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40
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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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.
Provably nearoptimal samplingbased policies for stochastic inventory control models
 Proceedings, 38th Annual ACM Symposium on Theory of Computing
, 2006
"... In this paper, we consider two fundamental inventory models, the singleperiod newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true ..."
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Cited by 16 (2 self)
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In this paper, we consider two fundamental inventory models, the singleperiod newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are given explicitly, these models are wellstudied and relatively straightforward to solve. However, in most reallife scenarios, the true demand distributions are not available or they are too complex to work with. Thus, a samplingdriven algorithmic framework is very attractive, both in practice and in theory. We shall describe how to compute samplingbased policies, that is, policies that are computed based only on observed samples of the demands without any access to, or assumptions on, the true demand distributions. Moreover, we establish bounds on the number of samples required to guarantee that with high probability, the expected cost of the samplingbased policies is arbitrarily close (i.e., with arbitrarily small relative error) compared to the expected cost of the optimal policies which have full access to the demand distributions. The bounds that we develop are general, easy to compute and do not depend at all on the specific demand distributions.
On the Power of Robust Solutions in TwoStage Stochastic and Adaptive Optimization Problems
"... informs doi 10.1287/moor.1090.0440 ..."
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Stochastic optimization on continuous domains with finitetime guarantees by Markov chain Monte Carlo Methods
"... We introduce bounds on the finitetime performance of Markov chain Monte Carlo (MCMC) algorithms in solving global stochastic optimization problems defined over continuous domains. It is shown that MCMC algorithms with finitetime guarantees can be developed with a proper choice of the target distri ..."
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Cited by 9 (3 self)
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We introduce bounds on the finitetime performance of Markov chain Monte Carlo (MCMC) algorithms in solving global stochastic optimization problems defined over continuous domains. It is shown that MCMC algorithms with finitetime guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm. This work is inspired by the concept of finitetime learning with known accuracy and confidence developed in statistical learning theory.
A Geometric Characterization of the Power of Finite Adaptability in Multistage Stochastic and Adaptive Optimization
"... In this paper, we show a significant role that geometric properties of uncertainty sets, such as symmetry, play in determining the power of robust and finitely adaptable solutions in multistage stochastic and adaptive optimization problems. We consider a fairly general class of multistage mixed in ..."
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Cited by 8 (5 self)
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In this paper, we show a significant role that geometric properties of uncertainty sets, such as symmetry, play in determining the power of robust and finitely adaptable solutions in multistage stochastic and adaptive optimization problems. We consider a fairly general class of multistage mixed integer stochastic and adaptive optimization problems and propose a good approximate solution policy with performance guarantees that depend on the geometric properties of the uncertainty sets. In particular, we show that a class of finitely adaptable solutions is a good approximation for both the multistage stochastic as well as the adaptive optimization problem. A finitely adaptable solution generalizes the notion of a static robust solution and specifies a small set of solutions for each stage and the solution policy implements the best solution from the given set depending on the realization of the uncertain parameters in past stages. Therefore, it is a tractable approximation to a fullyadaptable solution for the multistage problems. To the best of our knowledge, these are the first approximation results for the multistage problem in such generality. Moreover, the results and the proof techniques are quite general and also extend to include important constraints such as integrality and linear conic constraints.
Robust conjugate duality for convex optimization under uncertainty with application to data classification
, 2010
"... In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameteri ..."
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Cited by 5 (5 self)
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In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also is necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.
Multicut Benders Decomposition Algorithm for Process Supply Chain Planning under Uncertainty
, 2011
"... In this paper, we present a multicut version of the Benders decomposition method for solving twostage stochastic linear programming problems, including stochastic mixedinteger programs with only continuous recourse (twostage) variables. The main idea is to add one cut per realization of uncertaint ..."
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Cited by 4 (1 self)
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In this paper, we present a multicut version of the Benders decomposition method for solving twostage stochastic linear programming problems, including stochastic mixedinteger programs with only continuous recourse (twostage) variables. The main idea is to add one cut per realization of uncertainty to the master problem in each iteration, that is, as many Benders cuts as the number of scenarios added to the master problem in each iteration. Two examples are presented to illustrate the application of the proposed algorithm. One involves productiontransportation planning under demand uncertainty, and the other one involves multiperiod planning of global, multiproduct chemical supply chains under demand and freight rate uncertainty. Computational studies show that while both the standard and the multicut versions of the Benders decomposition method can solve largescale stochastic programming problems with reasonable computational effort, significant savings in CPU time can be achieved by using the proposed multicut algorithm.
Stochastic Optimization of Electricity Portfolios: Scenario Tree Modeling and Risk Management
"... Abstract We present recent developments in the field of stochastic programming with regard to application in power management. In particular we discuss issues of scenario tree modeling, i.e., appropriate discrete approximations of the underlying stochastic parameters. Moreover, we suggest risk avoid ..."
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Abstract We present recent developments in the field of stochastic programming with regard to application in power management. In particular we discuss issues of scenario tree modeling, i.e., appropriate discrete approximations of the underlying stochastic parameters. Moreover, we suggest risk avoidance strategies via the incooperation of socalled polyhedral risk functionals into stochastic programs. This approach, motivated through tractability of the resulting problems, is a constructive framework providing particular flexibility with respect to the dynamic aspects of risk. 1
A Tight Characterization of the Performance of Static Solutions in Twostage Adjustable Robust Linear Optimization
 SUBMITTED TO MATH PROGRAMMING
"... In this paper, we study the performance of static solutions for twostage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap. Computing an optimal adjustable robust optimization problem is often ..."
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In this paper, we study the performance of static solutions for twostage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap. Computing an optimal adjustable robust optimization problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters [16]. On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems. We show that for a fairly general class of uncertainty sets, a static solution is optimal for the twostage adjustable robust linear optimization problem. This is highly surprising in view of the usual perception about the conservativeness of static solutions. Furthermore, when a static solution is not optimal for the adjustable robust problem, we give a tight approximation bound on the performance of the static solution that is related to a measure of nonconvexity of a transformation of the uncertainty set. We also show that our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set [9,8].
The Design of Effective and Robust Supply Chain Networks
, 2009
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.