Functions

“…The Lebesgue measure of the set will be called its n-dimensional volume [Chap. 21] 31 and denoted Vol Q k = Ω k . We assume that the values of g k are from a uniform distribution over Q k for any realization of uncertainty.…”

Section: Impact Of Number Of Scenariosmentioning

confidence: 99%

“…The Lebesgue measure of the set will be called its n‐dimensional volume [Chap. 21] 31 and denoted

\mathrm{Vol}\kern3.0235pt {\mathbb{Q}}_k={\Omega}_k

. We assume that the values of

{g}_k

are from a uniform distribution over

{\mathbb{Q}}_k

for any realization of uncertainty.…”

Section: Local Reduction For Optimal Controlmentioning

confidence: 99%

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 proof of Lemma 4.3 uses a variant of the Hahn-Banach theorem, Minkowski's theorem, that ensures that two disjoint convex sets can be separated by a hyperplane. Only one variant of the Hahn-Banach theorem can be proved in ZF, the so-called Finite Extension Lemma (FEL) [Schechter 1997]. 21 But FEL is too weak to imply Minkowski's Theorem ( l is not a convex hull of the union of any of its hyperspaces and a finite number of points).…”

Section: Sketch Of Lewis' Argumentmentioning

confidence: 99%

Economic theory and the Alternative Set Theory AFA-+AD+DC

Tohmé

2009

Logic Journal of IGPL

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Many authors in the discipline as well as outsiders have claimed that the main results from Mathematical Economics are far removed from real world phenomena. A more precise version of this position is that one of the main reasons for this unrealistic stance is the use of the wrong formal tools. So, for example, it has been pointed out that the computability of choice functions as well as the existence of economic equilibria and of states of the world may not be ensured in general if the assumed set theory is ZFC. We will show that there exists a very natural set theory that overcomes some formal limitations of contemporary economic theory. A switch to an alternative set theory helps to obtain in a more natural way results widely accepted by mathematical economists. Moreover, alternative set-theoretical frameworks convey different intuitions about how agents behave when solving problems. We claim that AFA − +AD +DC is the adequate alternative set-theoretical universe for economic theory.

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Functions

Cited by 42 publications

References 0 publications

Asymptotic results for compound sums in Banach Spaces

Asymptotic results for compound sums in Banach Spaces

Automatic scenario generation for efficient solution of robust optimal control problems

Economic theory and the Alternative Set Theory AFA-+AD+DC

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