Scenarios and Policy Aggregation in Optimization Under Uncertainty

Rockafellar, R. T.; Wets, Roger J.-B.

doi:10.1287/moor.16.1.119

Cited by 1,065 publications

(587 citation statements)

References 17 publications

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“…In particular, if T = Of where f is a closed proper convex function, this problem reduces to that of minimizing f over V. One application of this method is the "progressive hedging" stochastic programming method of Rockafellar and Wets [51]. Consider now the operator…”

Section: Some Interesting Special Easesmentioning

confidence: 99%

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Eckstein¹,

Bertsekas

1992

Mathematical Programming

2,480

2,027

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This paper shows, by means of an operator called a splitting operator, that the Douglas-Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm, Therefore, applications of Douglas-Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new, generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

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Section: Some Interesting Special Easesmentioning

confidence: 99%

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Eckstein¹,

Bertsekas

1992

Mathematical Programming

2,480

2,027

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show abstract

“…Some methods for solving these problems approximately are closely related to our method; for example, probabilistic sampling or scenario approaches (see, e.g., [12][13][14]). …”

Section: Regularization and Stochastic Programmingmentioning

confidence: 99%

“…(The state deviation with nominal parameters is 0.04.) Histograms of the state deviation for both the nominal and the robust forces, over the 64 extreme points, are shown in figure (11), and state trajectories are shown in figure (12).…”

Section: Numerical Examplementioning

confidence: 99%

Cutting-set methods for robust convex optimization with pessimizing oracles

Mutapcic

Boyd

2009

Optimization Methods and Software

127

170

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We consider a general worst-case 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 worst-case analysis. With exact worst-case analysis, the method is shown to converge to a robust optimal point. With approximate worst-case 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 worst-case 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 warm-start 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.

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“…Let A (t) denote the set of scenarios selected to be included in set T (t) 1 (t ≥ 1). The sets T (t) 0 and T (t) 1 are constructed as follows: 0 , and by replacing constraints (15) and (16) in SMKP by constraints (23), (24), (26)- (28). Setting u 0 = min…”

Section: Outer Approximationmentioning

confidence: 99%

Mathematical programming approaches for generating p-efficient points

Lejeune¹,

Noyan²

2010

European Journal of Operational Research

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Probabilistically constrained problems, in which the random variables are finitely distributed, are nonconvex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.

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Scenarios and Policy Aggregation in Optimization Under Uncertainty

Cited by 1,065 publications

References 17 publications

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Cutting-set methods for robust convex optimization with pessimizing oracles

Mathematical programming approaches for generating p-efficient points

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