基于一维搜索和动态调节的非线性规划PSO算法

对非线性规划问题的处理通常采用罚函数法,使用罚函数法的困难在于参数的选取。本文提出了一种解非线性规划问题的新PSO算法(NSDPSO),该方法融入了一维搜索和动态调节技术,使NSDPSO很好地克服了标准PSO算法在前期收敛较快而在后期易陷入局部最优的缺陷。另外,文中还给出了一种新的适应度函数及选择算子,使算法在选择下一代时保持群体中不可行解的一定比例,这样不但能有效地增加群体的多样性,而且可以避免传统的过度惩罚,使群体向最优解逼近。最后的数据实验表明该算法对非线性规划问题求解是非常有效的。     

用罚函数求解线性双层规划的全局优化方法

用罚函数法将线性双层规划转化为带罚函数子项的双线性规划问题，由于其全局最优解可在约束域的极点上找到，利用对偶理论给出了一种求解该双线性规划的方法，并证明当罚因子大于某一正数时，双线性规划的解就是原线性双层规划的全局最优解。     

多目标参数线性规划

本文主要讨论利用一组参数变量求解多目标线性规划的方法,使得所有目标函数的凸线性组合最大,确定多目标规划非劣解的参数约束空间。如何利用参数空间的约束法确定多目标线性规划中目标函数的系数矩阵,以及如何利用参数空间分解法求多目标的非劣解。最后,文中附实例说明如何应用此方法来解决多目标线性规划的问题。     

线性规划标准型和整数线性规划最优解的两个注记

本文研究线性规划标准型的基本假设所蕴含的一些性质,并探讨整数线性规划最优解和其松弛问题最优解的关系.首先,分别讨论四种情形下线性规划最优解的性质,即无约束线性规划问题、仅有非负约束的线性规划问题、仅有等式约束的线性规划问题,以及标准线性规划问题系数矩阵的列向量有为零的情形等.然后,构造两族二维整数线性规划,其松弛问题的最优解与其(整数)最优解"相距甚远".     

一种具有非线性约束线性规划全局优化算法

本文提出了一种新的适用于处理非线性约束下线性规划问题的全局优化算法。该算法通过构造子问题来寻找优于当前局部最优解的可行解。该子问题可通过模拟退火算法来解决。通过求解一系列的子问题,当前最优解被不断地更新,最终求得全局最优解。最后,本算法应用于几个典型例题,并与罚函数法相比较,数值结果表明该算法是可行的,有效的。     

"准"精确惩罚函数法的渐近性分析

1引言众所周知,罚函数法在最优化理论与数值计算中占据着极其重要的位置,作为求解约束优化问题的一类重要方法,在上世纪五、六十年代曾经历一次发展高潮．近十几年来,伴随着对数障碍函数法在内点法中取得的成功,罚函数法的研究又呈现出一个小高潮[2,3,4]．在罚函数方法里,精确惩罚函数法有着非常吸引人的性质,即,当罚参数大于某个有限门槛值时,仅通过求解单个无约束罚问题便可得到原问题的最优解,从而省去了一般罚函数法解系列无约束优化问题的工作量．     

遗传算法求解约束非线性规划及Matlab实现

对于约束非线性规划问题,传统的方法:可行方向法、惩罚函数法计算烦琐且精度不高.用新兴的遗传算法来解决约束非线性规划,核心是惩罚函数的构造.以前的惩罚函数遗传算法有的精度较低,有的过于复杂.本文在两个定义的基础上构造了新的惩罚函数,并在新的惩罚函数的基础上,提出了一种解决约束非线性最优化问题的方法.通过两个例子应用Matlab说明了这个算法的可行性.     

一类全局优化问题的线性化方法

本文对带有不定二次约束且目标函数为非凸二次函数的最优化问题提出了一类新的确定型全局优化算法,通过对目标函数和约束函数的线性下界估计,建立了原规划的松弛线性规划,通过对松弛线性规划可行域的细分以及一系列松弛线性规划的求解过程,得到原问题的全局最优解.我们从理论上证明了算法能收敛到原问题的全局最优解.     

