Abstract and Applied Analysis

Volume 2012 (2012), Article ID 591612, 12 pages

doi:10.1155/2012/591612

Research Article

Homotopy Method for a General Multiobjective Programming Problem under Generalized Quasinormal Cone Condition

X. Zhao,1,2 S. G. Zhang,² Y. T. Yang,¹ and Q. H. Liu³

¹Department of Mathematics, Beihua University, Jilin 132013, China

²Department of Mathematics, Jilin University, Changchun 130001, China

³Institute of Applied Mathematics, Changchun University of Technology, Changchun 130012, China

Received 14 February 2012; Accepted 24 July 2012

Academic Editor: Irena Rachůnková

Copyright © 2012 X. Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A combined interior point homotopy continuation method is proposed for solving general multiobjective programming problem. We prove the existence and convergence of a smooth homotopy path from almost any interior initial interior point to a solution of the KKT system under some basic assumptions.

1. Introduction

In this paper, for any two vectors and in , we use the following conventions:

We consider the following multiobjective programming problem:

(MOP)

where , , and .

For , let and denote the nonnegative and positive orthant of . Respectively, let

and let

denote the active index set at a given point.

MOP has important application in many practical fields like production planning, structural designing, portfolio selection, and so forth. Research on it can be traced back to Pareto [1], Von Neumann and Morgenstern [2], and Koopmans [3] or even earlier. Especially, more and more attention has been paid to the homotopy method since Kellogg et al. [4], Smale [5], and Chow et al. [6] published the remarkable papers. The homotopy method now becomes an important tool for numerically solving complementary, variational inequalities, convex multiobjective programming, and nonlinear mathematical programming et al. [7–12] as a globally convergent method.

Among many methods, the weighed sum method is popular and efficient. It transforms the MOP to a single-objective programming [13]:

where is the weight vector.

Recently, Song and Yao [14] generalize the combined homotopy interior point method to the general multi-objective programming problem under the so-called normal cone condition instead of the convexity condition about the feasible set. In that paper, they proved the existence of the homotopy path under the following assumptions:

is nonempty and bounded; for all , the vectors are linearly independent; for all , .

In [14], the combined homotopy method was given as follows:

where , , , and . However, the solution simply yields for all . That is, is fixed. In fact, from the last equation, we have . According to this, we know that for all of .

That is, these methods are all solving the single-objective programming problem.

In [15], they present the concept of “positive linear independent” and weaken the assumptions than the ones in [14]. But in order to extend their results to a broader class of nonconvex multi-objective programming problems, we construct a new homotopy equation under generalized quasinorm cone condition in this paper and is not fixed in the calculation process.

The paper is organized as follows. In Section 2, we recall some notations and preliminaries results. In Section 3, we construct a new combined homotopy mapping and prove the existence and convergence of a smooth homotopy path from almost any interior initial point to the KKT points of MOP under some assumptions.  In Section 4, numerical results are given,which show that the method is feasible and effective.

2. Some Definitions and Properties

Definition 2.1. Let be an open set, and let be a smooth mapping. If for all , then is a regular value and is a regular point.

Definition 2.2. Let and . For any is said to be positive linear independent with respect to , if

implies that

where .

Lemma 2.3 (parametric form of the Sard theorem on a smooth manifold; see [16]). Let be smooth manifolds of dimensions . Respectively, let be a map, where . If is a regular value of , then for almost all , 0 is a regular value of .

Lemma 2.4 (inverse image theorem; see [17]). If 0 is a regular value of the mapping , then consists of some smooth manifolds.

Lemma 2.5 (classification theorem of one-dimensional manifold; see [17]). A one-dimensional smooth manifold is diffeomorphic to a unit circle or a unit interval.

The following four basic assumptions are commonly used in this paper:

is nonempty and bounded; for any and , there exists map and , such that is positive linear independent with respect to ; for any ,

(generalized quasinormal cone condition); for any , is nonsingular.

Remark 2.6. If satisfies the assumptions (A₁)–(A₃), then it necessarily satisfies the assumptions (C₁)–(C₄).

In fact, if we choose and , then it is easy to get the result. Clearly, if satisfies the assumptions (C₁)–(C₄), then it does not necessarily satisfies the assumptions (A₁)–(A₃).

3. Main Results

Let be a KKT point of MOP; our aim is to find , such that

where , , .

Meanwhile, the KKT system of MOP is (3.1a)–(3.1c).

For a convex multi-objective programming problem, the solution of the MOP can be obtained from the KKT system. And for a nonconvex multi-objective programming problem, it is significant that we can obtain a solution of the KKT system.

To solve the KKT system (3.1a)–(3.1c), we construct a homotopy equation as follows:

where , , , , , , and .

As , the homotopy equation (3.2) becomes

By the assumption , we get , . Since and , (3.3c) implies that . Equation (3.3d) shows that . That is, with respect to has only one solution .

