A General Scenario Theory for Nonconvex Optimization and Decision Making
TLDR
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 that is not necessarily the optimization of a cost function.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 that 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 nonconvex 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 subsample (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 that 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.read more
Citations
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Data-driven decision making in power systems with probabilistic guarantees: Theory and applications of chance-constrained optimization
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TL;DR: A comprehensive review on the applications of chance-constrained optimization in power systems and a critical comparison of existing methods based on numerical simulations, conducted on standard power system test cases are provided.
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Engineering analysis with probability boxes: A review on computational methods
Matthias G.R. Faes,Matthias G.R. Faes,Marco Daub,Marco Daub,Stefano Marelli,Edoardo Patelli,Edoardo Patelli,Michael Beer,Michael Beer,Michael Beer +9 more
TL;DR: A state-of-the art review for the modelling and propagation of p-boxes with a special focus on structural reliability analysis is given.
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Optimal Capacity Design and Operation of Energy Hub Systems
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Data-Driven Scenario Optimization for Automated Controller Tuning With Probabilistic Performance Guarantees
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TL;DR: In this article, the authors used non-convex scenario theory to provide a distribution-free bound on the probability of the closed-loop performance measures for automated controller tuning under uncertainty.
References
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Statistical learning theory
TL;DR: Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.
Proceedings ArticleDOI
YALMIP : a toolbox for modeling and optimization in MATLAB
TL;DR: Free MATLAB toolbox YALMIP is introduced, developed initially to model SDPs and solve these by interfacing eternal solvers by making development of optimization problems in general, and control oriented SDP problems in particular, extremely simple.
Book ChapterDOI
Graph Implementations for Nonsmooth Convex Programs
Michael C. Grant,Stephen Boyd +1 more
TL;DR: Graph implementations as mentioned in this paper is a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework, which allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved.
Book
Lectures on Stochastic Programming: Modeling and Theory
TL;DR: The authors dedicate this book to Julia, Benjamin, Daniel, Natan and Yael; to Tsonka, Konstatin and Marek; and to the Memory of Feliks, Maria, and Dentcho.
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