On the convergence of a new trust region algorithm

Summary. In this paper we present a new trust region algorithm for general nonlinear constrained optimization problems. The algorithm is based on the exact penalty function. Under very mild conditions, global convergence results for the algorithm are given. Local convergence properties are also studied. It is shown that the penalty parameter generated by the algorithm will be eventually not less than the norm of the Lagrange multipliers at the accumulation point. It is proved that the method is equivalent to the sequential quadratic programming method for all large , hence superlinearly convergent results of the SQP method can be applied. Numerical results are also reported. Received March 21, 1993     

A constraint linearization method for nondifferentiable convex minimization

Summary This paper presents a readily implementable algorithm for solving constrained minimization problems involving (possibly nonsmooth) convex functions. The constraints are handled as in the successive quadratic approximations methods for smooth problems. An exact penalty function is employed for stepsize selection. A scheme for automatic limitation of penalty growth is given. Global convergence of the algorithm is established, as well as finite termination for piecewise linear problems. Numerical experience is reported.Sponsored by Program CPBP 02.15     

Balance function for the optimal control problem

The balance-function concept for transforming constrained optimization problems into unconstrained optimization problems, for the purpose of finding numerical iterative solutions, is extended to the optimal control problem. This function is a combination orbalance between the penalty and Lagrange functions. It retains the advantages of the penalty function, while eliminating its numerical disadvantages. An algorithm is developed and applied to an orbit transfer problem, showing the feasibility and usefulness of this concept.These results are part of the author's doctoral thesis written under Professors H. Lo and D. Alspaugh of Purdue University.     

A new penalty function algorithm for convex quadratic programming

In this paper, we develop an exterior point algorithm for convex quadratic programming using a penalty function approach. Each iteration in the algorithm consists of a single Newton step followed by a reduction in the value of the penalty parameter. The points generated by the algorithm follow an exterior path that we define. Convergence of the algorithm is established. The proposed algorithm was motivated by the work of Al-Sultan and Murty on nearest point problems, a special quadratic program. A preliminary implementation of the algorithm produced encouraging results. In particular, the algorithm requires a small and almost constant number of iterations to solve the small to medium size problems tested.     

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.     

A non-interior continuation method for generalized linear complementarity problems

In this paper, we propose a non-interior continuation method for solving generalized linear complementarity problems (GLCP) introduced by Cottle and Dantzig. The method is based on a smoothing function derived from the exponential penalty function first introduced by Kort and Bertsekas for constrained minimization. This smoothing function can also be viewed as a natural extension of Chen-Mangasarian’s neural network smooth function. By using the smoothing function, we approximate GLCP as a family of parameterized smooth equations. An algorithm is presented to follow the smoothing path. Under suitable assumptions, it is shown that the algorithm is globally convergent and local Q-quadratically convergent. Few preliminary numerical results are also reported. Received September 3, 1997 / Revised version received April 27, 1999?Published online July 19, 1999     

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.     

非线性约束优化的光滑化平方根罚函数

介绍一种非线性约束优化的不可微平方根罚函数,为这种非光滑罚函数提出了一个新的光滑化函数和对应的罚优化问题,获得了原问题与光滑化罚优化问题目标之间的误差估计. 基于这种罚函数,提出了一个算法和收敛性证明,数值例子表明算法对解决非线性约束优化具有有效性.     

High-order Newton-penalty algorithms

Recent efforts in differentiable non-linear programming have been focused on interior point methods, akin to penalty and barrier algorithms. In this paper, we address the classical equality constrained program solved using the simple quadratic loss penalty function/algorithm. The suggestion to use extrapolations to track the differentiable trajectory associated with penalized subproblems goes back to the classic monograph of Fiacco & McCormick. This idea was further developed by Gould who obtained a two-steps quadratically convergent algorithm using prediction steps and Newton correction. Dussault interpreted the prediction step as a combined extrapolation with respect to the penalty parameter and the residual of the first order optimality conditions. Extrapolation with respect to the residual coincides with a Newton step.We explore here higher-order extrapolations, thus higher-order Newton-like methods. We first consider high-order variants of the Newton–Raphson method applied to non-linear systems of equations. Next, we obtain improved asymptotic convergence results for the quadratic loss penalty algorithm by using high-order extrapolation steps.     

Exactness and algorithm of an objective penalty function

Penalty function is an important tool in solving many constrained optimization problems in areas such as industrial design and management. In this paper, we study exactness and algorithm of an objective penalty function for inequality constrained optimization. In terms of exactness, this objective penalty function is at least as good as traditional exact penalty functions. Especially, in the case of a global solution, the exactness of the proposed objective penalty function shows a significant advantage. The sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved. Based on the objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the objective penalty function is proved for a local solution. An algorithm is presented in the paper in finding a local solution, with its convergence proved under some conditions. Finally, numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm.     

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.     