As , turns to the KKT system (3.1a)–(3.1c).

For a given , rewrite as . The zero-point set of is

Theorem 3.1. Suppose , and are three times continuous differentiable functions. In addition, let the assumptions (C₁)-(C₂) hold and twice times continuously differentiable functions. Then for almost all initial points , 0 is a regular value of and consists of some smooth curves. Among them, a smooth curve, say , is starting from .

Proof. Denote the Jacobi matrix of by . For any and , we have . Now, we consider the submatrix of .

For any ,

where .

We obtain that

That is, 0 is a regular value of . By parametric form of the Sard theorem, for almost all , 0 is a regular value of . By inverse image theorem, consists of some smooth curves. Since , there must be a smooth curve, denoted by , that starts from .

Theorem 3.2. Let assumptions (C₁)-(C₂) hold. For a given , if 0 is a regular value of , then the projection of the smooth curve on the component is bounded.

Proof. Suppose that the conclusion does not hold. Since (0,1] is bounded, there exists a sequence , such that

From the last equality of (3.2), we have

If we assume , this hypothesis implies

Since , , it follows that the second part in the left-hand side of some equations in (3.8) tends to infinity as . But the other two parts are bounded. This is impossible. Thus, the component is bounded.

Theorem 3.3. Let , and be three times continuous differentiable functions. In addition, let the assumptions (C₁)–(C₄) hold and twice times continuously differentiable functions. Then, for almost all of , contains a smooth curve , which starts from . As , the limit set of is nonempty and every point in is a solution of the KKT system (3.1a)–(3.1c).

Proof. From the homotopy equation (3.2), it is easy to see that . By Theorem 3.1, for almost all , 0 is a regular value of and contains a smooth curve starting from . By the classification theorem of one-dimensional smooth manifolds, is diffeomorphic to a unit circle or the unit interval .

Noticing that

Because , and by the assumption , we know that is nonsingular. Therefore, the smooth curve starts from diffeomorphic to .

Let be a limit point of ; only three cases are possible:

; ;.

Because has a unique solution , the case will not happen.

In case (b), because and are bounded sets and by the assumption , for any and , there exists map and such that, is positive linear independent with respect to . From the first equation of (3.2), we get that the component of is bounded.

If case holds, then there exists a sequence , such that

Because and are bounded, there exists a subsequence (denoted also by ) such that

By the third equation of (3.2), we have

Hence, the active index set is nonempty.

From the first equation of (3.2), it follows that

(i) When , rewrite (3.14) as

From the fact that is bounded for and by the assumptions and , when , we observe that

It is easy to see that the right-hand side of the equation is bounded. By the assumption , we have

where .

Then, we have

which contradicts the assumption .

(ii) When , rewrite (3.14) as

We know that, since and , are bounded as , the right-hand side of (3.19) is bounded. But by the assumption , if , then the left-hand side of (3.19) is infinite; this is a contradiction.

As a conclusion, is the only possible case, and is a solution of the KKT system.

Let be the arc-length of . We can parameterize with respect to .

Theorem 3.4. The homotopy path is determined by the following initial-value problem for the ordinary differential equation

The component of the solution point ,   for , is the solution of the KKT system.

4. Algorithm and Numerical Example

Algorithm 4.1. MOP’s Euler-Newton method.

Step 1. Give an initial point , an initial step length , and three small positive numbers . Let .

Step 2. Compute the direction of the predictor step.

(a)Compute a unit tangent vector of at .(b)Determine the direction of the predictor step. If the sign of the determinant is , take . If the sign of the determinant is , take .

Step 3. Compute a corrector point :

where

is the Moore-Penrose inverse of .

 If , let , and go to Step 4. If , let , and go to Step 4. If , let , and go to Step 3.

Step 4. If and , let and go to Step 2.

If and , let , go to Step 3, and recompute for the initial point .

If , let , go to Step 3, and recompute for the initial point .

If , and , then stop.

Example 4.2 (see [9]). Consider

The results are shown in Table 1.

Table 1: Results for Example 4.2.

(1.0000, 2.0000, 0.0000, 1.0000, 1.0000)
,

(−1.0000, −2.0000, 1.0000, 1.0000, 1.0000)
,

Now, we compare the same example with the one in [9] under the same initial point.

(1.0000, 2.0000, 0.0000, 1.0000, 1.0000)
,

(−1.0000, −2.0000, 1.0000, 1.0000, 1.0000)
,

Example 4.3. Consider

Since , it is easy to see that the assumption in [14] and the assumption in [15] are not satisfied at most points in feasible set. Hence, we introduce the functions and .

Let and . It is easily verified that the feasible set satisfies the assumptions (C₁)–(C₄). The results are shown in Table 2.

Table 2: Results for Example 4.3.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant no. 11171003), Key Project of Chinese Ministry of Education (Grant no. 211039), and the Jilin Province Natural Science Foundation (Grant no. 20101597).

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