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118
Convex Approximations of Chance Constrained Programs
"... We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given (close to one) probability, a system of randomly perturbed convex constraints. Our goal is to build a computationally tractable approximation of this (typically intractabl ..."
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Cited by 127 (6 self)
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We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given (close to one) probability, a system of randomly perturbed convex constraints. Our goal is to build a computationally tractable approximation of this (typically intractable) problem, i.e., an explicitly given convex optimization program with the feasible set contained in the one of the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables, we build a large deviations type approximation, referred to as ‘Bernstein approximation’, of the chance constrained problem. This approximation is convex, and thus efficiently solvable. We propose a simulationbased scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and wellknown scenario approximation approaches. Finally, we extend our construction to the case of ambiguously chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 110 (16 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
Second Order Cone Programming Approaches for Handling Missing and Uncertain Data
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... We propose a novel second order cone programming formulation for designing robust classifiers which can handle uncertainty in observations. Similar formulations are also derived for designing regression functions which are robust to uncertainties in the regression setting. The proposed formulations ..."
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Cited by 55 (9 self)
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We propose a novel second order cone programming formulation for designing robust classifiers which can handle uncertainty in observations. Similar formulations are also derived for designing regression functions which are robust to uncertainties in the regression setting. The proposed formulations are independent of the underlying distribution, requiring only the existence of second order moments. These formulations are then specialized to the case of missing values in observations for both classification and regression problems. Experiments show that the proposed formulations outperform imputation.
A sample approximation approach for optimization with probabilistic constraints
 IPCO 2007, Lecture Notes in Comput. Sci
, 2007
"... Abstract. We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with risk level larg ..."
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Cited by 50 (12 self)
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Abstract. We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.
Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications
 J OPTIM THEORY APPL (2009) 142: 399–416
, 2009
"... We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrai ..."
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Cited by 31 (1 self)
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We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained problems. Numerical experiments are performed to correctly tune the parameters involved in the SAA. In addition, we present a method for constructing statistical lower bounds for the optimal value of the considered problem and discuss how one should tune the underlying parameters. We apply the SAA to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem.
A Probabilistic ParticleControl Approximation of ChanceConstrained Stochastic Predictive Control
"... Abstract—Robotic systems need to be able to plan control actions that are robust to the inherent uncertainty in the real world. This uncertainty arises due to uncertain state estimation, disturbances, and modeling errors, as well as stochastic mode transitions such as component failures. Chancecons ..."
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Cited by 22 (1 self)
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Abstract—Robotic systems need to be able to plan control actions that are robust to the inherent uncertainty in the real world. This uncertainty arises due to uncertain state estimation, disturbances, and modeling errors, as well as stochastic mode transitions such as component failures. Chanceconstrained control takes into account uncertainty to ensure that the probability of failure, due to collision with obstacles, for example, is below a given threshold. In this paper, we present a novel method for chanceconstrained predictive stochastic control of dynamic systems. The method approximates the distribution of the system state using a finite number of particles. By expressing these particles in terms of the control variables, we are able to approximate the original stochastic control problem as a deterministic one; furthermore, the approximation becomes exact as the number of particles tends to infinity. This method applies to arbitrary noise distributions, and for systems with linear or jump Markov linear dynamics, we show that the approximate problem can be solved using efficient mixedinteger linearprogramming techniques. We also introduce an important weighting extension that enables the method to deal with lowprobability mode transitions such as failures. We demonstrate in simulation that the new method is able to control an aircraft in turbulence and can control a ground vehicle while being robust to brake failures. Index Terms—Chance constraints, hybrid discretecontinuous systems, nonholonomic motion planning, planning under stochastic uncertainty. I.
Probabilistic guarantees for the N1 security of systems with wind power generation
 International Conference on Probabilistic Methods Applied to Power Systems
, 2012
"... Abstract — This paper proposes a novel framework for designing a N1 secure generation dayahead dispatch for power systems with a high penetration of fluctuating power sources, e.g. wind or PV power. To achieve this, we integrate the security constraints in a DC optimal power flow optimization and ..."
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Cited by 20 (9 self)
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Abstract — This paper proposes a novel framework for designing a N1 secure generation dayahead dispatch for power systems with a high penetration of fluctuating power sources, e.g. wind or PV power. To achieve this, we integrate the security constraints in a DC optimal power flow optimization and formulate a stochastic program with chance constraints, which encode the probability of satisfying the transmission capacity constraints of the lines. To solve the resulting problem numerically, we transform the initial problem to a tractable one by using the so called scenario approach, which is based on sampling the uncertain parameter (in this paper the wind power) while keeping the desired probabilistic guarantees. To generate wind power scenarios a Markov chain based model is employed. To illustrate the effectiveness of the proposed technique we apply it to the IEEE 30bus network, and compare it with the solution of a deterministic variant of the problem, where the operator determines a secure generation dispatch based only on the available wind power forecast. A MonteCarlo simulation study is conducted to collect statistical results regarding the performance of our method.
Cuttingset methods for robust convex optimization with pessimizing oracles
 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, UNIVERSITY OF CALIFORNIA, SAN DIEGO. FROM
, 2011
"... We consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worstcase ana ..."
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Cited by 17 (5 self)
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We consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worstcase analysis. With exact worstcase analysis, the method is shown to converge to a robust optimal point. With approximate worstcase analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worstcase analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warmstart techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.
ChanceConstrained Optimal Path Planning with Obstacles
"... Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. For robust execution, we must take into account uncertainty, which arises due to uncertain localization, modeling errors, and disturbances. Prior work handled the case of setbounded uncertainty. We present here ..."
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Cited by 10 (0 self)
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Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. For robust execution, we must take into account uncertainty, which arises due to uncertain localization, modeling errors, and disturbances. Prior work handled the case of setbounded uncertainty. We present here a chanceconstrained approach, which uses instead a probabilistic representation of uncertainty. The new approach plans the future probabilistic distribution of the vehicle state so that the probability of failure is below a specified threshold. Failure occurs when the vehicle collides with an obstacle, or leaves an operatorspecified region. The key idea behind the approach is to use bounds on the probability of collision to show that, for linearGaussian systems, we can approximate the nonconvex chanceconstrained optimization problem as a Disjunctive Convex Program. This can be solved to global optimality using branchandbound techniques. In order to improve computation time, we introduce a customized solution method that returns almostoptimal solutions along with a hard bound on the level of suboptimality. We present an empirical validation with an aircraft obstacle avoidance example.