Nonlinear programming via an exact penalty function: Global analysis

In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.This work is supported in part by NSERC Grant No. A8639 and the U.S. Dept. of Energy.     

Second-order conditions for an exact penalty function

In this paper we give first- and second-order conditions to characterize a local minimizer of an exact penalty function. The form of this characterization gives support to the claim that the exact penalty function and the nonlinear programming problem are closely related.In addition, we demonstrate that there exist arguments for the penalty function from which there are no descent directions even though these points are not minimizers.This research is partially supported by the Natural Science and Engineering Research Council Grant No. A8639 and the U.S. Department of Energy.     

Exact penalty functions in nonlinear programming

It is shown that the existence of a strict local minimum satisfying the constraint qualification of [16] or McCormick's [12] second order sufficient optimality condition implies the existence of a class of exact local penalty functions (that is ones with a finite value of the penalty parameter) for a nonlinear programming problem. A lower bound to the penalty parameter is given by a norm of the optimal Lagrange multipliers which is dual to the norm used in the penalty function.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS74-20584 A02.     

A multiplier method with automatic limitation of penalty growth

This paper presents a multiplier method for solving optimization problems with equality and inequality constraints. The method realizes all the good features that were foreseen by R. Fletcher for this type of algorithm in the past, but which suffers from none of the drawbacks of the earlier attempts.Research sponsored by the National Science Foundation (RANN) Grant ENV76-04264 the National Science Foundation Grant ENG73-08214-A01 and the Sweden-American Foundation.     

A lower bound for the controlling parameters of the exact penalty functions

The purpose of this paper is to present new exact penalty functions and discuss their properties. A lower bound on the controlling parameters is given, for which above this value, the optimum of the exact penalty function coincides with the optimum of the nonlinear programming problem.This work was supported by the National Research Council of Canada under Grant A4414.     

Linear programming via a quadratic penalty function

We use quadratic penalty functions along with some recent ideas from linearl ₁ estimation to arrive at a new characterization of primal optimal solutions in linear programs. The algorithmic implications of this analysis are studied, and a new, finite penalty algorithm for linear programming is designed. Preliminary computational results are presented.Research supported by grant No. 11-0505 from the Danish Natural Science Research Council SNF.     

A globally convergent constrained quasi-Newton method with an augmented lagrangian type penalty function

The recently proposed quasi-Newton method for constrained optimization has very attractive local convergence properties. To force global convergnce of the method, a descent method which uses Zangwill's penalty function and an exact line search has been proposed by Han. In this paper a new method which adopts a differentiable penalty function and an approximate line is presented. The proposed penalty function has the form of the augmented Lagrangian function. An algorithm for updating parameters which appear in the penalty function is described. Global convergence of the given method is proved.     

A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example. This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.     

A two-parameter exact penalty function for nonlinear programming

A sequential quadratic programming algorithm for nonlinear programs using anl -exact penalty function is described. Numerical results are also presented. These results show that the algorithm is competitive with other exact penalty function based algorithms and that the inclusion of the second penalty parameter can be advantageous.     

On differentiable exact penalty functions

In this work, we study a differentiable exact penalty function for solving twice continuously differentiable inequality constrained optimization problems. Under certain assumptions on the parameters of the penalty function, we show the equivalence of the stationary points of this function and the Kuhn-Tucker points of the restricted problem as well as their extreme points. Numerical experiments are presented that corroborate the theory, and a rule is given for choosing the parameters of the penalty function.     

一个关于二次规划问题的分段线性同伦算法

本文发展了一个关于二次规划问题的分段线性同伦算法。该算法可看作是外点罚函数法的一个变体。凡是符合外点罚函数法收敛条件的二次规划问题用该算法均可经有限次轮回运算得到稳定解。大量的关于随机的凸二次规划问题的数值实验结果表明它的计算效率是高的，在某些条件下可能是多项式时间算法。     

Extensions of algorithmic senstivity calculations nonlinear programming using barrier and penalty functions

We provide a theoretical basis for approximating the sensitivity of a perturbed solution and the local optimalvalue function, using information generated by a sequential unconstrained minimization technique in the normal course of solving a mathematical program. We show that various algorithmic sensitivity results can be obtained without other assumptions than those needed for the corresponding nonalgorithmic results. Our results extend the algorithmic calculation of sensitivity information introduced by Fiacco, utilizing the logarithmic barrier function and quadratic penalty function     

A RECURSIVE QUADRATIC PROGRAMMING ALGORITHM THAT USES A NEW NONDIFFERENTIABLE PENALTY FUNCTIONS

In this paper, a recursive quadratic programming algorithm is proposed and studied. The line search functions used axe Han‘s nondifferentiable penalty functions with a second oeder penalty term. In order to avoid maratos effect Fukushima‘s mixed direction is used as the direction of line search. Finally, we prove the global convergence and the local second order convergence of the algorithm.     

Path-following barrier and penalty methods for linearly constrained problems

In the present paper some barrier and penalty methods (e.g. logarithmic barriers, SUMT, exponential penalties), which define a continuously differentiable primal and dual path, applied to linearly constrained convex problems are studied, in particular, the radius of convergence of Newton’s method depending on the barrier and penalty para-meter is estimated, Unlike using self-concordance properties the convergence bounds are derived by direct estimations of the solutions of the Newton equations. The obtained results establish parameter selection rules which guarantee the overall convergence of the considered barrier and penalty techniques with only a finite number of Newton steps at each parameter level. Moreover, the obtained estimates support scaling method which uses approximate dual multipliers as available in barrier and penalty methods     

Second order algorithms for the posynomial geometric programming dual,part I: Analysis

In this paper we analyse algorithms for the geometric dual of posynomial programming problems, that make explicit use of second order information. Out of two possible approaches to the problem, it is shown that one is almost always superior. Interestingly enough, it is the second, inferior approach that has dominated the geometric programming literature.This work was partially supported by the National Research Council of Canada, Grant A3552 and National Science Foundation Grant ENG78-21615.Earlier versions of this paper were presented at the Optimization Days Conference in Montreal (May 1976) and the TIMS meeting in Athens (July 1977).     

A note on solving nonlinear minimax problems via a differentiable penalty function

The note demonstrates that modeling a nonlinear minimax problem as a nonlinear programming problem and applying a classical differentiable penalty function to attempt to solve the problem can lead to convergence to a stationary point of the penalty function which is not a feasible point of the nonlinear programming problem. This occurred naturally in an application from statistical reliability theory. The note resolves the problem through modification of both the problem formulation and the iterative penalty function method.     

A polynomially bounded algorithm for a singly constrained quadratic program

This paper presents a characterization of the solutions of a singly constrained quadratic program. This characterization is then used in the development of a polynomially bounded algorithm for this class of problems.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming

One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal–dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.