Study on the Duality between MFP and ACP

Xiaojun Lei Department of Mathematics Tongren University Tongren 554300, China Tel: 86-856-523-0984 E-mail:xjleitrxy@163.com Zhian Liang Department of Statistics Shanghai Financial and Economics University Shanghai 200433, China Tel: 86-21-6591-3351 E-mail:Zhian L@163.com Abstract Under the generalized weak convexity of (F, d , , ρ α ), we studied the results of several sorts of duality type about the problem of multi-objective fractional programming (MFP), extended this results to the generalized arcwise connected hypothesis, established the optimized problem of arcwise connected area (ACP) and the optimal sufficient condition of 0 ) ( . . ) ( min ≤ ∈ x g t s x f S x under constraint condition, and gave the duality model, and obtained the conclusions of

Generally, when we solute an optimized problem, the feasible area is usually in the area with interior points, but in practical problems, it always doesn't possess this condition, for example, the feasible area of the problem is the following line-type figure without interior point which is seen Figure 1.Its feasible area is connected by curve S.So when we define the function in this feasible area, we can not consider its partial derivative or directional derivative, and the grads of the function.For these problems, in early 1970, Ortega and Rheinboldt (Ortega, 1970) put forward the concept of regional arcwise connection, after that, Avriel and Zang (Avriel, 1980, p.407-435) extended it as various generalized convexities.The arcwise connected function and various generalized functions possess very good local-global extremum property, and in this article, we mainly introduce the duality problem result under the generalized weak convexity of (F, d ,

Basic conclusions of MFP duality under generalized weak convexity of (F,
Here, we will give the conclusions of several duality problems about MFP.

Supposed (MFP)
is the real valued function on X, and h is m dimensional vector value function defined on X, and λ represent column vectors, and their component subscripts belong to k M .
Theorem 2.2 (strong duality): Supposed x is an effective solution of (MFP), and x fulfills the restrain condition (GGCQ) (Avriel, 1980, p.407-435), so In fact, because x is an effective solution of (MFP), and (GGCQ) exists on x , as a necessary and effective condition, ) of (MFD1) must exist and make ) ( Its result is contradictive with the conclusion of weak duality in Theorem 2.1, so ) , , ( λ τ u is an effective solution of (MFD1).

The optimal condition and duality of generalized arcwise connected function
After we give the weak duality and strong duality of (MFP) under some very weak generalized functions, now we consider the optimized problem which area is arcwise connection.
x , and it is marked as In this way, to , a continual arcwise connected function (ACF) f(x) on S can be denoted as .
(3----4) x exists and makes the following containment relationship come into existence.
So we call that f(x) is the puppet arcwise connected function on x 0 which is marked as PACF.
Under the same condition, if the containment relationship is 0 , so we call f(x) is the strong puppet arcwise connected function on x 0 which is marked as SPACF, and if , so we call f(x) is the strict strong puppet arcwise connected function on x 0 which is marked as STPACF.
If f(x) is PACF, SPACF and STPACF on any point of S, so we call f(x) is PACF, SPACF and STPACF on S. Prove: counterevidence.Supposed f(x) is SQACF and To any neighbor area of 0 x , we can always find 0 in this neighbor area when , that is contradictive with that 0 x is a local minimum point of f(x), so the theorem is proved.
To STQACF, there are following theorems.
Thus, from the definition of STPACF, we can obtain Prove: first, we prove the equation group is arcwise derivative and continual on solution of (MFD1), and the objective function values on the corresponding points of (MFP) and (MFD1) are equal, and if it fulfills the generalized convex inequation in Theorem 2.1, so ) , , ( λ τx is an effective solution of (MFD1).

.
If all hypotheses in Theorem 2.3 are fulfilled, so the corresponding and the objective function values of (MFP) and (MFD3) are respectively equal on x and ( v x , ,τ ), If the hypotheses and conditions in Theorem 2.5 are fulfilled, so the ( v x , can define the arcwise derivative concept of arcwise connected function.Definition 3.2: Supposed f(x) is the continual real valued function on the arcwise connected set 1 3: Supposed f(x) is the continual real valued function on the arcwise connected set n R S ⊆ , to any one point x in S, Theorem 3.1 (Zhiun, 2001): Supposed f(x) is the quasi-arcwise connected function QACF on an arcwise connected set local minimum point of f(x), so 0 x is a strict global minimum point of f(x) on S. Theorem 3.2 (Zhiun, 2001): Supposed f(x) is the strong quasi-arcwise connected function SQACF on an arcwise connected set local minimum point of f(x), so 0 x is the only strict global minimum point of f(x) on S.
Theorem 3.3: Supposed f(x) is the STQACF defined on an arcwise connected set of f(x), so 0x is the global minimum point of f(x) on S. Theorem 3.4: Supposed f(x) is the real valued continual funciton on an arcwise connected set f(x) is STPACF, so 0x is the global minimum point of f(x) on S. If f(x) is SPACF, so 0x is the only strict global minimum point of f(x) on S.Prove: supposed f(x) is STPACF, 0 − ) from (3.10), (3.11), (3.14) and (3.15), we can obtain, to enough big 1 0 from (3.16), (3.11), (3.18), to 1 feasible solution of (ACPD), and the objective function values of (ACP) and (ACPD) are equal on * x .If to every feasible ) , , ( 0 r r u of (ACPD), f(x) is STPACF on u point,

Figure
Figure 1.A Line-type Figure without Interior Point x 0 is the global minimum point of f(x) on S,