一种基于步长的SQP滤子法

本文将滤子法用到SQP方法中,通过减小违反约束度函数值和一个逼近目标函数的函数值来确定试探步是否被滤子接受.此方法不同于其他滤子法,它不需要减小信赖域半径而是通过改变步长因子来保证充分下降性.本文在一定条件下获得了全局收敛性.     

一类修改的带NCP函数信赖域滤子算法

本文针对非线性规划给出了一种修改的带NCP函数的信赖域滤子SQP算法,主要的修改之处是用NCP函数替代了滤子中约束违反度函数,而且进一步证明了这种修改的算法同样具有全局收敛性.     

一种求解非线性约束优化问题的无罚函数无滤子的方法

借助于强次可行方向法的思想和滤子法的思想,给出了一种求解非线性约束优化问题的无罚函数无滤子的方法.方法借助于广义投影技术产生搜索方向,直接通过原目标函数和约束违反度函数作为搜索函数来产生步长,有效地避免了消耗计算成本的恢复阶段.最后在适当的假设条件下,给出了算法的全局收敛性和有效性.     

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.     

一类求解非线性约束优化问题的线搜索渐缩滤子算法（英文）

针对非线性等式约束优化问题,本文给出一种新的线搜索滤子算法.算法中将非线性等式约束优化问题的最优性条件作为滤子,并在接受准则中加入渐缩函数,使得当线搜索试探步长减小时时滤子包络的越来越薄,从而使得试探步被接受程度更有弹性,不会被当前的迭代点拒绝.在适当的假设下,证明算法的全局收敛性,并给出算法初步的数值实验结果.     

不等式约束优化全局收敛的滤子线性方程组算法

设计了求解不等式约束非线性规划问题的一种新的滤子序列线性方程组算法,该算法每步迭代由减小约束违反度和目标函数值两部分构成.利用约束函数在某个中介点线性化的方法产生搜索方向.每步迭代仅需求解两个线性方程组,计算量较小.在一般条件下,证明了算法产生的无穷迭代点列所有聚点都是可行点并且所有聚点都是所求解问题的KKT点.     

带等式约束二次规划子问题的滤子SQP算法

一类求解非线性规划问题的滤子序列二次规划(SQP)方法被提出.为了提高收敛速度,给目标函数和约束违反度函数都设置了斜边界.二次规划子问题(QP)设置为两项:不等式约束QP和等式约束QP.两个子问题产生的搜索方向进行线性迭加后为算法的搜索方向.这样的设置可以改善收敛性,并调节算法运行中的一些不良效果.在较温和的条件下,可得到全局收敛性.     

解约束优化问题的相容SQP滤子方法

提出了解约束优化问题的一类相容SQP滤子算法.利用序列二次规划方法结合信赖域技术计算试探步,而用滤子接受准则选择接受试探步.对二次规划子问题的不相容问题,应用Powell1978年于文[9]提出的方法对其约束引进参数进行了可行化处理.在一般条件下,算法具有全局收敛性.最后,数值试验显示了较好的结果.     

一类可行的两阶段SQP方法

提出了解约束优化问题的一类可行的两阶段SQP滤子算法.利用一类两阶段序列二次规划方法计算试探步,而用滤子接受准则选择接受试探步.针对二次规划子问题的不可行问题,对其约束引进参数进行了可行化处理,可以省略可行恢复项,节省了计算时间.在一般条件下,算法具有全局收敛性.最后,数值试验显示了较好的结果.     

求解极小极大问题的非单调过滤算法

提出一种改进的求解极小极大问题的信赖域滤子方法,利用SQP子问题来求一个试探步,尾服用滤子来衡量是否接受试探步,避免了罚函数的使用;并且借用已有文献的思想, 使用了Lagrange函数作为效益函数和非单调技术,在适当的条件下,分析了算法的全局和局部收敛性,并进行了数值实验.     

Sequential penalty quadratic programming filter methods for nonlinear programming

Filter approaches, initially proposed by Fletcher and Leyffer in 2002, are recently attached importance to. If the objective function value or the constraint violation is reduced, this step is accepted by a filter, which is the basic idea of the filter. In this paper, the filter approach is employed in a sequential penalty quadratic programming (SlQP) algorithm which is similar to that of Yuan's. In every trial step, the step length is controlled by a trust region radius. In this work, our purpose is not to reduce the objective function and constraint violation. We reduce the degree of constraint violation and some function, and the function is closely related to the objective function. This algorithm requires neither Lagrangian multipliers nor the strong decrease condition. Meanwhile, in our SlQP filter there is no requirement of large penalty parameter. This method produces K-T points for the original problem.     

一种修改的非单调线搜索SQP算法

提出了一种解非线性规划问题的修改的非单调线搜索算法,并给出了它的全局收敛性证明.不需要用罚函数作为价值函数,也不用滤子和可行性恢复阶段.该算法是基于多目标优化的思想一个迭代点被接受当且仅当目标函数值或是约束违反度函数值有充分的下降.数值结果与LANCELOT作了比较,表明该算法是可靠的.     

非线性半定规划的逐次线性化柔性惩罚法

针对非线性不等式约束半定规划问题提出一种新的逐次线性化方法, 新算法既不要求罚函数单调下降, 也不使用过滤技巧, 尝试步的接受准则仅仅依赖于目标函数和约束违反度, 罚函数中对应于成功迭代点的罚因子不需要单调增加. 新算法或者要求违反约束度量有足够改善, 或者在约束违反度的一个合理范围内要求目标函数值充分下降, 在通常假设条件下, 分析了新算法的适定性及全局收敛性. 最后, 给出了非线性半定规划问题的数值试验结果, 结果表明了新算法的有效性.     

