Choosing descent directions in unconstrained minimization problems

Translated from Issledovaniya po Prikladnoi Matematike, No. 9, pp. 37–42, 1981.     

Equivalence of variational inequality problems to unconstrained minimization

In this paper we propose a class of merit functions for variational inequality problems (VI). Through these merit functions, the variational inequality problem is cast as unconstrained minimization problem. We estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of VI. This work was supported by the state key project “Scientific and Engineering Computing”.     

A rank-one algorithm for unconstrained function minimization

A rank-one algorithm is presented for unconstrained function minimization. The algorithm is a modified version of Davidon's variance algorithm and incorporates a limited line search. It is shown that the algorithm is a descent algorithm; for quadratic forms, it exhibits finite convergence, in certain cases. Numerical studies indicate that it is considerably superior to both the Davidon-Fletcher-Powell algorithm and the conjugate-gradient algorithm.     

Generalized conjugate directions for unconstrained function minimization

The theory of unconstrained optimization has led to the development of a large class of iterative methods that reduce to conjugate direction methods when they are applied to quadratic functions. The purpose of this paper is to give a generalization of conjugate directions to nonquadratic functions.     

Global convergence and stabilization of unconstrained minimization methods without derivatives

In this paper, acceptability criteria for the stepsize and global convergence conditions are established for unconstrained minimization methods employing only function values. On the basis of these results, the convergence of an implementable line search algorithm is proved and some global stabilization schemes are described.The authors would like to thank the anonymous referees for their useful suggestions.     

Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization

In this paper, we propose a new nonmonotone Armijo type line search and prove that the MBFGS method proposed by Li and Fukushima with this new line search converges globally for nonconvex minimization. Some numerical experiments show that this nonmonotone MBFGS method is efficient for the given test problems.     

Global convergence of a modified BFGS-type method for unconstrained non-convex minimization

To the unconstrained programme of non-convex function, this article give a modified BFGS algorithm associated with the general line search model. The idea of the algorithm is to modify the approximate Hessian matrix for obtaining the descent direction and guaranteeing the efficacious of the new quasi-Newton iteration equationB _k +1s _k =y _k ^* ,, wherey _k ^* is the sum ofy _k andA _k s _k , andA _k is some matrix. The global convergence properties of the algorithm associating with the general form of line search is proved.     

Smooth minimization of non-smooth functions

In this paper we propose a new approach for constructing efficient schemes for non-smooth convex optimization. It is based on a special smoothing technique, which can be applied to functions with explicit max-structure. Our approach can be considered as an alternative to black-box minimization. From the viewpoint of efficiency estimates, we manage to improve the traditional bounds on the number of iterations of the gradient schemes from keeping basically the complexity of each iteration unchanged.This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Ministers Office, Science Policy Programming. The scientific responsibility is assumed by the author.     

A filled function method with one parameter for unconstrained global optimization

The filled function method is considered as an efficient approach to solve the global optimization problems. In this paper, a new filled function method is proposed. Its main idea is as follows: a new continuously differentiable filled function with only one parameter is constructed for unconstrained global optimization when a minimizer of the objective function is found, then a minimizer of the filled function will be found in a lower basin of the objective function, thereafter, a better minimizer of the objective function will be found. The above process is repeated until the global optimal solution is found. The numerical experiments show the efficiency of the proposed filled function method.     

A new filled function method for unconstrained global optimization

In this paper, a new filled function which has better properties is proposed for identifying a global minimum point for a general class of nonlinear programming problems within a closed bounded domain. An algorithm for unconstrained global optimization is developed from the new filled function. Theoretical and numerical properties of the proposed filled function are investigated. The implementation of the algorithm on seven test problems is reported with satisfactory numerical results.     

A parameter free filled function for unconstrained global optimization

The filled function method is considered as an efficient method to find the global minimum of multidimensional functions. A number of filled functions were proposed recently, most of which have one or two adjustable parameters. However, there is no efficient criterion to choose the parameter appropriately. In this paper, we propose a filled function without parameter. And this function includes neither exponential terms nor logarithmic terms so it is superior to the traditional ones. Theories of the filled function are investigated. And an algorithm which does not compute gradients during minimizing the filled function is presented. Moreover, the numerical experiments demonstrate the efficiency of the proposed filled function.     

