The present paper develops a simple, easy to interpret algorithm for a large class of dynamic allocation problems with unknown, volatile demand. Potential applications include ad display problems and network revenue management problems. The algorithm operates in an online fashion and relies on reoptimization and forecast updates. The algorithm is robust (as witnessed by uniform worst-case guarantees for arbitrarily volatile demand) and in the event that demand volatility (or equivalently deviations in realized demand from forecasts) is not large, the method is simultaneously optimal. Computational experiments, including experiments with data from real-world problem instances, demonstrate the practicality and value of the approach. From a theoretical perspective, we introduce a new device—a balancing property—that allows us to understand the impact of changing bases in our scheme.

The goal of ranking and selection (R&S) procedures is to identify the best stochastic system from among a finite set of competing alternatives. Such procedures require constructing estimates of each system's performance, which can be obtained simultaneously by running multiple independent replications on a parallel computing platform. However, nontrivial statistical and implementation issues arise when designing R&S procedures for a parallel computing environment. Thus we propose several design principles for parallel R&S procedures that preserve statistical validity and maximize core utilization, especially when large numbers of alternatives or cores are involved. These principles are followed closely by our parallel Good Selection Procedure (GSP), which, under the assumption of normally distributed output, (i) guarantees to select a system in the indifference zone with high probability, (ii) runs efficiently on up to 1,024 parallel cores, and (iii) in an example uses smaller sample sizes compared to existing parallel procedures, particularly for large problems (over 10 6 alternatives). In our computational study we discuss two methods for implementing GSP on parallel computers, namely the Message-Passing Interface (MPI) and Hadoop MapReduce and show that the latter provides good protection against core failures at the expense of a significant drop in utilization due to periodic unavoidable synchronization.(Henderson and Pasupathy 2014). For overviews of methods to solve the SO problem, see, e.g., Fu (1994), Andradóttir (1998), Fu et al. (2005, Pasupathy and Ghosh (2013).We consider the case of SO on finite sets, in which the decision variables can be categorical, integer-ordered and finite, or a finite "grid" constructed from a continuous space. Formally, the SO problem on finite sets can be written aswhere S = {1, . . . , k} is a finite set of design points or "systems" indexed by i, and ξ is a random element used to model the stochastic nature of simulation experiments. (In the remainder of the paper we assume that µ 1 ≤ µ 2 ≤ · · · ≤ µ k , so that system k is the best.) The objective function µ : S → R cannot be computed exactly, but can be estimated using output from a stochastic simulation represented by X(·; ξ). While the feasible space S may have topology, as in the finite but integer-ordered case, we consider only methods to solve the SO problem in (1) that (i) do not exploit such topology or structural properties of the function, and that (ii) apply when the computational budget permits at least some simulation of every system. Such methods are called ranking and selection (R&S) procedures.R&S procedures are frequently used in simulation studies because structural properties, such as convexity, are difficult to verify for simulation models and rarely hold. They can also be used in conjunction with heuristic search procedures in a variety of ways (Pichitlamken et al. 2006, Boesel et al. 2003, making them useful even if not all systems can be simulated. See Kim and Nelson (2006a) for an excellent introdu...

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We show that interactive and noninteractive zero-knowledge are equivalent in the 'help model' of Ben-Or and Gutfreund (J. Cryptology, 2003). In this model, the shared reference string is generated by a probabilistic polynomial-time dealer who is given access to the statement to be proven. Our results do not rely on any unproven complexity assumptions and hold for statistical zero knowledge, for computational zero knowledge restricted to AM, and for quantum zero knowledge when the help is a pure quantum state.

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