Results 11  20
of
121
Decision making in an uncertain environment: the scenario based optimization approach, in
 Multiple Participant Decision Making, Advanced Knowledge International
, 2004
"... Abstract. A central issue arising in financial, engineering and, more generally, in many applicative endeavors is to make a decision in spite of an uncertain environment. Along a robust approach, the decision should be guaranteed to work well in all possible realizations of the uncertainty. A less ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Abstract. A central issue arising in financial, engineering and, more generally, in many applicative endeavors is to make a decision in spite of an uncertain environment. Along a robust approach, the decision should be guaranteed to work well in all possible realizations of the uncertainty. A less restrictive approach consists instead of requiring that the risk of failure associated to the decision should be small in some possibly probabilistic sense. From a mathematical viewpoint, the latter formulation leads to a chanceconstrained optimization program, i.e. to an optimization program subject to constraints in probability. Unfortunately, however, both the robust approach as well as the chanceconstrained approach are computationally intractable in general. In this paper, we present a computationally efficient methodology for dealing with uncertainty in optimization based on sampling a finite number of instances (or scenarios) of the uncertainty. In particular, we consider uncertain programs with convexity structure, and show that the scenariobased solution is, with high confidence, a feasible solution for the chanceconstrained problem. The proposed approach represents a viable way to address general convex decision making problems in a riskadjusted sense.
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 ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
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.
New formulations for optimization under stochastic dominance constraints
 SIAM Journal on Optimization
"... Stochastic dominance constraints allow a decisionmaker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Stochastic dominance constraints allow a decisionmaker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first and secondorder stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, and relaxing integrality in the firstorder formulation yields a secondorder formulation, demonstrating the tightness of this formulation. We also present a specialized branching strategy and heuristics which can be used with the new firstorder formulation. Computational tests illustrate the potential benefits of the new formulations.
A BranchandCut Decomposition Algorithm for Solving ChanceConstrained Mathematical Programs with Finite Support
, 2013
"... We present a new approach for exactly solving chanceconstrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods ar ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
We present a new approach for exactly solving chanceconstrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods are only able to find provably good solutions in certain very special cases. Our approach uses both decomposition, to enable processing subproblems corresponding to one possible outcome at a time, and integer programming techniques, to combine the results of these subproblems to yield strong valid inequalities. Computational results on a chanceconstrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixedinteger programming formulation and a simple decomposition approach that does not use strong valid inequalities. We also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels, as would be done when solving for the efficient frontier of risk and cost.
Solving chanceconstrained stochastic programs via sampling and integer programming
 2008 TUTORIALS IN OPERATIONS RESEARCH: STATEOFTHEART DECISIONMAKING TOOLS IN THE INFORMATIONINTENSIVE AGE
, 2008
"... Various applications in reliability and risk management give rise to optimization problems with constraints involving random parameters, which are required to be satisfied with a prespecified probability threshold. There are two main difficulties with such chanceconstrained problems. First, checki ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
Various applications in reliability and risk management give rise to optimization problems with constraints involving random parameters, which are required to be satisfied with a prespecified probability threshold. There are two main difficulties with such chanceconstrained problems. First, checking feasibility of a given candidate solution exactly is, in general, impossible since this requires evaluating quantiles of random functions. Second, the feasible region induced by chance constraints is, in general, nonconvex leading to severe optimization challenges. In this tutorial we discuss an approach based on solving approximating problems using Monte Carlo samples of the random data. This scheme can be used to yield both feasible solutions and statistical optimality bounds with high confidence using modest sample sizes. The approximating problem is itself a chanceconstrained problem, albeit with a finite distribution of modest support, and is an NPhard combinatorial optimization problem. We adopt integer programming based methods for its solution. In particular, we discuss a family valid inequalities for a integer programming formulations for a special but large class of chanceconstraint problems that have demonstrated significant computational advantages.
Giannakis, “Riskconstrained energy management with multiple wind farms
 in Proc. of 4th IEEEPES on Innovative Smart Grid Tech
, 2013
"... Abstract—To achieve the goal of high wind power penetration in future smart grids, economic energy management accounting for the stochastic nature of wind power is of paramount importance. Multiperiod economic dispatch and demandside management for power systems with multiple wind farms is consid ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
Abstract—To achieve the goal of high wind power penetration in future smart grids, economic energy management accounting for the stochastic nature of wind power is of paramount importance. Multiperiod economic dispatch and demandside management for power systems with multiple wind farms is considered in this paper. To address the challenge of intrinsically stochastic availability of the nondispatchable wind power, a chanceconstrained optimization problem is formulated to limit the risk of supplydemand imbalance based on the lossofload probability (LOLP). Since the spatiotemporal joint distribution of the wind power generation is intractable, a novel scenario approximation technique using Monte Carlo sampling is pursued. Enticingly, the problem structure is leveraged to obtain a samplesizefree problem formulation, thus making it possible to accommodate a very small LOLP requirement even with a long scheduling time horizon. Finally, to capture the temporal and spatial correlation among power outputs of multiple wind farms, an autoregressive model is introduced to generate the required samples based on wind speed distribution models as well as the windspeedtopoweroutput mappings. Numerical results are provided to corroborate the effectiveness of the novel approach. I.
