episciences.org_506_20230328193045898 20230328193045898 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Discrete Mathematics & Theoretical Computer Science 1365-8050 01 01 2010 Vol. 12 no. 2 On a Class of Optimal Stopping Problems with Mixed Constraints F. Thomas Bruss 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. 01 01 2010 506 https://hal.science/hal-00990470v1 10.46298/dmtcs.506 https://dmtcs.episciences.org/506 https://dmtcs.episciences.org/506/pdf https://dmtcs.episciences.org/506/pdf