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

116 N. Papamarkos / Appl. Math. Comput. 112 (2000) 113±124· The approximation problem (9) does not have a linear form.By using (1), the approximation problem (9) is rewritten in the followingform:maximize dsubject to! G…x k † 1 ‡ XMbj~B j …x k † XNa j~A j …x k †!6 n 0 ;…11†jˆ1jˆ1! 1 ‡ XMb j~B j …x k † ‡ dn 0 i 6 0 for k ˆ 1; 2; ...; K;jˆ1where n 0 P 0 and d > 0.The approximation problem, as formulated in (11), is not linear, but it maybe converted to a linear one via the transformations:n ˆ 1n 0 ;b 0 j ˆ bjn 0 ; for j ˆ 1; 2; ...; M;…12†a 0 j ˆ ajn 0 ; for j ˆ 1; 2; ...; N:Now, using (12), the approximation problem is reformulated asmaximize dsubject to!n ‡ XMb 0 ~ jB j …x k † XNa 0 ~ jA j …x k † 6 1;G k G kjˆ1n ‡ XMn ‡ XMjˆ1jˆ1b 0 j ~ B j …x k †!jˆ1‡ XNjˆ1a 0 j ~ A j …x k † 6 1;!b 0 ~ jB j …x k † ‡ d 6 0 for k ˆ 1; 2; ...; K;…13†where n 0 P 0 and d > 0.The above problem (13) shows that the approximation problem (11) can beeasily linearized by using relations (12). If some or all of the function coecientsmust take only integer values then an integer approximation problemmust be formulated.