A General Class of Convexification Transformation for the Noninferior Frontier of a Multiobjective Program

A general class of convexification transformations is proposed to convexify the noninferior frontier of a multiobjective program. We prove that under certain assumptions the noninferior frontier could be convexified completely or partly after transformation and then weighting method can be applied to identify the noninferior solutions. Numerical experiments are given to vindicate our results.


Introduction
In this paper, we consider the following multiobjective optimization problem: min , : . .0, 1 k j , 2, , , where k > 1 and x R  2 i is a decision vector, f C  2 j , are objective functions and 1, 2, , , be the feasible region in the decision space and be the feasible region in the objective space.A solution x  to Problem (P) is called noniferior solution if there is no other feasible solution x such that and is noninfe X x x X    be the set of all the noninferior solutions in the decision space and be the set of noninferior points in the objective space and . .0, 1, 2, , , is also called the noninferior frontier of Problem (P).An important problem in multiobjective optimization is to find the set of noninferior solutions.Many methods that are intended to identify noninferior solutions have been proposed such as the weighting method, weighting p-norm method, the ∞-norm method and the ξ-constraint method.Among these methods the weighting method is one of the simplest methods.In fact, the weighting method transforms multiple objectives into the following weighted sum by introducing weighting vector (w 1 ,•••w k ): is well-known that the optimal solution of Problem (SP) is the noninferior solution of Problem (P).Let  be an optimal solution of Problem (SP) with , then we have .By the definition of supporting hyperplane we know that there exists a supporting hyperplane of . Thus the existence of a supporting hyperplane at the noninferior solution in the    -space which separates all the noninferior points one side is a necessary condition to guarantee the successful finding of noninferior solutions of Problem (P) by using weighting method.However, in many nonconvex circumstances, supporting  Copyright © 2013 SciRes.AJOR hyperplane does not exist at some points of the noninferior frontier.Therefore, weighting method always fails to identify all the noninferior solutions in these cases.Recently, convexification method has been successfully adopted in many subjects of optimization.For example, in [1][2][3] a series of convexification methods are proposed to process some classes of global optimization problems with certain monotone properties and in [4,5] convexification schemes are presented to convexify the perturbation function and Lagrangian function in the dual search methods for nonlinear programming.In [6,7], a general convexification and concavification scheme are proposed for certain classes of monotone and nonmonotone optimization problems.The scheme converts the problems into classes of concave and reverse convex programming problems with better structures.Li et al derived a general convexification method for nonconvex minimization problems in [8].Their method transforms the problems into convex ones and thus the local techniques can be used to solve the new problems.A reciprocal transformation for the convexification of posynomial programs with positive variables are presented in [9].In [10][11][12], p-power and exponential generating method were used as a special convexification transformations and they proved that under certain assumptions, by applying the p-power or exponential generating method to objective function, the noninferior frontier of a multiobjective problem can be convexified completely or partly and then the weighting method can be applied to identify the noninferior solutions.However, due to the various forms of objective functions, p-power might not always serve as an efficient transformation.Thus the choice range of such transformations should be enlarged.
The main purpose of this paper is to present a class of general convexification transformation methods to convexify the noninferior frontier of a multiobjective problem.Compared with previous works, the major contributions of our paper are as follows:  We prove that the noninferior frontier could be convexified completely or partially by applying a more general transformation under certain assumptions.Also, we generalized the results in [10]. Our transformation further expands the class of multiobjective program that weighting method could solve by designing the transformation function based on the objective function.Our transformation can handle practical problems more efficiently than the one in [10] as well.
The paper is organized as follows: in Sections 2 and 3, a general form of transformation is proposed and then we prove that under some assumptions the noninferior frontier could be convexified completely or partly.In Section 4 some examples are given to vindicate our results.We give a conclusion about this paper in the last section.

