﻿A Variable Metric Algorithm with Broyden Rank One Modifications for Nonlinear Equality Constraints Optimization

Open Journal of Optimization
Vol.2 No.1(2013), Article ID:29488,5 pages DOI:10.4236/ojop.2013.21005

A Variable Metric Algorithm with Broyden Rank One Modifications for Nonlinear Equality Constraints Optimization

Chunyan Hu1, Zhibin Zhu2

1School of Electronic Engineering and Automation, Guilin University of Electronic Technology, Guilin, China

2School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China

Email: huchyel@hotmail.com

Received January 17, 2013; revised February 18, 2013; accepted March 8, 2013

Keywords: Equality Constrained Optimization; Variable Metric Algorithm; Broyden Rank One Modification; Superlinear Convergence

ABSTRACT

In this paper, a variable metric algorithm is proposed with Broyden rank one modifications for the equality constrained optimization. This method is viewed expansion in constrained optimization as the quasi-Newton method to unconstrained optimization. The theoretical analysis shows that local convergence can be induced under some suitable conditions. In the end, it is established an equivalent condition of superlinear convergence.

1. Introduction & Algorithm

This In this paper, it is proposed to consider the following nonlinear mathematical programming problem:

(1)

where are continuously differentiable functions. Denote the feasible set as follows:

Let be Lagrangian function of (1), and

If is a KKT point pair of Equation (1), then

i.e.,

(2)

where

At the point pair, the Newton’s iteration of (2) is defined as follows:

(3)

Later, a positive definite matrix is replaced for

by a lot of authors to develop some kinds of variable metric methods, such as sequential quadratic programming (SQP) methods [1-5], sequential systems of linear equations (SSLE) algorithms [6-8]. In general, the computational cost of those methods is large.

In this paper, a new variable metric method is presented, in which the following fact is based on: a positive definite matrixis replaced for the matrix

.

In the sequel, we describe the algorithm for the solution of (1). Denote

It is obvious that and from Equation (3), we have

(4)

To the system of linear Equations (4), like unconstrained optimization, is dealt with using Broyden rank one modifications as follows:

(5)

Now, the algorithm for the solution of Equation (1) can be stated as follows:

Algorithm A:

Step 1: Initialization: Given a starting point (i.e.,), and a initial positive definite matrix.

Step 2: Compute. If, stop;

Step 3: Compute;

Step 4: Let, and obtain according to (5). Set. Go back to step 2.

2. Convergence of Algorithm

If the algorithm stops at, then is a KKT point of (1). In the sequel, we suppose that algorithm generates an infinite sequence.

Four basic assumptions are given as follows:

A1 The feasible set is nonempty; The functions

are two-times continuously differentiable;

A2 For all, the vectors are linearly independent;

A3 and are bounded. There exists a KKT point pair, such that is positive definite;

A4 There exists a ball of radius about, where satisfy the Lipschitz condition on.

Lemma 1 [9] Let be continuously differentiable in some open and convex set, and is Lipschitz continuous in, then, it holds that

where is the Lipschitz constant. Moreover,

, it follows that

Lemma 2 [9] Let be continuously differentiable in some open and convex set, and is Lipschitz continuous in. Moreover, assume is invertible for some, then there exist some, such that for all, the fact

implies that

Lemma 3 [9] For operator, which satisfies:

R1 is continuously differentiable on;

R2 There exists a point, such that, and is reversible;

R3 is Lipschitz continuous at, i.e., there exists a constant, such that

Let, where, and it holds that

(6)

then there exist and, when

it is true that is meaning, and is linearly convergent to. Thereby, we can conclude that is superlinearly convergent to, if and only if

(7)

In the sequel, we prove the convergence Theorem as follows:

Theorem 1 If there exist constants and such that

then is meaning, and is linearly convergent to

, thereby and are linearly convergent to and respectively.

Proof: From Lemma 3, we only prove that and satisfy conditions R1, R2, R3 and the inequality (6). From assumption A1, it is obvious that is continuously differentiable. From A3, it holds that, and

While is positive definite, and are linearly independent. So, it is easy to see that is reversible. In addition,

From assumption A4, it holds that is Lipschitz continuous at. In a word, satisfies the conditions R1, R2, R3.

