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Randomized Sampling for Large ZeroSum Games
, 2012
"... This paper addresses the solution of large zerosum matrix games using randomized methods. We formalize a procedure, termed as the sampled security policy (SSP) algorithm, by which a player can compute policies that, with a high confidence, are security policies against an adversary using randomized ..."
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This paper addresses the solution of large zerosum matrix games using randomized methods. We formalize a procedure, termed as the sampled security policy (SSP) algorithm, by which a player can compute policies that, with a high confidence, are security policies against an adversary using randomized methods to explore the possible outcomes of the game. The SSP algorithm essentially consists of solving a stochastically sampled subgame that is much smaller than the original game. We also propose a randomized algorithm, termed as the sampled security value (SSV) algorithm, which computes a highconfidence securitylevel (i.e., worstcase outcome) for a given policy, which may or may not have been obtained using the SSP algorithm. For both the SSP and the SSV algorithms we provide results to determine how many samples are needed to guarantee a desired level of confidence. We start by providing results when the two players sample policies with the same distribution and subsequently extend these results to the case of mismatched distributions. We demonstrate the usefulness of these results in a hideandseek game that exhibits exponential complexity.
Randomized solutions to partial information dynamic zerosum games
 In: Proc. of the 2011 Amer. Contr. Conf
, 2011
"... Abstract — This paper presents randomized methods to solve partial information dynamic zerosum games. We extend the recently introduced sampled saddlepoint (SSP) algorithm, which provided probabilistic security guarantees in static zerosum matrix games. A straightforward extension to partial info ..."
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Abstract — This paper presents randomized methods to solve partial information dynamic zerosum games. We extend the recently introduced sampled saddlepoint (SSP) algorithm, which provided probabilistic security guarantees in static zerosum matrix games. A straightforward extension to partial information dynamic games is to apply the SSP algorithm to the matrix obtained by recording the outcomes of playing every policy of one player against every policy of the other player. However, the matrix so obtained has typically a very large size. This paper formalizes a novel extension of the SSP algorithm to partial information dynamic games, which does not require generating the entire matrix. We show that the bounds derived for the SSP algorithm in the static case, provide the same level of probabilistic security for a partial information dynamic game. The effectiveness of the procedure is demonstrated by solving a prototypical example of a board game with partial information, for which no deterministic security levels have been published.
A unified analysis of securityconstrained opf formulations considering uncertainty, risk, and controllability in single and multiarea systems
 in Bulk Power System Dynamics and Control  IX Optimization, Security and Control of the Emerging Power Grid (IREP), IREP Symposium, 2013
"... ©2013 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other wo ..."
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©2013 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
Adaptively Constrained Stochastic Model Predictive Control for ClosedLoop Constraint Satisfaction
"... discretetime linear systems subject to additive disturbances with chance constraints on the states and hard constraints on the inputs is considered. Current chance constrained MPC methods—based on analytic reformulations or on sampling approaches—tend to be conservative partly because they fail to ..."
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discretetime linear systems subject to additive disturbances with chance constraints on the states and hard constraints on the inputs is considered. Current chance constrained MPC methods—based on analytic reformulations or on sampling approaches—tend to be conservative partly because they fail to exploit the predefined violation level in closedloop. For many practical applications, this conservatism can lead to a loss in performance. We propose an adaptive SMPC scheme that starts with a standard conservative chance constrained formulation and then online adapts the formulation of constraints based on the experienced violation frequency. Using martingale theory we establish guarantees of convergence to the desired level of constraint violation in closedloop for a special class of linear systems. Comments are given on how to extend this to a broader class of (non)linear systems. The developed methodology is demonstrated with an illustrative example.
Design in the presence of uncertainty: the scenario approach
"... Abstract. In this chapter, we describe the "scenario approach" methodology to solve design problems in the presence of uncertainty. We focus on problems that can be reformulated as finitedimensional optimization problems, where a linear cost has to be minimized, subject to an uncertain c ..."
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Abstract. In this chapter, we describe the "scenario approach" methodology to solve design problems in the presence of uncertainty. We focus on problems that can be reformulated as finitedimensional optimization problems, where a linear cost has to be minimized, subject to an uncertain convex constraint. To account for the uncertain element, a probabilistic approach is adopted where the constraint satisfaction has to be only achieved over a set of uncertainty instances having probability larger than a given level (chanceconstrained optimization). Chanceconstrained optimization problems are hard to solve in general, and the scenario approach provides an effective methodology for obtaining an approximate solution via random sampling of the uncertain element. If the number of extractions is appropriately chosen, probabilistic guarantees on the feasibility level of the scenario solution can be provided. The objective of this chapter is to illustrate the scenario approach at a tutorial level, focusing mainly on algorithmic aspects. Some simple examples are used throughout the presentation to this purpose.
Performance assessment and design of abstracted models for stochastic hybrid systems through a randomized approach
, 2014
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Randomized minmax optimization: the exact risk of multiple cost levels
"... AbstractIn this paper, we present a theoretical result that applies to convex optimization problems in the presence of an uncertain stochastic parameter. We consider the minmax samplebased solution, i.e. the minmax solution computed over a finite sample of instances of the uncertain stochastic ..."
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AbstractIn this paper, we present a theoretical result that applies to convex optimization problems in the presence of an uncertain stochastic parameter. We consider the minmax samplebased solution, i.e. the minmax solution computed over a finite sample of instances of the uncertain stochastic parameter, and the costs incurred by this solution in correspondence of the sampled parameter instances. Our goal is to evaluate the risks associated to the various costs, where the risk associated to a cost is the probability that the cost is exceeded when a new uncertainty instance is seen. The theoretical result proven in this paper is that the risks form a random vector whose probability distribution is always an ordered Dirichlet distribution, irrespective of the probability measure of the uncertain stochastic parameter. This evaluation characterizes completely the risks associated to the costs, and represents a fullfledged result on the reliability of the minmax samplebased solution.
Digital Object Identifier: 10.1109/ACC.2015.7170853
"... Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting / republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighte ..."
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Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting / republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.