System Develofiment Corporation S a n t a Monica, California
An optimization problem which frequently a r i s e s in applications of mathematical programming is the following:
t fn (xn)l , A 5x1' . . . Lx, 5 B where f i a r e convex functions. In this paper, the function F i s studied and shown to satisfy F ( A , B ) = M (A) t N (B), where M and N a r e increasing and decreasing convex functions, respectively. A l s o , the functional equation F ( A , C ) = F ( A , B ) t F ( B , C ) -F ( B , B ) i s established. These results generalize to the continuous c a s e F ( A , B ) = min IT f ( t , x ( t ) ) dt , 0 with x ( t ) increasing and A 5 x (0) 5 x ( T ) 5 B . The results obtained in this paper a r e useful f o r reducing a n optimization problem with many variables t o one with fewer variables.
' I n c r e a s i n g ( d e c r e a s i n g ) w i l l a l w a y s m e a n n o n d e c r e a s i n g ( n o n i n c r e a s i n g ) . 2 F o r s u c h a c a s e s e e W. K a r u s h a n d A. V a z s o n y i , " M a t h e m a t i c a l P r o g r a m m i n g a n d E m p l o ym e n t Scheduling," Nav. R e s . Log. Q u a r t . 4: 297-320 (1957). 'Recall that convexity on P 5 x implies continuity only on P < x .