Sequential Design for Optimal Stopping Problems

We propose a new approach to solve optimal stopping problems via simulation. Working within the backward dynamic programming/Snell envelope framework, we augment the methodology of Longstaff-Schwartz that focuses on approximating the stopping strategy. Namely, we introduce adaptive generation of the stochastic grids anchoring the simulated sample paths of the underlying state process. This allows for active learning of the classifiers partitioning the state space into the continuation and stopping regions. To this end, we examine sequential design schemes that adaptively place new design points close to the stopping boundaries. We then discuss dynamic regression algorithms that can implement such recursive estimation and local refinement of the classifiers. The new algorithm is illustrated with a variety of numerical experiments, showing that an order of magnitude savings in terms of design size can be achieved. We also compare with existing benchmarks in the context of pricing multi-dimensional Bermudan options.Comment: 24 page

Learning Price-Elasticity of Smart Consumers in Power Distribution Systems

Demand Response is an emerging technology which will transform the power grid of tomorrow. It is revolutionary, not only because it will enable peak load shaving and will add resources to manage large distribution systems, but mainly because it will tap into an almost unexplored and extremely powerful pool of resources comprised of many small individual consumers on distribution grids. However, to utilize these resources effectively, the methods used to engage these resources must yield accurate and reliable control. A diversity of methods have been proposed to engage these new resources. As opposed to direct load control, many methods rely on consumers and/or loads responding to exogenous signals, typically in the form of energy pricing, originating from the utility or system operator. Here, we propose an open loop communication-lite method for estimating the price elasticity of many customers comprising a distribution system. We utilize a sparse linear regression method that relies on operator-controlled, inhomogeneous minor price variations, which will be fair to all the consumers. Our numerical experiments show that reliable estimation of individual and thus aggregated instantaneous elasticities is possible. We describe the limits of the reliable reconstruction as functions of the three key parameters of the system: (i) ratio of the number of communication slots (time units) per number of engaged consumers; (ii) level of sparsity (in consumer response); and (iii) signal-to-noise ratio.Comment: 6 pages, 5 figures, IEEE SmartGridComm 201

Social welfare and profit maximization from revealed preferences

Consider the seller's problem of finding optimal prices for her $n$ (divisible) goods when faced with a set of $m$ consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, with which subgradients of the dual function can be computed. We thereby obtain a simple subgradient-based algorithm for strongly concave valuations and convex cost, with query complexity $O(m^2/\epsilon^2)$, where $\epsilon$ is the additive difference between the social welfare induced by our algorithm and the optimum social welfare. We also study social welfare maximization under the online setting, specifically the random permutation model, where consumers arrive one-by-one in a random order. For the case where consumer valuations can be arbitrary continuous functions, we propose a price posting mechanism that achieves an expected social welfare up to an additive factor of $O(\sqrt{mn})$ from the maximum social welfare. Finally, for profit maximization (which may be non-convex in simple cases), we give nearly matching upper and lower bounds on the query complexity for separable valuations and cost (i.e., each good can be treated independently)