Risk and complexity in scenario optimization

Scenario optimization is a broad methodology to perform optimization based on empirical knowledge. One collects previous cases, called “scenarios”, for the set-up in which optimization is being performed, and makes a decision that is optimal for the cases that have been collected. For convex optimiz...

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Bibliographic Details
Published inMathematical programming Vol. 191; no. 1; pp. 243 - 279
Main Authors Garatti, S., Campi, M. C.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2022
Springer
Springer Nature B.V
Subjects
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Complexity
Convexity
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Risk assessment
Theoretical
Probabilistic constraints
90C25
62C12
Scenario approach
90C15
Stochastic optimization
Data-driven optimization
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Summary:Scenario optimization is a broad methodology to perform optimization based on empirical knowledge. One collects previous cases, called “scenarios”, for the set-up in which optimization is being performed, and makes a decision that is optimal for the cases that have been collected. For convex optimization, a solid theory has been developed that provides guarantees of performance, and constraint satisfaction, of the scenario solution. In this paper, we open a new direction of investigation: the risk that a performance is not achieved, or that constraints are violated, is studied jointly with the complexity (as precisely defined in the paper) of the solution. It is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk. This result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios; in particular, no extra knowledge on the distribution by which scenarios are generated is assumed, so that the result is broadly applicable. This deep-seated result unveils a fundamental and general structure of data-driven optimization and suggests practical approaches for risk assessment.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01446-4