A new filled function method for an unconstrained nonlinear equation

In this paper, we transform an unconstrained system of nonlinear equations into a special optimization problem. A new filled function is constructed by employing the special properties of the transformed optimization problem. Theoretical and numerical properties of the proposed filled function are investigated and a solution of the algorithm is proposed. Under some conditions, we can find a solution or an approximate solution to the system of nonlinear equations in finite iterations. The implementation of the algorithm on six test problems is reported with satisfactory numerical results.     

Gauss-Newton-based BFGS method with filter for unconstrained minimization

One class of the lately developed methods for solving optimization problems are filter methods. In this paper we attached a multidimensional filter to the Gauss-Newton-based BFGS method given by Li and Fukushima [D. Li, M. Fukushima, A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal of Numerical Analysis 37(1) (1999) 152-172] in order to reduce the number of backtracking steps. The proposed filter method for unconstrained minimization problems converges globally under the standard assumptions. It can also be successfully used in solving systems of symmetric nonlinear equations. Numerical results show reasonably good performance of the proposed algorithm.     

Global optimization of a nonconvex single facility location problem by sequential unconstrained convex minimization

The problem of maximizing the sum of certain composite functions, where each term is the composition of a convex decreasing function, bounded from below, with a convex function having compact level sets arises in certain single facility location problems with gauge distance functions. We show that this problem is equivalent to a convex maximization problem over a compact convex set and develop a specialized polyhedral annexation procedure to find a global solution for the case when the inside function is a polyhedral norm. As the problem was solved recently only for local solutions, this paper offers an algorithm for finding a global solution. Implementation and testing are not treated in this short communication.An earlier version of this paper appeared in the proceedings of a conference on Recent Advances in Global Optimization, C. Floudas and P. Pardalos, eds., Princeton University Press, 1991.     

GLOBAL COVERGENCE OF THE NON-QUASI-NEWTON METHOD FOR UNCONSTRAINED OPTIMIZATION PROBLEMS

In this paper, the non-quasi-Newton＇s family with inexact line search applied to unconstrained optimization problems is studied. A new update formula for non-quasi-Newton＇s family is proposed. It is proved that the constituted algorithm with either Wolfe-type or Armijotype line search converges globally and Q-superlinearly if the function to be minimized has Lipschitz continuous gradient.     

Global optimization conditions for certain nonconvex minimization problems

By means of suitable dual problems to the following global optimization problems: extremum{f(x): x M X}, wheref is a proper convex and lower-semicontinuous function andM a nonempty, arbitrary subset of a reflexive Banach spaceX, we derive necessary and sufficient optimality conditions for a global minimizer. The method is also applicable to other nonconvex problems and leads to at least necessary global optimality conditions.     

Global minimization algorithms for concave quadratic programming problems

Abstract: In this article, we consider the concave quadratic programming problem which is known to be NP hard. Based on the improved global optimality conditions by [Dür, M., Horst, R. and Locatelli, M., 1998, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217, 637–649] and [Hiriart-Urruty, J.B. and Ledyav, J.S., 1996, A note in the characterization of the global maxima of a convex function over a convex set, Journal of Convex Analysis, 3, 55–61], we develop a new approach for solving concave quadratic programming problems. The main idea of the algorithms is to generate a sequence of local minimizers either ending at a global optimal solution or at an approximate global optimal solution within a finite number of iterations. At each iteration of the algorithms we solve a number of linear programming problems with the same constraints of the original problem. We also present the convergence properties of the proposed algorithms under some conditions. The efficiency of the algorithms has been demonstrated with some numerical examples.     

Supermemory descent methods for unconstrained minimization

The supermemory gradient method of Cragg and Levy (Ref. 1) and the quasi-Newton methods with memory considered by Wolfe (Ref. 4) are shown to be special cases of a more general class of methods for unconstrained minimization which will be called supermemory descent methods. A subclass of the supermemory descent methods is the class of supermemory quasi-Newton methods. To illustrate the numerical effectiveness of supermemory quasi-Newton methods, some numerical experience with one such method is reported.The authors are indebted to Dr. H. Y. Huang for his helpful criticism of this paper.     

Global minimization of a generalized convex multiplicative function

This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.