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MULTI –OBJECTIVE  OPTIMIZATION
Kinds of Optimum and Relation between them

Dr Khin Maung Win, M.A. (Yangon, Myanmar), Ph.D. (Caen, France), Advisor, Mathematics Department, Yangon University

In this paper the definitions of Maximum, strong maximum and weak maximum relative to an ordering come are given. Then the relations between them and equivalent definitions in Rⁿ are stated.

Let X be a real Banach space and C be a convex cone which is neither the whole space nor  { 0 }   in X. For x, y belonging to X ,we may define a partial ordering relative C in the following way:

     x  ≥  y, relative to C, (written  x  ≥_c  y) if and only if  x – y  ∈  C.

We may simply write   x  ≥  y, if C is supposed to be understood, as in the case of R,  in which  C  is understood to be  [0,   ∞ ), so that  x  ≥  y    means    x – y  ∈ [  0, ∞) .

Now, let D be a non-empty sub set of X.  

      z  is said to be a strong maximum of D , if
         z  ∈ D  and  for all  x in D,  z  ≥  x.

      z is called a maximum of D, if    
         z  ∈  D    and   (z + C ) ∩ D  =  {  z} .

      z  is called a weak maximum of D,  if 
          z  ∈  D   and     (z + int.C ) ∩  D    =   Φ

We let   S- max_c D =  the set of all strong maximums of  D, relative to C

              max_c  D = the set of  maximums of D, relative to C

              W-max_c  D  = the set of weak maximums of D relative to C.

We may of course write  S-max D. max D and W-max D if C is supposed to be understood.

Then we have the following results:

1. If C is pointed, then   S-max D   ⊂   max D.

2. max D  ⊂   W-max D

3. S-max D   ⊂   W-max D

4. In R, ordered by    [ 0,  ∞ ), S-maxD, max D and W-max D are equivalent.

5. z  ∈ max D   if and only if there does not exist any other element  x in D such that  x ≥ z  .

6.  When  X =  Rⁿ  and  C  =  Rⁿ₊  , then
     z  ∈ max D if and only if there is no x in D such that  x_i ≥  z_i ,
     for all  i = 1,2,3.....n and  x_j   >  z_j , for some j=1,2,...n .

7.  When   X  =  R ⁿ  and    C =   Rⁿ₊, z  ∈ max D   if and only if
     x _i  > z _i  for some   x  in  D and for some  i     implies that there exists  j such that x _j< z_j .

8. When   X  =  Rⁿ and   C  =   R ⁿ_+,   z  ∈   W-max D   if and only if
    there is no   x   in  D  such that     x_i  > z _i, for all  i = 1, 2, 3, .....n

Bibliography

1. Aung Thu : Convex Cones and Vector Maximization. Ph.D. Thesis. Mathematics Department. Yangon University. 2001.

2. Chen (G.Y.) and Jahn (J) : Optimality Conditions for set valued Optimization .
   Reprint No. 217,Institute of Applied Mathematics, University of Erlangen-Nuremberg,1997.

3. Kyi Kyi Than:  Some Relations Between Optimums. Seminar Presentation.
    Mathematics Department .Yangon University ,October 2001.

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