Optimization Theory and Applications by Jochen Werner (auth.)

By Jochen Werner (auth.)

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Example text

M)} can be reduced to the solution of a linear program. An interesting theorem of STEINHAGEN (1922) connects the inradius of a convex set in lRn with its width. 1 Definition: Suppose P C lRn is non empty , convex and 51 compact. i) := sup {r > 0 ~ = ~(P) 3 x E P with B[Xir] c p} is the inradius of P. ii) w = w{P) :='inf 6{sup cTy - inf cTy} is the width of P. 2~~T~h~e~o~r~ern= P := {x E lRn : a iT x ~ b i (i=1, ••• ,m)} bounded. If r is the inradius and w the width of P, then w ~ -r • 2n1/2 if n is odd 2 (n+1 ) {n+2)1/ 2 if n is even.

The most important questions in optimization are: - Under what assumptions do solutions exists? - Which conditions are necessary for the existence of a solution? e. are they also sufficient? - How do the value and the solution set of an optimization problem change under perturbation of the data, that is of the objective function and the set of feasible solutions? 28 Unfortunately for reasons of time and space we shall not be able to treat this last question, but we shall treat the other two quite thoroughly.

A ii) Suppose (D) is feasible (say yEN) and (P) not feasible. Then Ax = b, x ~ 0 has no solution, so by the FARKAS-Lemma there is a z E mm with ATZ ~ 0 and bTz < O. If one defines yet) 1\ := y - tz, then yet) € N for all t TA T b Y - tb z Thus sup (D) ~ + = for ~ 0 and t ~ + =. = + =. iii) Follows from ii) for reasons of symmetry. 1. 2 Theorem (Existence Theorem) : Suppose given the r:rimal program (P) and the dual program (D) • Then we have: I f (P) is feasible and inf (P) > - = or (D) is feasible and sup (D) < + =, then (P) and (D) both have a solution and max (D) Proof: = min (P).

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