On the approximation of real rational functions via mixed-integer ...

120 N. Papamarkos / Appl. Math. Comput. 112 (2000) 113±124p j ˆ V ‡ a 0 jfor j ˆ 1; 2; ...; N;q j ˆ V ‡ b 0 jfor j ˆ 1; 2; ...; M;…20†then, the linear problem (19) is reformulated asmaximize dsubject toG kn ‡ XMjˆ1q j~B j …x k †jˆ1! XN! G k n ‡ XMq j~B j …x k †jˆ1jˆ1‡ XNp j~ Aj …x k †‡d k y 6 1;jˆ1p j~A j …x k †d k y 6 1;! n ‡ XMq j~B j …x k † ‡ d ‡ h k y 6 0 y ˆ 1; q n1n ‡ p j 6 V ; q n2n ‡ p j P V ; j 2‰1 ...NŠ; r m1 n ‡ q j 6 V ; r m2 n ‡ q j P V ; j 2‰1 ...MŠ;…21†where q n1; q n2and r m1 ; r m2 known integers, n 0 P 0, d > 0 and k ˆ 1; 2; ...; K.All the variables are positive, andh k ˆ V XMjˆ1~B j …x k †; …22†d k ˆG k V XM~B j …x k †‡V XNjˆ1jˆ1~A j …x k †ˆG k h k ‡ v XNjˆ1~A j …x k †: …23†Problem (21) is the linear programming problem that must be solved in eachnode. It corresponds to a total number of M ‡ N ‡ 3 variables and 3K ‡ D n ‡ 1linear constraints, where D n is the number of the additional linear constraints inthe node n. Usually, M ‡ N ‡ 3 3K ‡ D n ‡ 1, which means that the linearprogramming problem must be solved by its dual, using a method such as theRSA.According to the above analysis, the mixed integer linear rational approximationalgorithm consists of the following steps:Step 1. The integer and continuous coecients are de®ned and the LRAapproximation problem (13) is formulated. If it is known from the beginning,we can give an estimation of the global-integer solution.Step 2. The LRA problem is solved as a continuous variable problem. Let Xthe vector of variables that must take integer values in the ®nal mixed-integersolution.