advertisement

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 [email protected] [email protected] (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. 2