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

N. Papamarkos / Appl. Math. Comput. 112 (2000) 113±124 119tree search algorithm refers to the variables a j and b j , then, at each tree node,additional constraints of the form:a j 6 p;a j P p;b j 6 p;…17†b j P p:can be transformed into the following form:qn ‡ a 0 j 6 0;qn a 0 j 6 0;rn ‡ b 0 j 6 0;…18†rn b 0 j 6 0:This form is suitable for the linear programming problem (13) and thereforecan be added to it. Doing this, at each tree node a linear programming problemis solved in the following form:maximize dsubject to!n ‡ XMb 0 ~ jB j …x k † XNa 0 ~ jA j …x k † 6 1;G kjˆ1!jˆ1…19†0; G k n ‡ XMb 0 ~ jB j …x k † ‡ XNa 0 ~ jA j …x k † 6 1; r m1 n ‡ b 0 j 6 0; r m 2n b 0 j 6 j 2‰1 ...MŠ;jˆ1jˆ1! n ‡ XMb 0 ~ jB j …x k † ‡ d 6 0;jˆ1 q n1n ‡ a 0 j 6 0; q n 2n a 0 j6 0; j 2‰1 ...NŠ;where q n1; q n2and r m1 ; r m2 integers, n 0 P 0, d > 0 and k ˆ 1; 2; ...; K.It is noted that in the linear problem (19), the variables a 0 j and b0 jare unrestrictedin sign. On the other hand, linear programming algorithms, such asthe RSA, require only non-negative variables. A simple way to overcome thisdiculty is to shift the variables by a suitable shift constant V > 0. Therefore,if we de®ne