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We investigate a data-driven approach to constructing uncertainty sets for robust optimization problems, where the uncertain problem parameters are modeled as random variables whose joint probability distribution is not known. Relying only on independent samples drawn from this distribution, we provide a nonparametric method to estimate uncertainty sets whose probability mass is guaranteed to approximate a given target mass within a given tolerance with high confidence. The nonparametric estimators that we consider are also shown to obey distribution-free finite-sample performance bounds that imply their convergence in probability to the given target mass. In addition to being efficient to compute, the proposed estimators result in uncertainty sets that yield computationally tractable robust optimization problems for a large family of constraint functions. 

We consider the decentralized control of a discretetime, linear system subject to exogenous disturbances and polyhedral constraints on the state and input trajectories. The underlying system is composed of a finite collection of dynamically coupled subsystems, where each subsystem is assumed to have a dedicated local controller. The decentralization of information is expressed according to sparsity constraints on the state measurements that each local controller has access to. In this context, we investigate the design of decentralized controllers that are affinely parameterized in their measurement history. For problems with partially nested information structures, the optimization over such policy spaces is known to be convex. Convexity is not, however, guaranteed under more general (nonclassical) information structures in which the information available to one local controller can be affected by control actions that it cannot access or reconstruct. With the aim of alleviating the nonconvexity that arises in such problems, we propose an approach to decentralized control design where the information-coupling states are effectively treated as disturbances whose trajectories are constrained to take values in ellipsoidal contract sets whose location, scale, and orientation are jointly optimized with the underlying affine decentralized control policy. We establish a natural structural condition on the space of allowable contracts that facilitates the joint optimization over the control policy and the contract set via semidefinite programming. 

We consider the decentralized control of radial distribution systems with controllable photovoltaic inverters and energy storage resources. For such systems, we investigate the problem of designing fully decentralized controllers that minimize the expected cost of balancing demand, while guaranteeing the satisfaction of individual resource and distribution system voltage constraints. Employing a linear approximation of the branch flow model, we formulate this problem as the design of a decentralized disturbance-feedback controller that minimizes the expected value of a convex quadratic cost function, subject to robust convex quadratic constraints on the system state and input. As such problems are, in general, computationally intractable, we derive a tractable inner approximation to this decentralized control problem, which enables the efficient computation of an affine control policy via the solution of a finite-dimensional conic program. As affine policies are, in general, suboptimal for the family of systems considered, we provide an efficient method to bound their suboptimality via the optimal solution of another finite-dimensional conic program. A case study of a 12 kV radial distribution system demonstrates that decentralized affine controllers can perform close to optimal. 

We consider the setting in which an electric power utility seeks to curtail its peak electricity demand by offering a fixed group of customers a uniform price for reductions in consumption relative to their predetermined baselines. The underlying demand curve, which describes the aggregate reduction in consumption in response to the offered price, is assumed to be affine and subject to unobservable random shocks. Assuming that both the parameters of the demand curve and the distribution of the random shocks are initially unknown to the utility, we investigate the extent to which the utility might dynamically adjust its offered prices to maximize its cumulative risk-sensitive payoff over a finite number of T days. In order to do so effectively, the utility must design its pricing policy to balance the tradeoff between the need to learn the unknown demand model (exploration) and maximize its payoff (exploitation) over time. In this paper, we propose such a pricing policy, which is shown to exhibit an expected payoff loss over T days that is at most O( p T), relative to an oracle pricing policy that knows the underlying demand model. Moreover, the proposed pricing policy is shown to yield a sequence of prices that converge to the oracle optimal prices in the mean square sense.