Method For Nonlinear Programming Problems Research Articles

A Family of Descent Functions for Constrained Optimization

In order to achieve a robust implementation of methods for nonlinear programming problems, it is necessary to devise a procedure which can be used to test whether or not a prospective step would yield a “better” approximation to the solution than the current iterate. In this paper, we present a family of descent or merit functions which are shown to be compatible with local Q-superlinear convergence of Newton and quasi-Newton methods. A simple algorithm is used to verify that good descent and convergence properties are possible using this merit function.

On the numerical feasibility of continuation methods for nonlinear programming problems

In the present paper the radius of convergence of a class of locally convergent nonlinear programming algorithms (containing Robinson's and Wilson's methods) applied to a parametric nonlinear programming problem is estimated. A consequence is the numerical feasibility of globalizations of Robinson's and Wilson's methods by means of continuation techniques.

Revisions of constraint approximations in the successive QP method for nonlinear programming problems

In the last few years the successive quadratic programming methods proposed by Han and Powell have been widely recognized as excellent means for solving nonlinea programming problems. However, there remain some questions about their linear approximations to the constraints from both theoretical and empirical points of view. In this paper, we propose two revisions of the linear approximation to the constraints and show that the directions generated by the revisions are also descent directions of exact penalty functions of nonlinear programming problems. The new technique can cope better with bad starting points than the usual one.

Considerations of a Penalty Shifting Method for Nonlinear Programming Problems

Ordinary Lagrange function and penalty function methods are typical methods for solving nonlinear programing problems with constraints. The application of the ordinary Lagrange function method is in general limited to convex programing problems. The penalty function method can be applied to nonconvex programing problems, but may suffer from numerical difficulties. The penalty shifting method is superior to the above two methods, since it can be applied to nonconvex programing problems without suffering from numerical difficulties. In this paper two geometric interpretations for the penalty shifting method are mainly given, so that its characteristics may be more clarified. First, the correction rules of two parameters and the relation between one of the parameters and the existence of a saddle point are investigated by the ordinary geometric interpretation. Secondly, the operation of the parameters is also investigated by a new geometric interpretation, and finally, the result for a nonconvex programing problem is illustrated by two geometric interpretations.

A second-order method for the general nonlinear programming problem

This paper presents a multiplier-type method for nonlinear programming problems with both equality and inequality constraints. Slack variables are used for the inequalities. The penalty coefficient is adjusted automatically, and the method converges quadratically to points satisfying second-order conditions.

Zur methode der approximierenden optimiernng

To meet difficulties of convergence and “stepsize-choosing” within the Method of Approximation Programming (MAP) of GRIFFITH and STEWART a modified approximation method for nonlinear programming problems is given. It is proved to be convergent for convex problems. The running times and gained numerical experience is discussed for numerous test problems.

On penalty function methods for nonlinear programming problems

1, 2,...,p (1.1) by transforming it into a sequence of unconstrained minimisation problems. In the form considered here this approach is due to Carrol [l]. This work has been extensively developed by Fiacco and McCormick whose book [2] can be taken as defining the present state of the subject. Our main aim is to study questions relating to the rate of convergence of penalty function algorithms. This work is the subject of the second section of the paper, and it is illustrated numerically in the third. Let R = {x;fi(x) > 0,

Ableitungsfreiebleitungsfreie Verfahren für nichtlineare Optimierungsprobleme

In the paper the authors develop approximation methods for nonlinear programming problems using function-values only. The idea is to approximate the objective function by a quadratic interpolating polynomial. It is shown how to construct methods of an arbitrarily high convergence rate. By aid of the efficiency index methods are selected from the class in view which require a minimal number of function-values in order to guarantee a given exactness.

A decomposition method for structured linear and nonlinear programs

A decomposition method for nonlinear programming problems with structured linear constraints is described. The structure of the constraint matrix is assumed to be block diagonal with a few coupling constraints or variables, or both. The method is further specialized for linear objective functions. An algorithm for performing post optimality analysis—ranging and parametric programming—for such structured linear programs is included. Some computational experience and results for the linear case are presented.

Conjugate Gradient Method for Nonlinear Programming Problems with Linear Constraints

Conjugate Gradient Method for Nonlinear Programming Problems with Linear Constraints