求解约束优化问题的一种新的遗传算法

提出一种新的求解约束优化问题的遗传算法,算法通过重新定义可行解与不可行解的适应度函数分别对它们进行选择,有效避免了惩罚函数法引入参数所带来的困难,重新设计的交叉算子使得算法对解空间的寻优范围扩大了.数值实验结果表明算法具有较好的鲁棒性,且对最优解位于约束边界上的一类问题具有很大优势.     

极小扰动约束非线性规划的最优性条件

考虑当目标函数在约束条件下的最优值作扰动时，使各约束作极小扰动的非线性规划问题．文中引进了极小扰动约束规划的极小扰动有效解概念．利用把问题归为一个相应的多目标规划问题，给出了极小扰动约束有效解的最优性条件．     

非线性-线性二层规划问题的罚函数方法

利用下层问题的K-T最优性条件将下层为线性规划的一类非线性二层规划转化成相应的单层规划,同时取下层问题的互补条件为罚项,构造了该类非线性二层规划的罚问题.通过对相应罚问题性质的分析,得到了该类非线性二层规划问题的最优性条件,同时设计了该类二层规划问题的求解方法.数值结果表明该方法是可行、有效的.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

A dual-relax penalty function approach for solving nonlinear bilevel programming with linear lower level problem

The penalty function method, presented many years ago, is an important numerical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty function approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.     

一种新的逼近精确罚函数的罚函数及性质

针对可微非线性规划问题提出了一个新的逼近精确罚函数的罚函数形式,给出了近似逼近算法与渐进算法,并证明了近似算法所得序列若有聚点,则必为原问题最优解. 在较弱的假设条件下,证明了算法所得的极小点列有界,且其聚点均为原问题的最优解,并得到在Mangasarian-Fromovitz约束条件下,经过有限次迭代所得的极小点为可行点.     

一个修正的罚函数方法

本文通过给出的一个修正的罚函数,把约束非线性规划问题转化为无约束非线性规划问题.我们讨论了原问题与相应的罚问题局部最优解和全局最优解之间的关系,并给出了乘子参数和罚参数与迭代点之间的关系,最后给出了一个简单算法,数值试验表明算法是有效的.     

An algorithm for solving linearly constrained minimax problems

In this study, we propose an algorithm for solving a minimax problem over a polyhedral set defined in terms of a system of linear inequalities. At each iteration a direction is found by solving a quadratic programming problem and then a suitable step size along that direction is taken through an extension of Armijo's approximate line search technique. We show that each accumulation point is a Kuhn-Tucker solution and give a condition that guarantees convergence of the whole sequence of iterations. Through the use of an exact penalty function, the algorithm can be used for solving constrained nonlinear programming. In this case, our algorithm resembles that of Han, but differs from it both in the direction-finding and the line search steps.     

等式约束优化问题的一类新的简单光滑精确罚函数

精确罚函数方法是求解优化问题的一类经典方法,传统的精确罚函数不可能既是简单的又是光滑的,这里简单的是指罚函数中不包含目标函数和约束函数的梯度信息。针对等式约束问题提出了不同与传统罚函数的一类新的简单光滑罚函数并证明了它是精确的。给出了以新的罚函数为基础的罚函数方法并用数值例子说明算法是可行的。     

A penalty function method based on bilevel programming for solving inverse optimal value problems

In this work, we reformulate the inverse optimal value problem equivalently as a corresponding nonlinear bilevel programming (BLP) problem. For the nonlinear BLP problem, the duality gap of the lower level problem is appended to the upper level objective with a penalty, and then a penalized problem is obtained. On the basis of the concept of partial calmness, we prove that the penalty function is exact. Then, an algorithm is proposed and an inverse optimal value problem is resolved to illustrate the algorithm.     

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.     

Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.