Multi-objective Pareto-optimal control: an application to wastewater management

Álvarez-Vázquez, Lino J.; García-Chan, N.; Martínez, A.; Vázquez-Méndez, Miguel E.

doi:10.1007/s10589-008-9190-9

Cited by 28 publications

(10 citation statements)

References 20 publications

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“…In next two sections we will study in detail the problems (5) and (6). In particular we will prove that, for each b ∈ U ad , the problem (5) admits a unique solution (which guarantees that the above definition is correct), and we will prove that our problem admits, at least, one Stackelberg strategy (the problem (6) admits, at least, one solution).…”

Section: Description and Formulation Of The Problemmentioning

confidence: 93%

“…Conversely, if the system is managed by different organizations, the problem becomes a multi-objective character, and the concept of optimal strategy depends on the particular circumstances of each situation. Recently, the authors have studied some of these scenarios both from a cooperative viewpoint [6,7] as from a non-cooperative one [8].…”

Section: Introductionmentioning

confidence: 99%

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Stackelberg strategies for wastewater management

Álvarez-Vázquez

García-Chan

Martínez

et al. 2015

Journal of Computational and Applied Mathematics

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a b s t r a c tManagement of wastewater has become a critical environmental problem during past decades, specially for the particular case of urban areas. In this work we use numerical simulation, and combine optimal control theory of partial differential equations with multi-objective optimization techniques to formulate and solve this type of problems from a new original hierarchical viewpoint with a leader and a follower (Stackelberg strategies). Specifically, the main aim of this paper deals with applying these Stackelberg techniques to the environmental problem related to determining the optimal location and the optimal purification profile when building a new treatment plant in a wastewater depuration system. We analyze the control problem, introduce a numerical algorithm, and show the possibilities of our approach by solving a realistic problem posed in the Estuary of Vigo (NW Spain).

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Section: Description and Formulation Of The Problemmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Stackelberg strategies for wastewater management

Álvarez-Vázquez

García-Chan

Martínez

et al. 2015

Journal of Computational and Applied Mathematics

Self Cite

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“…Problem. Find a Pareto equilibrium associated to (3) and 4, that is, a control f ∈ L 2 (ω) N such that there is no f satisfying…”

Section: Definition Of Pareto Equilibriummentioning

confidence: 99%

“…They are very important from the theoretical and practical viewpoints and appear frequently in the applications. For some previous works on the subject, see for instance [3].…”

mentioning

confidence: 99%

Theoretical and numerical results for some bi-objective optimal control problems

Fernández-Cara¹,

Marín-Gayte²

2020

Communications on Pure &Amp; Applied Analysis

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This article deals with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear cases. We prove the existence of equilibria, we deduce appropriate optimality systems, we present some iterative algorithms and we establish convergence results. Then, we analyze the existence and characterization of Pareto equilibria for the Navier-Stokes equations. Here, we use the formalism of Dubovitskii and Milyutin. In this framework, we also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.

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“…Additionally, it has been shown in Logist et al (2010a that the integration of optimal control methods with these scalarization approaches leads to an e cient solution of challenging MultiObjective Optimal Control Problems (MOOCPs). This field of research has 35 been particularly active in the last decade and new methods to e ciently tackle non-convex optimal control problems are currently developed (Yalçma Kaya & Maurer (2014)) and applied to di↵erent engineering fields (Alvarez-Vázquez et al (2010)). …”

Section: Introductionmentioning

confidence: 99%

Interactive NBI and (E)NNC methods for the progressive exploration of the criteria space in multi-objective optimization and optimal control

Vallerio

Vercammen

Impe

et al. 2015

Computers & Chemical Engineering

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A wide range of problems arising from real world applications present multiple and conflicting objectives to be simultaneously optimized. However, this multi-objective nature is too often neglected. Multi-objective optimization proved to be a powerful tool to correctly describe the trade-o↵s among conflicting objectives in a set of optimal solutions known as the Pareto set. This paper introduces an interactive method to solve multi-objective problems based on geometric considerations. The method returns a wider Pareto set, at a negligible computational cost, when compared to existing methods. The interactivity also allows the decision-maker to explore only relevant parts of the Pareto set. The extreme solutions yield insightful considerations on the generation of the scalarization parameters for the Normal Boundary Intersection and the Enhanced Normalized Normal Constraints methods. The proposed method is applied to: (i) three scalar multi-objective problems, (ii) the multi-objective optimal control of a tubular and (iii) a fed-batch reactor.

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Multi-objective Pareto-optimal control: an application to wastewater management

Abstract: Multi-objective, Pareto, Optimal control, Wastewater,

Cited by 28 publications

References 20 publications

Stackelberg strategies for wastewater management

Stackelberg strategies for wastewater management

Theoretical and numerical results for some bi-objective optimal control problems

Interactive NBI and (E)NNC methods for the progressive exploration of the criteria space in multi-objective optimization and optimal control

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