定点之间的关系,证明了强平稳点与KKT点的等价性.最后数值结果表明,序列二次规划方法处理这类问题是可行的.     

求解非线性单调方程组的一种无导数投影算法

推广了一种修正的CG_DESCENT共轭梯度方法,并建立了一种有效求解非线性单调方程组问题的无导数投影算法.在适当的线搜索条件下,证明了算法的全局收敛性.由于新算法不需要借助任何导数信息,故它适应于求解大规模非光滑的非线性单调方程组问题.大量的数值试验表明,新算法对给定的测试问题是有效的.     

解非线性约束优化问题的依赖域算法

本文对一般非线性约束优化问题提出了一个信赖域算法,导出了等价的KKT条件.在试探步满足适当条件下,证明了算法的全局收敛性,并进行了数值试验.     

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 Simple Primal-Dual Feasible Interior-Point Method for Nonlinear Programming with Monotone Descent

We propose and analyze a primal-dual interior point method of the feasible type, with the additional property that the objective function decreases at each iteration. A distinctive feature of the method is the use of different barrier parameter values for each constraint, with the purpose of better steering the constructed sequence away from non-KKT stationary points. Assets of the proposed scheme include relative simplicity of the algorithm and of the convergence analysis, strong global and local convergence properties, and good performance in preliminary tests. In addition, the initial point is allowed to lie on the boundary of the feasible set.     

Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints

Recently, some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints were proved by Luo, Pang, and Ralph (Ref. 1). In this paper, we show that those results remain valid under some other mild conditions. One of these conditions, called strong convexity with order , is discussed in detail.     

Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs

In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed. Necessary and sufficient conditions for approximate strong duality results are derived. Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given. Under certain conditions, it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program.     

On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization

In this paper we propose two methods for smoothing a nonsmooth square-root exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original optimization problem. We develop an algorithm for solving the optimization problem based on the smoothed penalty function and prove the convergence of the algorithm. The efficiency of the smoothed penalty function is illustrated with some numerical examples, which show that the algorithm seems efficient.     

约束优化问题的修正共轭梯度投影算法

对闭凸集约束的非线性规划问题构造了一个修正共轭梯度投影下降算法,在去掉迭代点列有界的条件下,分析了算法的全局收敛性.新算法与共轭梯度参数结合,给出了三类结合共轭梯度参数的修正共轭梯度投影算法.数值例子表明算法是有效的.     

A SQP Method for Inequality Constrained Optimization

Abstract In this paper, a new SQP method for inequality constrained optimization is proposed and the globalconvergence is obtained under very mild conditions.     

求解MINLP的几类罚函数方法的分析和探讨

针对混合整数非线性约束优化问题(MINLP)的一般形式,通过罚函数的方法,给出了它的几种等价形式,并证明了最优解的等价性.将约束优化问题转化成更容易求解的无约束非线性优化问题,并把混合整数规划转化成非整数优化问题,从而将MINLP的求解简化为求解一个连续的无约束非线性优化问题,进而可用已有的一般无约束优化算法进行求解.     

互补约束均衡问题一个新的磨光技术

研究了一类带非线性互补约束的均衡问题．借助于逐步逼近思想,构造了一个在求解意义上与原问题等价的磨光非线性规划．从而保证一些经典的标准优化算法可以应用到该类优化问题上．最后提出了两个算法模型并分析了其全局收敛性．