Abstract
Let X(1),X(2),...,X(n) be independent, identically distributed uniform random variables on [0, 1]. We can observe the outcomes sequentially and must select online at least r of them, and, moreover, in expectation at least mu >= r. Here mu need not be integer. We see X(k) as the cost of selecting item k and want to minimize the expected total cost under the described combined (r, mu)-constraint. We will see that an optimal selection strategy exists on the set S(n) of all selection strategies for which the decision at instant k may depend on the value X(k), on the number N(k) of selections up to time k and of the number n - k of forthcoming observations. Let sigma(r,mu)(n) be the corresponding S(n)-optimal selection strategy and v(r,mu)(n) its value. The main goal of this paper is to determine these and to understand the limiting behavior of v(r,mu)(n). After discussion of the specific character of this combination of two types of constraints we conclude that the S(n)-problem has a recursive structure and solve it in terms of a double recursion. Our interest will then focus on the limiting behavior of nv(r,mu)(n) as n -> infinity. This sequence converges and its limit allows for the interpretation of a normalized limiting cost L (r, mu) of the (r, mu)-constraint. Our main result is that L(r, mu) = g(r) ((mu - r)(2)/(2)) where g(r) is the r(th) iterate of the function g(x) = 1 + x + root 1 + 2x. Our motivation to study mixed-constraints problems is indicated by several examples of possible applications. We also shortly discuss the intricacy of the expectational part of the constraint if we try to extend the class of strategies S n to the set of full-history-dependent and/or randomized strategies.
Highlights
This study is dedicated to Philippe Flajolet, INRIA, Dr h.c. of the Universite Libre de Bruxelles, at the occasion of his 60th birthday for his outstanding contributions to the Theory and Applications of Algorithms.Let X1, X2, · · ·, Xn be i.i.d
The same holds for the selection problem of Baryshnikov and Gnedin (2000) of vector-valued observations and the competitive selection problem with two decision makers studied by Enns and Ferenstein (1985) and Bruss et al (1998)
One special case of the described problem has been studied in the literature, and, interestingly, it appeared before secretary problems started to attract attention
Summary
On a Class of Optimal Stopping Problems with Mixed Constraints. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Our interest will focus on the limiting behavior of nvr,μ(n) as n → ∞. This sequence converges and its limit allows for the interpretation of a insotrhmearlitzheidtelrimateitionfgthcoesftuLnc(tri,oμn)go(fxt)h=e (1r,+μ)x-c+on√str1ai+nt.2xO.ur main result is that L(r, μ) = gr((μ − r)2/2) where gr. We shortly discuss the intricacy of the expectational part of the constraint if we try to extend the class of strategies Sn to the set of full-history-dependent and/or randomized strategies
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