Global and local convergence of a class of penalty-free-type methods for nonlinear programming

We present a class of trust region algorithms without using a penalty function or a filter for nonlinear inequality constrained optimization and analyze their global and local convergence. In each iteration, the algorithms reduce the value of objective function or the measure of constraints violation according to the relationship between optimality and feasibility. A sequence of steps focused on improving optimality is referred to as an f-loop, while some restoration phase focuses on improving feasibility and is called an h-loop. In an f-loop, the algorithms compute trial step by solving a classic QP subproblem rather than using composite-step strategy. Global convergence is ensured by requiring the constraints violation of each iteration not to exceed an progressively tighter bound on constraints violation. By using a second order correction strategy based on active set identification technique, Marato’s effect is avoided and fast local convergence is shown. The preliminary numerical results are encouraging.     

非线性半定规划一个全局收敛的无罚无滤子SSDP算法

提出了一个求解非线性半定规划的无罚函数无滤子序列二次半定规划(SSDP)算法. 算法每次迭代只需求解一个二次半定规划子问题确定搜索方向; 非单调线搜索保证目标函数或约束违反度函数的充分下降, 从而产生新的迭代点. 在适当的假设条件下, 证明了算法的全局收敛性. 最后给出了初步的数值实验结果.     

A penalty-free method with line search for nonlinear equality constrained optimization

A new line search method is introduced for solving nonlinear equality constrained optimization problems. It does not use any penalty function or a filter. At each iteration, the trial step is determined such that either the value of the objective function or the measure of the constraint violation is sufficiently reduced. Under usual assumptions, it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible, and there exists at least one limit point that is a stationary point for the problem. A simple modification of the algorithm by introducing second order correction steps is presented. It is shown that the modified method does not suffer from the Maratos’ effect, so that it converges superlinearly. The preliminary numerical results are reported.     

Global and local convergence of a nonmonotone SQP method for constrained nonlinear optimization

In this paper, we propose a robust sequential quadratic programming (SQP) method for nonlinear programming without using any explicit penalty function and filter. The method embeds the modified QP subproblem proposed by Burke and Han (Math Program 43:277–303, 1989) for the search direction, which overcomes the common difficulty in the traditional SQP methods, namely the inconsistency of the quadratic programming subproblems. A non-monotonic technique is employed further in a framework in which the trial point is accepted whenever there is a sufficient relaxed reduction of the objective function or the constraint violation function. A forcing sequence possibly tending to zero is introduced to control the constraint violation dynamically, which is able to prevent the constraint violation from over-relaxing and plays a crucial role in global convergence and the local fast convergence as well. We prove that the method converges globally without the Mangasarian–Fromovitz constraint qualification (MFCQ). In particular, we show that any feasible limit point that satisfies the relaxed constant positive linear dependence constraint qualification is also a Karush–Kuhn–Tucker point. Under the strict MFCQ and the second order sufficient condition, furthermore, we establish the superlinear convergence. Preliminary numerical results show the efficiency of our method.     

Approximate Greatest Descent Methods for Optimization with Equality Constraints

In an optimization problem with equality constraints the optimal value function divides the state space into two parts. At a point where the objective function is less than the optimal value, a good iteration must increase the value of the objective function. Thus, a good iteration must be a balance between increasing or decreasing the objective function and decreasing a constraint violation function. This implies that at a point where the constraint violation function is large, we should construct noninferior solutions relative to points in a local search region. By definition, an accessory function is a linear combination of the objective function and a constraint violation function. We show that a way to construct an acceptable iteration, at a point where the constraint violation function is large, is to minimize an accessory function. We develop a two-phases method. In Phase I some constraints may not be approximately satisfied or the current point is not close to the solution. Iterations are generated by minimizing an accessory function. Once all the constraints are approximately satisfied, the initial values of the Lagrange multipliers are defined. A test with a merit function is used to determine whether or not the current point and the Lagrange multipliers are both close to the optimal solution. If not, Phase I is continued. If otherwise, Phase II is activated and the Newton method is used to compute the optimal solution and fast convergence is achieved.     

Worst-Case Violation of Sampled Convex Programs for Optimization with Uncertainty

A deterministic approach called robust optimization has been recently proposed to deal with optimization problems including inexact data, i.e., uncertainty. The basic idea of robust optimization is to seek a solution that is guaranteed to perform well in terms of feasibility and near-optimality for all possible realizations of the uncertain input data. To solve robust optimization problems, Calafiore and Campi have proposed a randomized approach based on sampling of constraints, where the number of samples is determined so that only a small portion of the original constraints is violated by the randomized solution. Our main concern is not only the probability of violation, but also the degree of violation, i.e., the worst-case violation. We derive an upper bound of the worst-case violation for the sampled convex programs and consider the relation between the probability of violation and the worst-case violation. The probability of violation and the degree of violation are simultaneously bounded by a prescribed value when the number of random samples is large enough. In addition, a confidence interval of the optimal value is obtained when the objective function includes uncertainty. Our method is applicable to not only a bounded uncertainty set but also an unbounded one. Hence, the scope of our method includes random sampling following an unbounded distribution such as the normal distribution.