Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition

In this paper, we propose a homotopy continuous method (HCM) for solving a weak efficient solution of multiobjective optimization problem (MOP) with feasible set unbounded condition, which is arising in Economical Distributions, Engineering Decisions, Resource Allocations and other field of mathematical economics and engineering problems. Under the suitable assumption, it is proved to globally converge to a weak efficient solution of (MOP), if its x-branch has no weak infinite solution.


Introduction
Mathematical modeling of real-life, economics and engineering problems usually results in optimization and decision systems, such as multiobjective optimization systems, linear and nonlinear optimization systems, global optimization systems and others.Many mathematical formulation of economics and decisions contain multiobjective optimization problems, which arise in use of choosing projects in Bid Decision, developing production plan by Enterprise and requiring human resource in Management; for more details see ( [1,2]) and the reference cited therein.Therefore, it is very actually meaning subject how we find the KKT points of multiobjective optimization problem.Hence we consider the following multiobjective programming problem with inequality constraints: , , , : , , , : are twice times continuously differentiable functions.
Since 1981, Garcia and Zangwill [3] firstly used homotopy method to study convex programming problem, which makes the method become a powerful tool in dealing with various programming problems.In 1988, Megiddo [4] and Kojima [5] et al. discovered that the Karmakar interior point method was a kind of path following method for solving linear programming.Since then the interior path-following method has been extensively used for solving mathematical programming problems.In 1994, Lin, Yu and Feng [6] constructed a new interior point method-combined homotopy interior point method (CHIP method), formed by Newton homotopy and linearly homotopy-for solving the KKT point of convex nonlinear programming.Subsequently, Lin, Li and Yu [7], without strictly convexity of the logarithmic barrier function, showed that the iteration points generated by CHIP, also converged to the KKT points of optimization problem.In 2003, CHIP method was generalized to convex multiobjective optimization problem by Lin, Zhu and Sheng [8].They constructively proved the existence of KKT system solution for corresponding purification problem.
In 2008, Song and Yao [9] further generalized the results of [8,10].They constructed a new combined homotopy mapping.A smooth bounded homotopy path was obtained under the normal cone condition and weaker Mangasarian-Fromovitz constraint qualification.However, up till now the convergence of homotopy path in literature related above is obtained under the condition that the feasible set is nonempty and bounded.Recently by adapting the combined homotopy method developed in [11,12], a homotopy method was proposed in [13,14] for variational inequalities on unbounded sets.
In this paper, we will discuss about homotopy methods for MOP on unbounded set.Under conditions which are commonly used in the literature, a smooth path from a given interior point set to a solution of MOP will be proven exist.The paper is organized as follows.In Section 2, we recall some preliminaries results, formulate an equivalent form of KKT system for MOP and list some lemmas from differential topology which will be used in this paper.In Section 3, we proved in detail existence and convergence of the smooth path under a weak condition.
Throughout the paper, let :  n be the strictly feasible set of MOP.In addition, we denote the index set of at x by represent the nonnegative and positive orthant of , respectively.n R

Preliminaries
As we know, the solutions for a multiobjective programming problem are referred to variously as efficient, Pareto-optimal and nondominated solution [15].In this paper, we will refer to a solution of a multiobjective programming problem as an efficient solution.
Definition 2.1 [15] x  is an efficient solution of multiobjective programming problem, if there is no x holds.In [9], a homotopy method for MOP with bounded  was given.In this paper, we will discuss MOP with  which is not necessarily bounded.It is well known that, if x is an efficient solution of MOP, under some constraint qualifications (e.g.Kuhn and Tucker constraint qualification [16]), MOP satisfies KKT constraint condition at x (see [10,15]): where , , , We call that a point x satisfying the KKT condition ( 1 For solving the KKT system, we s ch a ve ear ctor

