رد قیقحت یناریا نمجنا یللملا نیب سنارفنک نیلوا 1386 ایلمع ت

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‫اولین کنفرانس بین المللی انجمن ایرانی تحقیق در‬
1386 ‫ بهمن‬--- ‫ دانشگاه کیش‬--- ‫عملیات‬
A differential model for solving an optimization problem with intervalvalued objective function
Sohrab Effati 1, Morteza Pakdaman2
Department of mathematics, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran
Abstract:
In this paper we apply a differential model (neural network model) for solving an
optimization problem with interval valued objective function. To construct the
differential model, here we used Karush-Kuhn-Tucker optimality conditions.
1. Model definition
Suppose  denotes the set of all closed and bounded intervals in R. So if a, b be two
closed and bounded intervals ( a, b   ), we denote them with a  [a L , a U ] and
b  [b L , bU ] and we have the following operations in  :
[ka L , kaU ] if k  0
a  b  [a L  b L , a U  bU ],  a  [a U ,a L ], ka   U
[ka , ka L ] f k  0
Where k is a real number. We define the following Hausdorff metric in  :
d H (a, b)  max{| a L  b L |, | aU  bU |}.
Also a function f : R n   is called an interval-valued function, i.e. for each
X  R n , f(X) is a closed interval in R..
Proposition1. 1. ([see [1]) If f be an interval valued function , then f is
continuous at x  R n if and only if f L and f U are continuous at x.
Definition1. 1. Let X be an open set in R . An interval valued function f with
f ( x)  [ f L ( x), f U ( x)] is called weakly differentiable at x 0 if the real valued
functions f
L
and f U are differentiable at x 0 . Now we define a partial ordering on
 ; If a, b   then we write a  LU b if and only if a L  b L and a U  bU . Also we
write a  LU b if and only if a  LU b and a  b . Following this we call f, LUconvex at x’ if f (x'(1   ) x)  LU f ( x' )  (1   ) f ( x) for   (0,1) and x  X .
Proposition1. 2 ([see [1]) f is LU-convex at x if and only if f L and f U are
convex at x.
Consider the following optimization problem ( with interval valued objective
function):
minimize
f(X)  f(x 1 , x 2 ,..., x n )  [f L (x 1 , x 2 ,..., x n ), f U (x 1 , x 2 ,..., x n )]  [f L (X), f U (X)]
subject to
X  (x 1 , x 2 ,..., x n )    R n ,
1
2
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(1)
1
‫اولین کنفرانس بین المللی انجمن ایرانی تحقیق در‬
1386 ‫ بهمن‬--- ‫ دانشگاه کیش‬--- ‫عملیات‬
Where  is the feasible set which is a convex subset of R n . Also (1) can be
formulated as follows:
minimize
f(X)  f(x 1 , x 2 ,..., x n )  [f L (x 1 , x 2 ,..., x n ), f U (x 1 , x 2 ,..., x n )]  [f L (X), f U (X)]
subject to
g i (X)  0 , i  1,2,..., m
(2)
The meaning of minimization in (2) follows from the partial ordering '  LU ' .
Definition1. 2. Let x * be a feasible solution. We say that x * is a type1 solution of
problem (2) if there exists no x  X such that f ( x)  LU f ( x * ).
Theorem1. 1. (see [1]) Suppose that the objective function f : R n   is LUconvex and weakly differentiable at x * and g i , i  1,2,..., m satisfy the KKT
assumptions at x * (see [2 ) if there exist Lagrange multipliers 0  L , U  R and
0   i  R (i=1,2,…,m) such that :
m
(i ) L f L ( x * )  U f U ( x * )    i g i ( x * )  0;
i 1
(ii )  i g i ( x )  0 for all i  1,2,..., m
*
Then x * is a type1 solution of problem (2).
2. Solving Method
Wu (see [1]) proved the KKT optimality conditions for problem (2) (theorem 1.1).
Using these conditions we propose the following differential model for solving problem
(2):
L
U
 dx j
g ( x)
( x) m
L f ( x )
U f
 

  y i i , j  1,2,..,n.

x j
x j
x j
 dt
i 1
(3)

 dy i
 dt   g i ( x), y i  0 i  1,2,..., m.
Theorem1.2. If the dynamical system (3) converges to a stable state x(.) and y(.)
then the state x(.) is convergent to the optimal solution ( type1 solution) of problem (3).
Proof. (The proof is similar to [3]).
3. Conclusions
Here using KKT optimality conditions we proposed a differential system which is
convergent to type1 solution. The proof of it’s convergence ( proof of theorem 1.2) is
similar to the proof of theorem 1 from [3].
References
[1]. H. C .Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with intervalvalued objective function, European Journal of Operational Research, 176 (2007) 46-59.
[2]. M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming, Wiley, NY, 1993.
[3]. S. Effati, M. Baymani, A new nonlinear neural network for solving convex nonlinear programming
problems, Applied Mathematics and Computation, 168 (2005) 1370-1379.
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