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 smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

An exact penalty on bilevel programs with linear vector optimization lower level

We are interested in a class of linear bilevel programs where the upper level is a linear scalar optimization problem and the lower level is a linear multi-objective optimization problem. We approach this problem via an exact penalty method. Then, we propose an algorithm illustrated by numerical examples.     

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.     

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.     

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.     

Double penalty method for bilevel optimization problems

A penalty function method approach for solving a constrained bilevel optimization problem is proposed. In the algorithm, both the upper level and the lower level problems are approximated by minimization problems of augmented objective functions. A convergence theorem is presented. The method is applicable to the non-singleton lower-level reaction set case. Constraint qualifications which imply the assumptions of the general convergence theorem are given.A part of this paper was presented in a talk at the 11th Symposium on Mathematical Programming with Data Perturbations, Washington, DC, May 1989.     

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.     

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.     

Semivectorial Bilevel Optimization Problem: Penalty Approach

We consider a bilevel optimization problem where the upper level is a scalar optimization problem and the lower level is a vector optimization problem. For the lower level, we deal with weakly efficient solutions. We approach our problem using a suitable penalty function which vanishes over the weakly efficient solutions of the lower-level vector optimization problem and which is nonnegative over its feasible set. Then, we use an exterior penalty method.Communicated by H. P. Benson(Formerly Serban Bolintinéanu) Professor, University of New Caledonia, ERIM, Nouméa, New Caledonia. This author thanks the University of Naples Federico II for its support and the Department of Mathematics and Statistics for its hospitality.     

An improved penalty function method for solving constrained parameter optimization problems

An effective algorithm is described for solving the general constrained parameter optimization problem. The method is quasi-second-order and requires only function and gradient information. An exterior point penalty function method is used to transform the constrained problem into a sequence of unconstrained problems. The penalty weightr is chosen as a function of the pointx such that the sequence of optimization problems is computationally easy. A rank-one optimization algorithm is developed that takes advantage of the special properties of the augmented performance index. The optimization algorithm accounts for the usual difficulties associated with discontinuous second derivatives of the augmented index. Finite convergence is exhibited for a quadratic performance index with linear constraints; accelerated convergence is demonstrated for nonquadratic indices and nonlinear constraints. A computer program has been written to implement the algorithm and its performance is illustrated in fourteen test problems.     

On the application of the residual method for the correction of inconsistent problems of convex programming

For the correction of a convex programming problem with potentially inconsistent constraint system (an improper problem), we apply the residual method, which is a standard regularization procedure for ill-posed optimization models. A problem statement typical for the residual method is reduced to a minimization problem for an appropriate penalty function. We apply two classical penalty functions: the quadratic penalty function and the exact Eremin-Zangwill penalty function. For each of the approaches, we establish convergence conditions and bounds for the approximation error.     

On smoothing exact penalty functions for nonlinear constrained optimization problems

In the paper, we give a smoothing approximation to the nondifferentiable exact penalty function for nonlinear constrained optimization problems. Error estimations are obtained among the optimal objective function values of the smoothed penalty problems, of the nonsmooth penalty problem and of the original problem. An algorithm based on our smoothing function is given, which is showed to be globally convergent under some mild conditions.     

On solving simple bilevel programs with a nonconvex lower level program

In this paper, we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint. By using the value function of the lower level program, we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constraint. To deal with such a nonsmooth and nonconvex optimization problem, we design a smoothing projected gradient algorithm for a general optimization problem with a nonsmooth inequality constraint and a convex set constraint. We show that, if the sequence of penalty parameters is bounded then any accumulation point is a stationary point of the nonsmooth optimization problem and, if the generated sequence is convergent and the extended Mangasarian-Fromovitz constraint qualification holds at the limit then the limit point is a stationary point of the nonsmooth optimization problem. We apply the smoothing projected gradient algorithm to the bilevel program if a calmness condition holds and to an approximate bilevel program otherwise. Preliminary numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

不等式约束优化问题的低阶精确罚函数的光滑化算法

对不等式约束优化问题提出了一个低阶精确罚函数的光滑化算法. 首先给出了光滑罚问题、非光滑罚问题及原问题的目标函数值之间的误差估计,进而在弱的假
设之下证明了光滑罚问题的全局最优解是原问题的近似全局最优解. 最后给出了一个基于光滑罚函数的求解原问题的算法,证明了算法的收敛性,并给出数值算例说明算法的可行性.     

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.     

An exact penalty function method for nonlinear mixed discrete programming problems

In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.     

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.     

An augmented penalty function method with penalty parameter updates for nonconvex optimization

Given an augmented Lagrangian scheme for a general optimization problem, we use an epsilon subgradient step for improving the dual function. This can be seen as an update for an augmented penalty method, which is more stable because it does not force the penalty parameter to tend to infinity. We establish for this update primal-dual convergence for our augmented penalty method. As illustration, we apply our method to the test-bed kissing number problem.     

低阶精确罚函数的一种光滑化逼近

本文对不等式约束优化问题给出了低阶精确罚函数的一种光滑化逼近.提出了通过搜索光滑化后的罚问题的全局解而得到原优化问题的近似全局解的算法.给出了几个数值例子以说明所提出的光滑化方法的有效性.