. Then Problem (P) is transformed to a new problem which reads
Let cp X  be the sets of all the noninferior solutions for Problem (CP), then , with at least one strict inequality.Without loss of generality, we assume , with strict inequality holds at .Then x  is not a noninferior point which contradicts the assumption.So, . The noninferior frontier of Problem (CP) can be expressed as , , , , .
, where 0 x  is the minimum eigenvalue of   H x .Further, we make the following assumptions: I)  is a twice continuous differentiable function is negative for all x X   , and is twice differentiable and where M is a positive number.III) is a compact set.Then we have the following theorem: Suppose that assumptions I)-III) are satisfied.Then there exists a 0 such that when the noninferior frontier of Problem (CP) is convex.
From (2.1) and (2.2), we have that is a positive definite matrix if and only if p k is a positive definite matrix.

  B x
We assume that 0 is a positive definite matrix when 0 .Therefore, the noninferior frontier of Problem (CP) is convex when and we complete the proof.
Corollary 2.2 Suppose that assumptions I)-III) are satisfied.Then there is a 1 such that the supporting hyperplane exists everywhere on the noninferior frontier of Problem (CP) when . 1 Proof.By theorem 2.2, there exists a 1 such that the noninferior frontier is convex everywhere when 1 , and then by [13] we know that the supporting hyperplane exists everywhere.
Further by the discussion above, we can obtain the following corollary: Corollary 2.2 Suppose that assumptions I)-III) hold.Then x  is a noninferior solution of Problem (P) if and only if there exists a 1 such that is an optimal solution of Problem (CP) with w w when ., ,  : . .( ) 0, 1, , .

, it is easy to verify that
satisfies assumption II).The primal problem could be transformed into the following problem: Note that (2.3) is exactly the transformation proposed in [10].
Remark 2. We can derive other types of transformations by constructing many specific function forms satisfying assumption II).For example, each of functions

Local Convexification of the Noninferior Frontier
In some cases, assumptions I)-III) may not hold simultaneously.For instance, x  might not be twice continuous differentiable or      . In these circumstances, it might be difficult to globally convexify the noninferior frontier, however, we can achieve the local convexification of the noninferior frontier.Assume that assumptions I)-III) hold in a compact neighborhood , then there exists a 1 such that the Hessian matrix Proof.Theorem 3.1 could be vindicated by the similar way used in the proof of Theorem 2.2.
Then similar to corollaries 2.1 and 2.2, we have the following corollaries: The noninferior frontier of Problem (1) is Figure 1 depicts the feasible region of Problem (1).From Figure 1 we know that the noninferior frontier of Problem (1) is not convex, thus weighting method would not identify all the noninferior solutions in this case.
Let p and , then the Problem (1) could be transformed to the following problem: Figure 2 depicts the noninferior frontier of Problem (2).Clearly, the noninferior frontier of Problem ( 2) is convex and then we can identify all the noninferior solution of Problem (2) by applying the weighting method.Also, it's worthwhile to point out that compared to the transformation we used, the p-power transformation may produce a much more complicate problem.
Obviously, as shown in Figure 4, the noninferior frontier of Problem (4) is locally convex and then we can identify part of the noninferior solutions of Problem (4) by applying the weighting method.

Conclusion
As one of the simplest methods to identify the noninferior solutions of multiobjective problems, weighting method fails in many nonconvex cases.In this paper, a general class of convexification transformations is presented and we prove that the transformation could con-  vexify the noninferior frontier completely or partly under assumptions and then weighting method can be used successfully.This paper expands greatly the class of multiobjective programs that weighting method can cope with and provides more specific transformations to tackle practical problems efficiently.

1 Remark 1 .
If we set p p

Corollary 3 . 1 Example 1 :
Suppose that assumptions I)-III) hold in a compact neighborhood Consider the following example:

Figure 3
Figure 3 depicts the feasible region of Problem (3).From Figure 3 we know that the noninferior frontier of Problem (3) is nonconvex.Note that the noninferior frontier of Problem (3) can be expressed as     f x x   And the hessian matrix of  x 