Exact Pareto Optimal Search for Multi-Task Learning and Multi-Criteria Decision-Making

Given multiple non-convex objective functions and objective-specific weights, Chebyshev scalarization (CS) is a well-known approach to obtain an Exact Pareto Optimal (EPO), i.e., a solution on the Pareto front (PF) that intersects the ray defined by the inverse of the weights. First-order optimizers that use the CS formulation to find EPO solutions encounter practical problems of oscillations and stagnation that affect convergence. Moreover, when initialized with a PO solution, they do not guarantee a controlled trajectory that lies completely on the PF. These shortcomings lead to modeling limitations and computational inefficiency in multi-task learning (MTL) and multi-criteria decision-making (MCDM) methods that utilize CS for their underlying non-convex multi-objective optimization (MOO). To address these shortcomings, we design a new MOO method, EPO Search. We prove that EPO Search converges to an EPO solution and empirically illustrate its computational efficiency and robustness to initialization. When initialized on the PF, EPO Search can trace the PF and converge to the required EPO solution at a linear rate of convergence. Using EPO Search we develop new algorithms: PESA-EPO for approximating the PF in a posteriori MCDM, and GP-EPO for preference elicitation in interactive MCDM; experiments on benchmark datasets confirm their advantages over competing alternatives. EPO Search scales linearly with the number of decision variables which enables its use for training deep networks. Empirical results on real data from personalized medicine, e-commerce and hydrometeorology demonstrate the efficacy of EPO Search for deep MTL

Multicriteria optimization: scalarization tecniques

In this work, we briefly present the notations about multicriteria optimization problem and then focus our attention on scalarization techniques. Multicriteria optimization presents issues at the analyst that are very different from the usual problem that we can find when we deal with a single objective optimization problem; for this reason, we need new tools and definitions for solving it. One of this issue consist in ordering a vector of functions and for this purpose we need to introduce the concept of cone and all related theorems. The core of the work, however, is about scalarization methods. Those techniques consist in transforming the multiobjective optimization problem into a scalar one that is easier to solve. The most important and used method – the weighted sum method – is presented in depth, together with another general method – the Pascoletti-Serafini one – that is also used to highlight the pitfalls of other scalarization techniques. During the work, various issues about multicriteria optimization are introduced and discussed

Multi-objective optimal control problems with application to energy management systems

Orientadora: Dra. Elizabeth Wegner KarasCoorientadoras: Dra. Claudia Sagastizabal, Dra. Hasnaa ZidaniDissertação (mestrado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa : Curitiba, 22/02/2019Inclui referências: p. 122-130Resumo: Este trabalho investiga problemas de controle ótimo em tem po contínuo. Em horizonte de tem po finito, apresentamos um a nova abordagem ao analisar problemas com objetivos de diferentes naturezas, que precisam ser minimizados simultaneamente. Um objetivo esta na forma clássica de Bolza e o outro e definido como um a funcão de maximo. Baseados na teoria de Hamilton-Jacobi-Bellman caracterizamos a fronteira de Pareto fraca e a fronteira de Pareto para tais problemas. Prim eiram ente, definimos um problema de controle átim o auxiliar sem restricoes de estado e mostramos que a fronteira de Pareto fraca íe um subconjunto do conjunto de nível zero da funcao valor correspondente. Em seguida, com um a abordagem geometrica estabelecemos a caracterizaçao da fronteira de Pareto. Alguns resultados numericos sao considerados para m ostrar a relevancia do nosso metodo. Os bons resultados obtidos para horizonte de tem po finito nos m otivaram a investigar problemas de controle íotimo multiobjetivo com horizonte de tem po infinito. Com um a abordagem similar, caracterizamos a fronteira de Pareto para essa classe de problemas. Introduzimos um metodo, baseado no princípio da programacão dinamica, para reconstrucão de trajetorias de problemas de controle otimo com restricçoães de estado e horizonte de tem po infinito. A teoria íe aplicada em sistemas de gestãao de energia. Para problemas de energia simples, mas ainda representativos, que minimizam custo de geracao e emissão de CO2, comparamos a habilidade de diferentes baterias como substituto para o mecanismo de deslocamento de dem anda de ponta (load shaving). Com a resoluçcãao do problema multiobjetivo íe possível obter um a relaçcaão entre a minimizacao dos custos de geraçao de energia e de emissao de gás carbônico das usinas term oeletricas consideradas no modelo. P a la v ra s-c h a v e : Controle otimo multi-objetivo; caracterizaçao da fronteira de Pareto; abordagem de Hamilton-Jacobi-Bellman; baterias para armazenamento de energia; resposta a demanda.Abstract: In this work we investigate optim al control problems in continuous time. A novel theory is developed for finite horizon problems w ith two objectives of different nature th a t need to be minimized simultaneously. One objective is in the classical Bolza form and the other one is defined as a maximum function. Based on the Hamilton-Jacobi-Bellman framework we characterize the weak Pareto front and the Pareto front for such problems. First we define an auxiliary optim al control problem w ithout state constraints and show th a t the weak Pareto front is a subset of the zero level set of the corresponding value function. Then w ith a geometrical approach we establish a characterization of the Pareto front. Some numerical examples are considered to show the interest of our proposal. The encouraging results obtained with the finite horizon m otivated us to investigate infinite horizon multi-objective optim al control problems and characterize the corresponding Pareto front. Additionally, we introduce a m ethod, based on the dynamical programming principle, to reconstruct optim al trajectories for infinite horizon control problems w ith state constraints. The theory is applied to energy management systems. We compare the ability of different batteries as a substitute of the load shaving mechanism in smoothing the load peaks, for simple, yet representative, power mix systems w ith two different objectives. The multi-objective approach makes it possible to obtain a compromise between the minimization of generation costs and the carbon emissions of the therm al power plants in the mix. K e y w o rd s: M ulti-Objective optim al control problems; Pareto front characterization; Hamilton-Jacobi-Bellman approach; energy management systems; battery energy storage systems; dem and response

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios