TABLE OF CONTENTS Until now, we did not consider the question of how to solve optimization problems of the types we have encountered. This is the issue we address in this and in the next lectures.
All optimization programs we dealt with are covered by the following universal form of a mathematical programming program:
where
n( p) is the design dimension of problem ( p),
X( p) ? R n( p) is the feasible domain of the problem, and
p 0( x): R n( p) ? R is the objective of ( p).
The mathematical programming form ( p) of an optimization program is most convenient for investigating solvability issues in optimization, so at this point we switch to this form. Note that the optimization programs we dealt with in the previous lectures-the conic programs
where K ? R m is closed convex pointed cone with a nonempty interior-can be easily written in the MP form with
The MP programs obtained from the conic ones possess a very important characteristic feature: they are convex.
A mathematical programming program (p) is called convex if
the domain X(p) of the program is a convex set: whenever x, x ? ? X(p), the segment { y = ? x + (1 ? ?) x ? 0 ? ? ?
