Oliver Stein's Bi-Level Strategies in Semi-Infinite Programming PDF

By Oliver Stein

ISBN-10: 1441991646

ISBN-13: 9781441991645

ISBN-10: 146134817X

ISBN-13: 9781461348177

Semi-infinite optimization is a vibrant box of lively learn. lately semi­ endless optimization in a basic shape has attracted loads of consciousness, not just as a result of its wonderful structural elements, but in addition as a result of huge variety of purposes which might be formulated as normal semi-infinite courses. the purpose of this e-book is to focus on structural points of normal semi-infinite programming, to formulate optimality stipulations which take this constitution under consideration, and to offer a conceptually new resolution technique. actually, less than sure assumptions common semi-infinite courses should be solved successfully while their bi-Ievel constitution is exploited effectively. After a short creation with a few ancient historical past in bankruptcy 1 we be­ gin our presentation by means of a motivation for the looks of normal and common semi-infinite optimization difficulties in purposes. bankruptcy 2 lists a couple of difficulties from engineering and economics which provide upward thrust to semi-infinite types, together with (reverse) Chebyshev approximation, minimax difficulties, ro­ bust optimization, layout centering, disorder minimization difficulties for operator equations, and disjunctive programming.

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23 enables us to prove the following basic proposition which generalizes a well-known fact from finite programming. For a given point x E lRn the set Io(x) = {i E II ct'i(X) = 0 } is called the active index set at X. 8) iE10(x) where we adopt the usual convention n0 = lRn. e. 11). 11. A local description of M around if by active constraints Proof. 9) is trivial. Assume that the inclusion ":::>" does not hold. DVitisct'i(xV) ~ 0, i E Io(x),and there exists an index iv E 18(x) such that ct'dxV) > O.

In terms of semi-continuity, the latter is just the outer semi-continuity of Y. t. yEY(X) = {YEIRI(xy-1)y=0} . 10. Although 9 is continuous. and Y is outer semi-continuous. it is not hard to see that the optimal value function cp(x) = SUPyEY(x) g(x, y) of this problem is not upper semi-continuous at x=o. 10. 18 is that Y is not bounded around x = O. On the other hand, an unbounded set-valued mapping like Y(x) = {y E JRI y = x} would not harm the upper semi-continuity of the optimal value function.

For all other choices of 8 we obtain a more general nonempty and compact convex set Yo , but we still deal with a standard semi-infinite optimization problem. 3 Finally we can also consider the case in which the risk aversion of the decision maker depends on the point x. If for instance his risk aversion increases when the values x j deviate from 1I n, j = 1, ... t. J J ,x>O j=l with Of course, the modified term 8(x) can only take effect iii, (J'j , j = 1, .. 1. 3 we will treat these examples numerically.

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Bi-Level Strategies in Semi-Infinite Programming by Oliver Stein

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