General Information

In this paper, a new sequential penalty algorithm, based on the L_infin exact penalty function, is proposed for a general nonlinear constrained optimization problem. The algorithm has the following characteristics: it can start from an arbitrary initial point; the feasibility of the subproblem is guaranteed; the penalty parameter is adjusted automatically; global convergence without any regularity assumption is proved. The update formula of the penalty parameter is new. It is proved that the algorithm proposed in this paper behaves equivalently to the standard SQP method after sufficiently many iterations. Hence, the local convergence results of the standard SQP method can be applied to this algorithm. Preliminary numerical experiments show the efficiency and stability of the algorithm.     

Constrained Nonconvex Nonsmooth Optimization via Proximal Bundle Method

In this paper, we consider a constrained nonconvex nonsmooth optimization, in which both objective and constraint functions may not be convex or smooth. With the help of the penalty function, we transform the problem into an unconstrained one and design an algorithm in proximal bundle method in which local convexification of the penalty function is utilized to deal with it. We show that, if adding a special constraint qualification, the penalty function can be an exact one, and the sequence generated by our algorithm converges to the KKT points of the problem under a moderate assumption. Finally, some illustrative examples are given to show the good performance of our algorithm.     

A solution method for semivectorial bilevel programming problem via penalty method

In this paper, we address a class of semivectorial bilevel programming problem in which the upper level is a scalar optimization problem and the lower level is a linear multi-objective optimization problem. Then, we present a new penalty function method, which includes two different penalty parameters, for solving such a problem. Furthermore, we give a simple algorithm. Numerical examples show that the proposed algorithm is feasible.     

A new Lagrangian net algorithm for solving max-bisection problems

The max-bisection problem is an NP-hard combinatorial optimization problem. In this paper, a new Lagrangian net algorithm is proposed to solve max-bisection problems. First, we relax the bisection constraints to the objective function by introducing the penalty function method. Second, a bisection solution is calculated by a discrete Hopfield neural network (DHNN). The increasing penalty factor can help the DHNN to escape from the local minimum and to get a satisfying bisection. The convergence analysis of the proposed algorithm is also presented. Finally, numerical results of large-scale G-set problems show that the proposed method can find a better optimal solutions.     

Exact barrier function methods for Lipschitz programs

The purpose of this paper is twofold. First we consider a class of nondifferentiable penalty functions for constrained Lipschitz programs and then we show how these penalty functions can be employed to solve a constrained Lipschitz program. The penalty functions considered incorporate a barrier term which makes their value go to infinity on the boundary of a perturbation of the feasible set. Exploiting this fact it is possible to prove, under mild compactness and regularity assumptions, a complete correspondence between the unconstrained minimization of the penalty functions and the solution of the constrained program, thus showing that the penalty functions are exact according to the definition introduced in [17]. Motivated by these results, we propose some algorithm models and study their convergence properties. We show that, even when the assumptions used to establish the exactness of the penalty functions are not satisfied, every limit point of the sequence produced by a basic algorithm model is an extended stationary point according to the definition given in [8]. Then, based on this analysis and on the one previously carried out on the penalty functions, we study the consequence on the convergence properties of increasingly demanding assumptions. In particular we show that under the same assumptions used to establish the exactness properties of the penalty functions, it is possible to guarantee that a limit point at least exists, and that any such limit point is a KKT point for the constrained problem.This research has been partially supported by the National Research Program on Metodi di Ottimizzazione per le Decisioni, Ministero dell' Università e della Ricerca Scientifica e Tecnologica.     

一个解非线性0-1整数规划问题基于罚函数的混合粒子群优化算法

利用罚函数思想把非线性0-1整数规划问题转化为无约束最优化问题,然后把粒子群优化和罚函数方法结合构造出一个基于罚函数的混合粒子群优化算法,数值结果表明所提出的算法是有效的.     

Parallel-machine scheduling with release dates and rejection

In this paper, we consider the parallel-machine scheduling problem with release dates and rejection. A job is either rejected, in which case a rejection penalty has to be paid, or accepted and processed on one of the m identical parallel machines. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. When m is a fixed constant, we provide a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme for the problem. When m is arbitrary, we present a 2-approximation algorithm for the problem.     

An exact penalty method for weak linear bilevel programming problem

In this paper, we present an exact penalty method, which is different from the existing penalty method, for solving weak linear bilevel programming problem. Then, we establish an existence result of solutions for such a problem. Finally, we propose an algorithm and give two examples to illustrate its feasibility.     

Primal and Dual Penalty Methods for Contact Problems with Geometrical Non-linearities

The nonlinearity caused by two or more bodies in contact is often source of computational difficulties. Probably the most popular solution method is based on direct iterations with the non-penetration conditions imposed by the penalty method [1]. The method enables treatment of other non-linearities such as in the case of large displacements. In this paper we are concerned with application of a variant of the FETI domain decomposition method that enforces feasibility of Lagrange multipliers by the penalty [5]. The dual penalty method, which has been shown to be optimal for small displacements, is used in inner loop of the algorithm that treats large displacements. We give results of numerical experiments that demonstrate high efficiency of the FETI method with the dual penalty. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)