Objective function approximations in mathematical programming

Mathematical programming applications often require an objective function to be approximated by one of simpler form so that an available computational approach can be used. An a priori bound is derived on the amount of error (suitably defined) which such an approximation can induce. This leads to a natural criterion for selecting the "best" approximation from any given class. We show that this criterion is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. This gains access to the rich legacy on Chebyshev approximation techniques, to which we add some new methods for cases of particular interest in mathematical programming. Some results relating to post-computational bounds are also obtained. M o s t a p p l i c a t i o n s of m a t h e m a t i c a l p r o g r a m m i n g require the m o d e l e r to e x e r c i s e s o m e d i s c r e t i o n in e s t i m a t i n g or a p p r o x i m a t i n g the o b j e c t i v e f u n c t i o n to be o p t i m i z e d. W e give a simple a priori b o u n d relating the a m o u n t of o b j e c t i v e f u n c t i o n a p p r o x i m a t i o n error to the a m o u n t of error t h e r e b y i n d u c e d in the solution of the c o r r e s p o n d i n g o p t i m i z a t i o n p r o b l e m. This f u r n i s h e s a natural criterion to guide the c h o i c e of an e s t i m a t e d or a p p r o x i m a t e o b j e c …