Elsevier

Applied Mathematics and Computation

Volume 226, 1 January 2014, Pages 198-211
Applied Mathematics and Computation

Stochastic perturbation of reduced gradient & GRG methods for nonconvex programming problems

https://doi.org/10.1016/j.amc.2013.10.024Get rights and content

Abstract

In this paper, we consider nonconvex differentiable programming under linear and nonlinear differentiable constraints. A reduced gradient and GRG (generalized reduced gradient) descent methods involving stochastic perturbation are proposed and we give a mathematical result establishing the convergence to a global minimizer. Numerical examples are given in order to show that the method is effective to calculate. Namely, we consider classical tests such as the statistical problem, the octagon problem, the mixture problem and an application to the linear optimal control servomotor problem.

Introduction

We consider the following nonconvex programming problemminimizef(X)subject tog(X)=0XRn,where f:RnR and g:RnRm are continuously differentiable (for nonsmooth case, see for instance [7]).

Feasible methods for constrained optimization, such as, for instance, reduced gradient and GRG play an important role in practicing, and are extensively used in technological applications [23]. Nevertheless, nonconvexity introduces difficulties and the standard methods do not ensure convergence to aglobal minimizer. When dealing with nonconvex optimization, an alternative is the transformation of local descent methods into global ones by the adjonction of random perturbations (see, for instance, [24], [26], [6]). Taking into account constraints involve supplementary difficulties (see, for instance, [27], [7]). In this work, we are mainly interested in the situations where, on one hand, f is nonconvex and, on the other hand, nonlinear constraints are involved. Here, the main techniques considered for these situations are reduced gradient method or GRG method via stochastic perturbation. It is worth noticing that some variants of the generalized gradient method reduce, in the case where all the constraints are linear, to the reduced gradient method, and some other variant, in the case of linear programming, to the Dantzig simplex method.

Numerical solutions for the problem (1) are often obtained by using sequential quadratic programming (see, for instance, [21]); other methods for nonlinear programming (see, for instance [16], [22]); GRG method (see, for instance, [13]). All these methods generate a sequence {Xk}k0, where X0 is an initial feasible point and, for each k>0, a new feasible point Xk+1 is obtained from Xk by using an operator Qk (see Section 3). Thus, these methods correspond to iterations having the form:Xk+1=Qk(Xk),k=0,1,As previously observed, a fundamental difficulty arises due to the lack of convexity; the convergence of the sequence {Xk}k0 to a global minimizer is not ensured in the general situation under consideration. In order to prevent convergence to a local minimum, various modifications of these basic methods have been introduced in the literature. For instance, we can find in the literature modifications of the basic descent methods [5], [26], [27], [6], [7]; stochastic methods combined to penalty functions [25] and simulated annealing [3].

We introduce in this paper a different approach, inspired from the method of stochastic perturbations introduced in [24] for unconstrained minimization of continuously differentiable functions and adapted to linearly constrained problems in [7].

In such a method, the sequence of real vectors {Xk}k0Rn is replaced by a random vectors sequence of random vectors {Xˆk}k0 taking their values on Rn and the iterations are modified as follows:Xˆk+1=Qk(Xˆk)+Pk,k=0,1,where Pk is a suitable random variable, usually referred as the stochastic perturbation. The sequence {Pk}k0 must converge to zero slowly enough in order to prevent convergence of the sequence {Xˆk}k0 to a local minimum (see Section 4).

The reduced gradient method and GRG method are recalled in Section 3, the notations are introduced in Section 2 and the results of some numerical experiments are given in Section 5.

Section snippets

Notations and assumptions

We denote by:

  • E=Rn, the n-dimensional positive real Euclidean space,

  • X stands for (x1,,xn)tE,

  • g(X) is the column vector whose components are g1(X),,gm(X)t,

  • X=(x12++xn2)1/2 the Euclidean norm of X,A=sup{AX:X=1}

  • At the transpose matrix associated to A.

LetC={XE|g(X)=0}.The objective function is f:ER, its upper bound on C is denoted by l:l=maxCf.Let us introduceCα=SαC;whereSα={XE|f(X)α}.We assume thatfis twice continuously differentiable onEα<l:Cαis not empty,closed and boundedα<l

Reduced gradient method

Let us start by considering linear constraints. We refer to the following version of the problem (1) involving linear constraints:minimizef(X),subject toAX=b,X0,where A is m×n matrix and bRm. If necessary, we add the artificial variables in order to obtain a matrix A with full row rank. Thus, we assume that any m columns of A are linearly independent, and every extreme point of the feasible region has m strictly positives variables.

We introduce basic and nonbasic variables according toA=[B,N]X

Stochastic perturbation of reduced gradient & GRG methods

Let us recall thatmax{f(x):xC}=-min{-f(x):xC}.This equality implies that we may consider the determination of maximizers instead of minimizers. In addition, f is concave if and only if -f is convex. Thus, it is equivalent to consider the minimization of -f under convexity assumptions and the maximization of f under concavity assumptions. In addition, we may consider an ascent direction dk at step k, instead of descent directions -dk.

In order to give a formal proof of convergence, we consider

Numerical results

In order to apply the method presented in (27), we start at the initial value Xˆ0=X0C. At step k0,Xˆk is known and Xˆk+1 is determined.

We generate ksto the number of perturbation, the iterations are stopped when Xk is a Kuhn–Tucker point. We denote by kend the value of k when the iterations are stopped (it corresponds to the number of evaluations of the gradient of f). The optimal value and optimal point are fopt and Xopt respectively. The perturbation is normally distributed and samples are

Concluding remarks

A reduced gradient & GRG methods are presented for linear and nonlinear constraints, involving the adjunction of a stochastic perturbation. This approach leads to a stochastic descent method where the deterministic sequence generated by the reduced gradient & GRG methods are replaced by a sequence of random variables.

Stochastic perturbation of reduced gradient & GRG methods converge to global minimum for all differential objective function, but reduced gradient & GRG methods converge to local

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