Now, we prove that, for and generated according to (5), (6) is satisfied.

In fact, from (5), we have

(8)

So,

While

and from Lemma 1, we have

(9)

So,

i.e., (6) is true, thereby, from Lemma 3, we get

The claim holds.

Theorem 2 Under above-mentioned assumptions, if, in Theorem 1 satisfy that

(10)

then is superlinearly convergent to, i.e.,

Proof. From Lemma 3, we only prove that and

satisfy (7). Denote, norm, Frobenius norm. From (8),

(11)

Since

So,

(12)

In addition, from (6), (10), using the method of mathematical induction, it is not difficult to prove that

(13)

thereby,

(14)

Thus, from (9), (12), (13), we have

i.e. is convergent, thereby,

From Theorem 2, we don’t conclude that is superlinearly convergent to, i.e.,

In the sequel, we discuss one condition which assures that is superlinearly convergent to.

Lemma 4 Let be a function defined by

where

Then, , if and only if

Proof. Obviously, using the triangle inequality, it holds that

In addition, from A1, we know that is continuously differentiable, and

(15)

It holds that is nonsingular. In fact, let, and

the facts that has full rank, is semi-positive definite and is positive definite imply that

So,

Thereby, from A4 and Lemma 2, there exist , such that

So,

.

The claim holds.

Theorem 3 Under above-mentioned conditions,

is superlinearly convergent toif and only if

and i.e.,

(16)

Proof. The sufficiency: Suppose that (16) holds. We only prove that

From (4), we have

So, the fact implies that

thereby,

From (15), we have

So,

While

In addition, according to Lemma 1, we have

so, from (16), it holds that

The necessity: Suppose that

i.e.,

On one hand,

So, in order to prove that

While

According to Lemma 1, we have

and

So,

On the other hand, the fact implies that

Let, then

It is easy to see that

Thereby,

The claim holds.

3. Acknowledgements

This work was supported in part by NNSF (No. 11061 011) of China and Guangxi Fund for Distinguished Young Scholars (2012GXSFFA060003).

REFERENCES

1. M. J. D. Powell, “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,” In: G. A. Watson, Ed., Numerical Analysis, Springer-Verlag, Berlin, 1978, pp. 144-157. doi:10.1007/BFb0067703
2. S.-P. Han, “Superlinearly Convergent Variable Metric Algorithm for General Nolinear Programming Problem,” Mathematical Programming, Vol. 11, No. 1, 1976, pp. 263-282. doi:10.1007/BF01580395
3. D. Q. Mayne and E. Polak, “A Superlinearly Convergent Algorithm for Constrained Optimization Problems,” Mathematical Programming Studies, Vol. 16, 1982, pp. 45- 61. doi:10.1007/BFb0120947
4. J.-B. Jian, C.-M. Tang, Q.-J. Hu and H.-Y. Zheng, “A Feasible Descent SQP Algorithm for General Constrained Optimization without Strict Complementarity,” Journal of Computational and Applied Mathematics, Vol. 180, No. 2, 2005, pp. 391-412. doi:10.1016/j.cam.2004.11.008
5. Z. Jin and Y. Q. Wang, “A Simple Feasible SQP Method for Inequality Constrained Optimization with Global and Superlinear Convergence,” Journal of Computational and Applied Mathematics, Vol. 233, No. 1, 2010, pp. 3060- 3073. doi:10.1016/j.cam.2009.11.061
6. Y.-F. Yang, D.-H. Li and L. Q. Qi, “A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization,” SIAM Journal on Optimization, Vol. 13, No. 4, 2003, pp. 1222-1244. doi:10.1137/S1052623401383881
7. Z. B. Zhu, “An Interior Point Type QP-Free Algorithm with Superlinear Convergence for Inequality Constrained Optimization,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1201-1212. doi:10.1016/j.apm.2006.04.019
8. J. B. Jian, D. L. Han and Q. J. Xu, “A New Sequential Systems of Linear Equations Algorithm of Feasible Descent for Inequality Constrained Optimization,” Acta Mathematica Sinica, English Series, Vol. 26, No. 12, 2010, pp. 2399-2420. doi:10.1007/s10114-010-7432-0
9. Y. Yuan and W. Sun, “Theory and Methods of Optimization,” Science Press, Beijing, 1997.