A General Scenario Theory for Nonconvex Optimization and Decision Making

Abstract

The scenario approach is a general methodology for data-driven optimization that has attracted a great deal of attention in the past few years. It prescribes that one collects a record of previous cases (scenarios) from the same setup in which optimization is being conducted and makes a decision which attains optimality for the seen cases. Scenario optimization is by now very well understood for convex problems, where a theory exists that rigorously certifies the generalization properties of the solution, that is, the ability of the solution to perform well in connection to new situations. This theory supports the scenario methodology and justifies its use. This paper considers nonconvex problems. While other contributions in the non-convex setup already exist, we here take a major departure from previous approaches. We suggest that the generalization level is evaluated only after the solution is found and its complexity in terms of the length of a support sub-sample (a notion precisely introduced in this paper) is assessed. As a consequence, the generalization level is stochastic and adjusted case by case to the available scenarios. This fact is key to obtain tight results. The approach adopted in this paper applies not only to optimization, but also to generic decision problems where the solution is obtained according to a rule which is not necessarily the optimization of a cost function. Accordingly, in our presentation we adopt a general stance of which optimization is just seen as a particular case.

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“…The following proposition establishes a basic property of any compression associated to the mapping (9).…”

Data-Driven Scenario Optimization for Automated Controller Tuning With Probabilistic Performance Guarantees

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Providing Grid Services With Heat Pumps: A Review

“…In this method, the capacity determination must not violate temperature constraints under a set of disturbance scenarios that are developed based on historical conditions. By satisfying a certain number of these scenarios, the controller can provide the flexibility it offers with a certain confidence level [85]. While this can be computationally intensive, scenario-based optimization can provide a less conservative flexibility capacity than robust optimization while still considering uncertainty.…”