求解带箱子集约束的非光滑全局优化问题的填充函数方法

提出了一个求解带箱子集约束的非光滑全局优化问题的填充函数方法.构造的填充函数只包含一个参数,且此参数在迭代过程中容易调节.分析了填充函数的理论性质,在此基础上设计了填充函数算法.数值计算验证了该算法的有效性.     

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non‐linear minimization problem, usually with a Tikhonov‐like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal‐to‐noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non‐linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.     

带等式约束的光滑优化问题的一类新的精确罚函数

罚函数方法是将约束优化问题转化为无约束优化问题的主要方法之一. 不包含目标函数和约束函数梯度信息的罚函数, 称为简单罚函数. 对传统精确罚函数而言, 如果它是简单的就一定是非光滑的; 如果它是光滑的, 就一定不是简单的. 针对等式约束优化问题, 提出一类新的简单罚函数, 该罚函数通过增加一个新的变量来控制罚项. 证明了此罚函数的光滑性和精确性, 并给出了一种解决等式约束优化问题的罚函数算法. 数值结果表明, 该算法对于求解等式约束优化问题是可行的.     

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．     

线性与非线性规划算法与理论

线性规划与非线性规划是数学规划中经典而重要的研究方向. 主要介绍该研究方向的背景知识,并介绍线性规划、无约束优化和约束优化的最新算法与理论以及一些前沿与热点问题. 交替方向乘子法是一类求解带结构的约束优化问题的方法,近年来倍受重视. 全局优化是一个对于应用优化领域非常重要的研究方向. 因此也试图介绍这两个方面的一些最新研究进展和问题.     

Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique.     

An efficient feasible SQP algorithm for inequality constrained optimization

In this paper, an efficient feasible SQP method is proposed to solve nonlinear inequality constrained optimization problems. Here, a new modified method is presented to obtain the revised feasible descent direction. Per single iteration, it is only necessary to solve one QP subproblem and a system of linear equations with only a subset of the constraints estimated as active. In addition, its global and superlinear convergence are obtained under some suitable conditions.     

Inexact projected gradient method for vector optimization

In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graña Drummond and Svaiter. In the constrained setting, the method we present extends the exact one proposed by Graña Drummond and Iusem, since it admits relative errors on the search directions. At each iteration, a decrease of the objective value is obtained by means of an Armijo-like rule. The convergence results of this new method extend those obtained by Fukuda and Graña Drummond for the exact version. For partial orders induced by both pointed and nonpointed cones, under some reasonable hypotheses, global convergence to weakly efficient points of all sequences generated by the inexact projected gradient method is established for convex (respect to the ordering cone) objective functions. In the convergence analysis we also establish a connection between the so-called weighting method and the one we propose.     

解线性约束优化问题的新锥模型信赖域法

本文提出了一个解线性等式约束优化问题的新锥模型信赖域方法.论文采用零空间技术消除了新锥模型子问题中的线性等式约束,用折线法求解转换后的子问题,并给出了解线性等式约束优化问题的信赖域方法.论文提出并证明了该方法的全局收敛性,并给出了该方法解线性等式约束优化问题的数值实验.理论和数值实验结果表明新锥模型信赖域方法是有效的,这给出了用新锥模型进一步研究非线性优化的基础.     

Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions

We present a new strategy for the constrained global optimization of expensive black box functions using response surface models. A response surface model is simply a multivariate approximation of a continuous black box function which is used as a surrogate model for optimization in situations where function evaluations are computationally expensive. Prior global optimization methods that utilize response surface models were limited to box-constrained problems, but the new method can easily incorporate general nonlinear constraints. In the proposed method, which we refer to as the Constrained Optimization using Response Surfaces (CORS) Method, the next point for costly function evaluation is chosen to be the one that minimizes the current response surface model subject to the given constraints and to additional constraints that the point be of some distance from previously evaluated points. The distance requirement is allowed to cycle, starting from a high value (global search) and ending with a low value (local search). The purpose of the constraint is to drive the method towards unexplored regions of the domain and to prevent the premature convergence of the method to some point which may not even be a local minimizer of the black box function. The new method can be shown to converge to the global minimizer of any continuous function on a compact set regardless of the response surface model that is used. Finally, we considered two particular implementations of the CORS method which utilize a radial basis function model (CORS-RBF) and applied it on the box-constrained Dixon–Szegö test functions and on a simple nonlinearly constrained test function. The results indicate that the CORS-RBF algorithms are competitive with existing global optimization algorithms for costly functions on the box-constrained test problems. The results also show that the CORS-RBF algorithms are better than other algorithms for constrained global optimization on the nonlinearly constrained test problem.     

解约束非凸规划问题的同伦方法的收敛性定理

本文在利用组合内点同伦方法求解约束非凸规划问题时,得到了一些新的收敛性定理.证明了同伦映射为正则映射的条件下,选取合适的同伦方程,用此同伦方法得到的K-K-T点一定是问题局部最优解.     

A modified Hestenes–Stiefel projection method for constrained nonlinear equations and its linear convergence rate

The Hestenes–Stiefel (HS) method is an efficient method for solving large-scale unconstrained optimization problems. In this paper, we extend the HS method to solve constrained nonlinear equations, and propose a modified HS projection method, which combines the modified HS method proposed by Zhang et al. with the projection method developed by Solodov and Svaiter. Under some mild assumptions, we show that the new method is globally convergent with an Armijo line search. Moreover, the R-linear convergence rate of the new method is established. Some preliminary numerical results show that the new method is efficient even for large-scale constrained nonlinear equations.     

An algorithm for solving global optimization problems with nonlinear constraints

In this paper we propose an algorithm using only the values of the objective function and constraints for solving one-dimensional global optimization problems where both the objective function and constraints are Lipschitzean and nonlinear. The constrained problem is reduced to an unconstrained one by the index scheme. To solve the reduced problem a new method with local tuning on the behavior of the objective function and constraints over different sectors of the search region is proposed. Sufficient conditions of global convergence are established. We also present results of some numerical experiments.     

A trust region affine scaling method for bound constrained optimization

We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

Gauss–Newton methods with approximate projections for solving constrained nonlinear least squares problems

This paper is concerned with algorithms for solving constrained nonlinear least squares problems. We first propose a local Gauss–Newton method with approximate projections for solving the aforementioned problems and study, by using a general majorant condition, its convergence results, including results on its rate. By combining the latter method and a nonmonotone line search strategy, we then propose a global algorithm and analyze its convergence results. Finally, some preliminary numerical experiments are reported in order to illustrate the advantages of the new schemes.     

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.     

On the computation of linearly constrained stationary points

We study a numerical method for the computation of linearly constrained stationary points. The proposed method can be interpreted as a projected gradient method with constant stepsize in which one allows perturbations in the admissible set and controls these perturbations in each iteration. The method is applicable to some classes of overdetermined problems to which the projected gradient method may not be directly applicable. Illustrative numerical examples are given.     

不等式约束优化的非单调可行信赖域-SQP算法

本文讨论不等式约束优化问题,给出一个信赖域方法与SQP方法相结合的新的可行算法,算法中采用了压缩技术,使得QP子问题产生的搜索方向尽可能为可行方向,并且采用了高阶校正的方法来克服算法产生的Maratos效应现象.在适当的条件下,证明了算法的全局收敛性和超线性收敛性.数值结果表明算法是有效的.     

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。