Objective function approximations in mathematical

Mathematical Programming 13 (1977) 23-37. North-Holland Publishing Company

OBJECTIVE FUNCTION APPROXIMATIONS IN MATHEMATICAL PROGRAMMING* A r t h u r M. G E O F F R I O N University of California at Los. Angeles, CA, U.S.A. Received 27 September 1976 Revised manuscript received 10 January 1977 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 postcomputational bounds are also obtained.

Keywords: Approximation, Errorbounds, Modeling.

M o s t a p p l i c a t i o n s o f m a t h e m a t i c a l p r o g r a m m i n g r e q u i r e t h e 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 o r a p p r o x i m a t i n g t h e o b j e c t i v e f u n c t i o n to b e o p t i m i z e d . W e g i v e a s i m p l e a p r i o r i b o u n d r e l a t i n g t h e 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 e r r o r to t h e a m o u n t o f e r r o r t h e r e b y i n d u c e d in t h e s o l u t i o n of t h e 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 . T h i s f u r n i s h e s a n a t u r a l c r i t e r i o n to g u i d e t h e c h o i c e of a n 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 t i v e f u n c t i o n . The criterion can often be applied via simple graphical constructions that we d e v e l o p f o r t h e c a s e o f l i n e a r s e p a r a b i l i t y , a n d w e s h o w t h a t it is g e n e r a l l y e q u i v a l e n t to t h e f a m i l i a r C h e b y s h e v c r i t e r i o n - w h i c h thereby provides d i r e c t a c c e s s to a p o w e r f u l a r r a y of e s t a b l i s h e d r e s u l t s a n d t e c h n i q u e s f o r t h e g e n e r a l case. In a d d i t i o n to a p r i o r i e r r o r b o u n d s , w h i c h f a c i l i t a t e t h e d e s i g n o f a n o b j e c t i v e f u n c t i o n before d o i n g a n y o p t i m i z a t i o n , w e a l s o d i s c u s s t h e t i g h t e r e r r o r b o u n d s a v a i l a b l e after a n o p t i m i z a t i o n h a s b e e n p e r f o r m e d . T h i s l e a d s to a n a t u r a l s u b j e c t i v e t i e - b r e a k i n g rule f o r u s e in c o n j u n c t i o n w i t h t h e p r i m a r y c r i t e r i o n a n d also to t h e n o t i o n of " r e t r o f i t " o b j e c t i v e f u n c t i o n s : i m p r o v e d h y b r i d s b e t w e e n the approximation actually used and the true unapproximated objective function. S i n c e u s i n g a r e t r o f i t o b j e c t i v e f u n c t i o n in p l a c e o f t h e a p p r o x i m a t e o n e w o u l d * This paper was partially supported by the National Science Foundation and by the Office of Naval Research, and was the basis for a plenary lecture delivered at the IX International Symposium on Mathematical Programming in Budapest, Hungary, August 1976. 23

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Arthur M. Geoffrion/Objectivefunction approximation

not destroy the optimality of the solution to the approximating problem, the analyst has the option of interpreting the optimization results as though they had been obtained using the retrofit objective function. The results of Section 1 are applied in a related paper [41 to obtain new aggregation results in a specific applications context.

1. B a s i c r e s u l t s

Let the following two optimization problems be given: (P) Minimize f(x), subject to x E X (~')

Minimize f(x),

subject to x E X,

where X is an arbitrary non-empty set and f and f are both real-valued functions bounded below on X. Interpret (P) as the "true" problem and (15) as the "approximating" problem in the sense that an approximate objective function is used in place of f. What can be said about the relationship between (P) and (15) when the difference between f and f can be bounded on X ? In the absence of further assumptions guaranteeing the existence of optimal solutions, it is necessary to phrase the answer to this question in terms of epsilon-optimal solutions, that is, in terms of feasible solutions having an objective function value known only to be within epsilon of the true infimal value. Let v(P) denote the infimal value of (P) and similarly for v(15). 1 (Objective Function Approximation). Let E and g be scalars (not necessarily nonnegative) satisfying

Theorem

- E < - f ( x ) - f ( x ) -O, any e-optimal solution 2 of (15) will necessarily be (e +_~+ g)optimal in (P). Proof.

Writing the first inequality of (1) as f(x)