 
, ,  is called a Kuhn-Tucker vector of MOP.Usually, we will solve its KKT system of MOP instead of solving MOP directly.
The following lemmas from differential topogy will be used in the next section.
Definition 2.2 [17] Let X and Y be topological space, : , 0 ,1 be continuous mapping and To solve (2), the homotopy mapping H is given by [7] as follow: where and Sometimes, we rewrite That is, the equation with respect to  has only on ti As e solu on.0   , the solution of the Equation t of KKT system (2).
(3) is just tha s, for Thu a given 0  , the zero-point of the mapping H constructed above is the homotopy mapping between the trivial mapping of 0  and KKT system (2) of MOP.
develop our main result, we need the following basic definitions and lemmas in topology.By the definition of C r differential manifold [17], we know 0  is a n-dimensional differential manifold.For the definition of product manifold [17], Definition 2.3 [17] Le ential manifold with dimV p  t U, V be differ , and :

(Classification Theorem of One-Dimennal Ma
with Boundary [17]) Each connecte co f some smooth manifolds.sio nifold d pa he objective function of optimizaex, there exists a rt of a one-dimensional manifold with boundary is homeomorphic either to a unit circle or to a unit interval.

Main Results
According to [15], as t tion problem is conv , and fo ny give , there exis 0 and 0 The following example illustrates the meaning of Definition 3.1.
is a n at weak for MO r any given Theorem 3.1 Consider the homotopy mapping H of MOP constructed as (3).Suppose the following four conditions are satisfied: (A) 0  is nonempty (Slater condition); is a solution of KKT system (2).
To prove Theorem 3.1, we nee ults.given d to prove the following three res For a where I is an identical matrix.By a simple computation, we have For , we have , and hence . Thus, 0 is the regular value o .By For any lows: Thus, we have The properties of norm y the following equality holds, that is, given a start point r any y   , we construct two set as fol- Take any .From the first Equation (3) multiplied by x y  , we obtained that By the convexity of  , 1,2, , j g x j m   and 0 0 y   , we k (9) From the ( 7), ( 8) and the second equation in (3), we simplify the Equation ( 8), that is now: and the equality ( 7), (10) From the second equality of (10), we obtain that t exists some index i such that here Since where denotes the ), ty of (10 The following is divided two parts: From the condition (C), . Hence (13) becom It is obviously that the coefficient in left-hand side itive finite quantities as value.m 3.1 im of equality ( 14) must be pos Otherwise, the condition (B) of Theore plies

and for any 0
x  , the leftha n with the (14).Let nd side of equality ( 14) is infinite.This is contradictio finite value of the right-hand for the equality Hence we have x    and 0 0 x   , the equalit ulti-Since y (15) m plied by However, the . This contrad Thus, we obtain 0 2) When

 
1 , we rewrite (12) as In since 0 is a regular value of fact, if matrix of H at and , the Jacobian only one solution in that there must be quence ere exists a sequence must be some Fr y Equation (10 s a contradiction.Second, we assume that om the second equality of homotop ), we can see . There also exists and from lity in (10), we have the third equa If there exists s  such that   0

Acknowledgements
The authors are grateful to Professor W. Song and the MOP with inequality constraints.The method needs much less restrictive condition an mputational work compared with traditional method.It is shown that the is glob convergent and computational tractability for solving MOP on unbounded set.re and suggesti thank editors and review r val ing the quality of the paper.
the feasible set of MOP, and   0 :

, 0 H
   for convenience.Because f and onppings, H is also continuous mapping.When g are c tinuous ma

Theorem 3 . 2
Lemma 3.1 guarantees that the f H has a good geometric structure.The followi theorem is the key of boundedness for the homotopy generated by (3), which is the main result of this paper. .zero-point set o ng path Suppose that the condition (C) of Theorem 3.1 holds.There exists solution at infinity to MOP by Defi 3.1.If MOP has no weak solution at infinity, the x-compo- We use proof by contradiction.Suppose th at condition (D),   is a solution of the KKT system.Remark 3.1 If   is a bounded set, the cond n (D) of Theorem 3.1 holds obviously.Hence, the result of Th m 3.1 i e one in [ but also give a kind of numerical algorithm, that is, the solution can be obtained by tracing numerically the homotopy path itio eore mplies th 9].Above all, we do not only prove the existence of solution for KKT equation, s represents arc length parameter of 0   .Then   s  is determined by the following initial value problem to the ordinary differential equation: