Bifurcation of Extremals in Optimal Control by Jacob Kogan PDF

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II Minimize F(x) XE{XI, ... 49) The trivial complete search algorithm, which consists in evaluating F(x) for all x E {Xl, ... , xm} is only applicable for moderate size m of the search space. As more efficient ways of organizing the search in large sets we consider· in more detail: 1. 1) 2. 1 Branch and Bound search The idea behind Branch and Bound procedures is to organize the search in such a way that larger sets of possible candidates can be excluded from detailed search because one has evidence that they do not contain the optimizer.

Price to pay for the recursiveness. The estimate Xn is much simpler than med(6, ... ; en): The computational complexity is O(n) and the storage complexity is 0(1). A sequential estimate can be based on an estimate of the density at the median. Let 9n be a consistent estimate of g( med (0)). e. the recursive estimate Xn stopped at 1"f satisfies = lim P{IX T , e-+O - med (G)I < f} == 1- 0:. - 18 CHAPTER 1. 5 The black-box and the white-box approach The simplest method of solving stochastic optimization problems is to use an optimization procedure which needs only function values (and not first or even second derivatives).

14). Bounds can be easily established if convexity assumptions are fulfilled. Suppose that the support of p is contained in a compact, convex set C E ~m. 2) H(x, IE(p)) :S J H(x, w) dp(w), where IE(p) = f w dp(w). 3), there is a probability measure concentrated on the set of extremals Be C, such that f H(x, w) dp(w) :S f H(x, w) dpext(w). (This inequality is also called the Edmundson - Madansky inequality). 13). The bounds may be improved, if C is iteratively decomposed into a union of compact convex sets C = Ui Ci and the inequalities are used for each Ci separately.

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