H ∞ Controller Design for Spectral MIMO Models by Convex Optimization ✩
"... A new method for robust fixedorder H ∞ controller design by convex optimization for multivariable systems is investigated. Linear TimeInvariant MultiInput MultiOutput (LTIMIMO) systems represented by a set of complex values in the frequency domain are considered. It is shown that the Generalize ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
A new method for robust fixedorder H ∞ controller design by convex optimization for multivariable systems is investigated. Linear TimeInvariant MultiInput MultiOutput (LTIMIMO) systems represented by a set of complex values in the frequency domain are considered. It is shown that the Generalized Nyquist Stability criterion can be approximated by a set of convex constraints with respect to the parameters of a multivariable linearly parameterized controller in the Nyquist diagram. The diagonal elements of the controller are tuned to satisfy the desired performances, while simultaneously, the offdiagonal elements are designed to decouple the system. Multimodel uncertainty can be directly considered in the proposed approach by increasing the number of constraints. The simulation examples illustrate the effectiveness of the proposed approach.
Riskaverse stochastic optimization: Probabilisticallyconstrained models and algorithms for blackbox distributions
 In SODA
, 2011
"... We consider various stochastic models that incorporate the notion of riskaverseness into the standard 2stage recourse model, and develop novel techniques for solving the algorithmic problems arising in these models. A key notable feature of our work that distinguishes it from work in some other re ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We consider various stochastic models that incorporate the notion of riskaverseness into the standard 2stage recourse model, and develop novel techniques for solving the algorithmic problems arising in these models. A key notable feature of our work that distinguishes it from work in some other related models, such as the (standard) budget model and the (demand) robust model, is that we obtain results in the blackbox setting, that is, where one is given only sampling access to the underlying distribution. Our first model, which we call the riskaverse budget model, incorporates the notion of riskaverseness via a probabilistic constraint that restricts the probability (according to the underlying distribution) with which the secondstage cost may exceed a given budget B to at most a given input threshold ρ. We also a consider a closelyrelated model that we call the riskaverse robust model, where we seek to minimize the firststage cost and the (1 − ρ)quantile (according to the distribution) of the secondstage cost. We obtain approximation algorithms for a variety of combinatorial optimization problems including the set cover, vertex cover, multicut on trees, and facility location problems, in the riskaverse budget and robust models with blackbox distributions. Our main contribution is to devise a fully polynomial approximation scheme for solving the LPrelaxations of a widevariety of riskaverse budgeted problems. Complementing this, we give a simple rounding procedure that shows that one can exploit existing LPbased approximation algorithms for the 2stagestochastic and/or deterministic counterpart of the problem to round the fractional solution and obtain an approximation algorithm for the riskaverse problem. To the best of our knowledge, these are the first approximation results for problems involving probabilistic constraints and blackbox distributions. A notable feature of our scheme is that it extends easily
Calafiore, Notes on the scenario design approach
 IEEE Trans. Automat. Control
"... Abstract—The scenario optimization method developed in [5] is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design, that would otherwise be hard to tackle via standard deterministic ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
Abstract—The scenario optimization method developed in [5] is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design, that would otherwise be hard to tackle via standard deterministic techniques. In this note, we explore some further aspects of the scenario methodology, and present two results pertaining to the tightness of the sample complexity bounds. We also state a new theorem that enables the user to make apriori probabilistic claims on the scenario solution, with one level of probability only. Index Terms—Probabilistic robustness, randomized algorithms, robust control, robust convex optimization, scenario design. I. PRELIMINARIES Recently, techniques based on uncertainty randomization have gained increasing favor among both control theoreticians and practitioners. Theoreticians are attracted by the solid foundations of these methods, rooting in the theory of probability, optimization and stochastic processes, while practitioners are interested in their relative simplicity of practical implementation. An uptodate description of this body of techniques, along with applications to control analysis and design problems and many pointers to the literature, can be found in the texts [7], [19]. Among these techniques, the socalled scenario design method developed in [5] permits one to effectively solve control design problems that can be cast in the form of a convex optimization program with uncertain constraints. A significant class of control problems indeed fall in this framework, see for instance the discussion and examples in [5]. In what follows, we briefly review the essential points of the scenario optimization approach of [5] in order to prepare the terrain for our further discussion. We also refer the reader to the recent contributions [1], [2], [9], [10] for further information on the scenario approach. Scenario Optimization: Consider an uncertain convex optimization problem of the form min subject to:
Gain scheduled controller design by linear programming
 in Proceedings of the European Control Conference
, 2007
"... Abstract—A linear programming approach is proposed to tune fixedorder linearly parameterized gainscheduled controllers for stable SISO Linear Parameter Varying (LPV) plants. The method is based on the shaping of the openloop transfer functions in the Nyquist diagram with constraints on the robust ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
Abstract—A linear programming approach is proposed to tune fixedorder linearly parameterized gainscheduled controllers for stable SISO Linear Parameter Varying (LPV) plants. The method is based on the shaping of the openloop transfer functions in the Nyquist diagram with constraints on the robustness margins and on the lower approximation of the crossover frequency. Two optimization problems are considered: optimization for robustness and optimization for performance. This method directly computes a gainscheduled controller from a set of frequencydomain models in different operating points or from an LPV model and no interpolation is needed. In terms of closedloop performance, this approach leads to extremely good results. However, closedloop stability is ensured only locally and for slow variations of the scheduling parameters. A stability analysis should be performed for fast variations of the scheduling parameters. An application to a highprecision doubleaxis positioning system illustrates the effectiveness of the proposed approach. I.