The scenario approach: A tool at the service of data-driven decision making

Campi, Marco C.; Carè, Algo; Garatti, Simone

doi:10.1016/j.arcontrol.2021.10.004

Cited by 33 publications

(36 citation statements)

References 89 publications

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“…In the framework of the scenario approach (also called scenario optimization) [8], [9], [11], the solution of the robust optimization problem in (1) is approximated with a finite number of samples. More precisely, we randomly generate N ∈ Z + samples, denoted by (δ 1 , δ 2 , • • • , δ N ) ∈ ∆ N , from some probability measure with the support ∆, and formulate the following scenario convex program (SCP)…”

Section: Preliminariesmentioning

confidence: 99%

“…The next lemma states the equivalence between the RCP in (1) and the max-min formulation as defined in (11).…”

Section: B Max-min Reformulationmentioning

confidence: 99%

“…Lemma 2: Consider Problems ( 1) and ( 11) for any N ∈ Z + with the optimal values J ⋆ and g * N respectively. Suppose Assumptions 1 -3 hold, then it holds that J ⋆ = g * N for any N ≥ d; Proof: From the definitions in ( 1) and (11), it holds that g N (ω) ≤ J ⋆ for any ω ∈ ∆ N , which implies that g * N ≤ J ⋆ for any N ∈ Z + . As {g * N } N ∈Z + is a monotonically increasing sequence, bounded from above by J ⋆ , we only need to show that g * d = J ⋆ .…”

Section: B Max-min Reformulationmentioning

confidence: 99%

“…An alternative approach for handling uncertainties is called the scenario approach or scenario optimization, which is somehow at the intersection of stochastic optimization and robust optimization as it is able to provide solutions to robust optimization and chance-constrained stochastic optimization problems in a probabilistic sense, see, e.g., some early works [8], [9]. We refer to [10], [11] for a comprehensive view on this subject.…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis

Wang¹,

Jungers²

2022

Preprint

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This paper is concerned with objective value performance of the scenario approach for robust convex optimization. A novel method is proposed to derive probabilistic bounds for the objective value from scenario programs with a finite number of samples. This method relies on a maxmin reformulation and the concept of complexity of robust optimization problems. With additional continuity and regularity conditions, via sensitivity analysis, we also provide explicit bounds which outperform an existing result in the literature. To illustrate the improvements of our results, we also provide a numerical example.

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Section: Preliminariesmentioning

confidence: 99%

“…The next lemma states the equivalence between the RCP in (1) and the max-min formulation as defined in (11).…”

Section: B Max-min Reformulationmentioning

confidence: 99%

Section: B Max-min Reformulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis

Wang¹,

Jungers²

2022

Preprint

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show abstract

“…We solve the problem using a sampling-based approach. Starting with a set of solution vectors, we use techniques from scenario optimization, a data-driven methodology for solving stochastic optimization problems [17,20]. As such, we construct the prediction region from the solution to an optimization problem.…”

Section: Introductionmentioning

confidence: 99%

Sampling-Based Verification of CTMCs with Uncertain Rates

Badings,

Jansen,

Junges

et al. 2022

Preprint

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We employ uncertain parametric CTMCs with parametric transition rates and a prior on the parameter values. The prior encodes uncertainty about the actual transition rates, while the parameters allow dependencies between transition rates. Sampling the parameter values from the prior distribution then yields a standard CTMC, for which we may compute relevant reachability probabilities. We provide a principled solution, based on a technique called scenario-optimization, to the following problem: From a finite set of parameter samples and a user-specified confidence level, compute prediction regions on the reachability probabilities. The prediction regions should (with high probability) contain the reachability probabilities of a CTMC induced by any additional sample. To boost the scalability of the approach, we employ standard abstraction techniques and adapt our methodology to support approximate reachability probabilities. Experiments with various well-known benchmarks show the applicability of the approach.

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Automatic scenario generation for efficient solution of robust optimal control problems

Zagorowska,

Falugi,

O'Dwyer

et al. 2023

Intl J Robust & Nonlinear

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Existing methods for nonlinear robust control often use scenario‐based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time‐varying uncertainty. This paper draws from local reduction methods used in semi‐infinite optimization to solve robust optimal control problems with parametric and time‐varying uncertainty. By iteratively adding interim worst‐case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst‐case constraint violations. In particular, we present examples where local reduction methods find worst‐case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time‐varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time‐varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.

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The scenario approach: A tool at the service of data-driven decision making

Cited by 33 publications

References 89 publications

Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis

Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis

Sampling-Based Verification of CTMCs with Uncertain Rates

Automatic scenario generation for efficient solution of robust optimal control problems

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