Exact penalty functions in nonlinear programming

In this paper some new theoretic results on piecewise differentiable exact penalty functions are presented. Sufficient conditions are given for the existence of exact penalty functions for inequality constrained problems more general than concave and several classes of such functions are presented.This research was partially supported by a grant from the Office of Naval Research; contract number N00014-67-A-0321-0003 (NR047-095).     

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.     

Interior penalty functions and duality in linear programming

Logarithmic additive terms of barrier type with a penalty parameter are included in the Lagrange function of a linear programming problem. As a result, the problem of searching for saddle points of the modified Lagrangian becomes unconstrained (the saddle point is sought with respect to the whole space of primal and dual variables). Theorems on the asymptotic convergence to the desired solution and analogs of the duality theorems for the arising optimization minimax and maximin problems are formulated.     

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.     

Generalized K-T conditions and penalty functions for quasidifferentiable programming

In this paper,a quasidifferentiable programming problem with inequality constraintsis considered. First,a general form of optimality conditions for this problem is glven,which contains the results of Luderer,Kuntz and Scholtes. Next,a new generalized K-T condition is derived. The new optimality condition doesn‘t use Luderer‘s regularity assumption and ita Lagrangian multipliers don‘t depend on the particular elements in the superdifferentials of the object function and constraint functions, Finally,a penalty function for the prohlem is studied. Sufficient conditions of the penalty function attaining a global minimum are obtained.     

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.     

A nonlinear programming algorithm based on non-coercive penalty functions

 We consider first the differentiable nonlinear programming problem and study the asymptotic behavior of methods based on a family of penalty functions that approximate asymptotically the usual exact penalty function. We associate two parameters to these functions: one is used to control the slope and the other controls the deviation from the exact penalty. We propose a method that does not change the slope for feasible iterates and show that for problems satisfying the Mangasarian-Fromovitz constraint qualification all iterates will remain feasible after a finite number of iterations. The same results are obtained for non-smooth convex problems under a Slater qualification condition. Received: September 2000 / Accepted: June 2002 Published online: March 21, 2003 Research partially supported by CAPES, Brazil Research partially supported by CNPq, Brazil, and CONICIT, Venezuela. Mathematics Subject Classification (1991): 20E28, 20G40, 20C20     

Exact penalty functions and calmness for mathematical programming under nonlinear perturbations

In the present paper, the effects of nonlinear perturbations of constraint systems are considered over the relationship between calmness and exact penalization, within the context of mathematical programming with equilibrium constraints. Two counterexamples are provided showing that the crucial link between the existence of penalty functions and the property of calmness for perturbed problems is broken in the presence of general perturbations. Then, some properties from variational analysis are singled out, which are able to restore to a certain extent the broken link. Consequently, conditions on the value function associated to perturbed optimization problems are investigated in order to guarantee the occurrence of the above properties.     

Multiobjective dynamic programming

In this paper the method of dynamic programming is carried to recurrence computations of sets of optimal values for discrete dynamic systems and vector-valued objective functions.     

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 differentiable exact penalty functions

In this paper, a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied. The line search functions used are approximations to Fletcher's differentiable exact penalty function. Global convergence and local superlinear convergence results are proved, and some numerical results are given.     

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.     

Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

Lower bounds of controlling parameters of exact penalty functions in locally Lipschitz programming

In this note, lower bounds of penalty parameters of general exact penalty functions in locally Lipschitz programming are directly derived from Rosenberg's results.     

On solving three classes of nonlinear programming problems via simple differentiable penalty functions

We consider the following classes of nonlinear programming problems: the minimization of smooth functions subject to general constraints and simple bounds on the variables; the nonlinearl ₁-problem; and the minimax problem. Numerically reliable methods for solving problems in each of these classes, based upon exploiting the structure of the problem in constructing simple differentiable penalty functions, are presented.This research was made possible by NSERC Grant No. A8442.The author would like to thank Mrs. J. Selwood of the Department of Combinatories and Optimization, University of Waterloo, Ontario, Canada for her excellent typesetting.This work was carried out in the Department of Combinatories and Optimization, University of Waterloo, Waterloo, Ontario, Canada.     

Multiobjective programming with new invexities

(Φ, ρ)-invexity and (Φ, ρ) ^w -invexity generalize known invexity type properties and have been introduced with the intent of extending most of theoretical results in mathematical programming. Here, we push this approach further, to obtain authentic extensions of previously known optimality and duality results in multiobjective programming.     

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.     

A new class of exact penalty functions and penalty algorithms

For nonlinear programming problems, we propose a new class of smooth exact penalty functions, which includes both barrier-type and exterior-type penalty functions as special cases. We develop necessary and sufficient conditions for exact penalty property and inverse proposition of exact penalization, respectively. Furthermore, we establish the equivalent relationship between these penalty functions and classical simple exact penalty functions in the sense of exactness property. In addition, a feasible penalty function algorithm is proposed. The convergence analysis of the algorithm is presented, including the global convergence property and finite termination property. Finally, numerical results are reported.     

Nonlinear programming without a penalty function

In this paper the solution of nonlinear programming problems by a Sequential Quadratic Programming (SQP) trust-region algorithm is considered. The aim of the present work is to promote global convergence without the need to use a penalty function. Instead, a new concept of a “filter” is introduced which allows a step to be accepted if it reduces either the objective function or the constraint violation function. Numerical tests on a wide range of test problems are very encouraging and the new algorithm compares favourably with LANCELOT and an implementation of Sl₁QP. Received: October 17, 1997 / Accepted: August 17, 2000?Published online September 3, 2001