Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure

Pareto front for chemotherapy scheludes

In this paper we consider a multi-objective approach to chemother-apy optimization. We assume that the dynamics of the tumor are mod-eling for the stochastic Gompertz growth model with a linear cell-losseffect . We consider fuzzy constraints for the problem. The primaryobjective is to eradicate the tumor (curative treatment) , the secondobjective of cancer chemotherapy is to prolong the patient survival .time maintaining a reasonable quality of life during the palliation pe-riod. Numerical results are presented for a particular case (bladdercancer).Fil: Barrea, Andres Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Hernandez, Matias Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

On Continuation Methods for Non-Linear Bi-Objective Optimization: Certified Interval-Based Approach

The global optimization of constrained Non-Linear Bi-Objective Optimization problems (MO) aims at covering their Pareto-optimal front which is in general a manifold in R^2. Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarizing framework which transforms the bi-objective problem into a parametric mono-objective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. In this paper, we provide a survey on continuation techniques in global optimization methods for MO, which allow discovering large portions of locally Pareto-optimal solutions. We also propose a rigorous active set management strategy on top of a previously proposed certified continuation method based on interval analysis, and illustrate it on a challenging bi-objective problem

Numerical and Evolutionary Optimization 2020

This book was established after the 8th International Workshop on Numerical and Evolutionary Optimization (NEO), representing a collection of papers on the intersection of the two research areas covered at this workshop: numerical optimization and evolutionary search techniques. While focusing on the design of fast and reliable methods lying across these two paradigms, the resulting techniques are strongly applicable to a broad class of real-world problems, such as pattern recognition, routing, energy, lines of production, prediction, and modeling, among others. This volume is intended to serve as a useful reference for mathematicians, engineers, and computer scientists to explore current issues and solutions emerging from these mathematical and computational methods and their applications

Métodos de escalarización en optimización multiobjetivo

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.La vida inevitablemente involucra la toma de decisiones y elecciones y es natural querer que estas sean las mejores posibles, en otras palabras, sean óptimas. La dificultad aquí radica en el conflicto (al menos parcial) entre nuestros diversos objetivos y metas. Los problemas con múltiples objetivos y criterios son generalmente conocidos como problemas de optimización multiobjetivo. A lo largo de este trabajo, se presentaron los conceptos necesarios y algunos métodos para resolver problemas de optimización multiobjetivo. Resolver un problema de optimización multiobjetivo significa encontrar el conjunto de soluciones Pareto optimal o frente de Pareto. Los métodos fueron divididos en cuatro categorías según la articulación de preferencias de un tomador de decisiones. De cada método se estudiaron las ventajas y desventajas y se seleccionaron tres métodos para estudiar con mayor profundidad. Los métodos seleccionados fueron métodos de escalarización; el método de sumas ponderadas, restricción ε y métricas ponderadas, que además fueron implementados para generar una aproximación del frente de Pareto. Se seleccionaron problemas test para generar aproximaciones de sus frentes de Pareto y analizar los resultados obtenidos. Se encontró que ningún método es superior que otro. La selección de un método específico depende del tipo de información que proporciona el problema, las preferencias del usuario, los requisitos de la solución y la capacidad de cómputo.Life inevitably involves decision making and choices and it is natural to want them to be the best possible, in other words, to be optimal. The difficulty here lies in the (at least partial) conflict between our various goals and objectives. Problems with multiple objectives and criteria are generally known as multiobjective optimization problems. Throughout this work, the necessary concepts and some methods to solve multiobjective optimization problems were presented. Solving a multiobjective optimization problem means finding the optimal Pareto solution set or Pareto front. The methods were divided into four categories according to the articulation of preferences of a decision maker. The advantages and disadvantages of each method were studied, and three methods were selected for further study. The selected methods were scalarization methods; the method of weighted sums, ε -constraint, and weighted metrics, which were also implemented to generate an approximation of the Pareto front. Test problems were selected to generate approximations of their Pareto fronts and to analyze the results obtained. It was found that no one method is superior to another. Selecting a specific method depends on the type of information the problem provides, the user preferences, the solution requirements, and the computational capacity.Fil: Fonseca, Rocío Guadalupe. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina

逐次探索による多目的最適化および宇宙往還機の複合領域概念設計への応用

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 土屋 武司, 東京大学教授 鈴木 真二, 東京大学教授 鈴木 宏二郎, 東京大学准教授 今村 太郎, 防衛大学校准教授 横山 信宏University of Tokyo(東京大学

Aproximación numérica equiespaciada de la variedad y el frente de Pareto para problemas de optimización o multiobjetivo

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física, 2011.En este trabajo se presenta un algoritmo que tiene aplicación en problemas de optimización multiobjetivo convexo irrestricto. Este método de continuación global, desarrollado en [16], hace uso del método de Newton y de restricciones adicionales de equiespaciado para proporcionar una aproximación numérica uniformemente espaciada de la variedad de Pareto (conjunto de soluciones eficientes) o del frente de Pareto (conjunto de puntos no dominados). Las ventajas que presenta el método son su bajo costo de implementación, el muestreo uniforme obtenido de los puntos óptimos y la posible paralelización del procedimiento computacional. Se dan ejemplos aplicados a funciones de testeo para ver la performance del método. Para motivar esta presentación se introduce el concepto de optimización simple y posteriormente se dan ejemplos donde los objetivos de interés están en conflicto, lo cual hace imposible, sin información adicional, definir una unica solución óptima. Puesto que se considera optimalidad en el sentido de Pareto, se define eficiencia y nodominancia de Pareto junto con los principales resultados teóricos de optimización multiobjetivo.Un poco de historia -- Optimización simple o de un objetivo -- Optimización multiobjetivo -- Conos y órdenes relacionados -- Soluciones eficientes y puntos no dominados -- Eficiencia y no dominancia propia -- Caracterización de soluciones eficientes -- Conectividad de conjuntos no dominados -- Método de continuación global -- Problemas de testeo

Fast computation of equispaced Pareto manifolds and Pareto fronts for multiobjective optimization problems

Abstract In this paper we consider the problem of generating a well sampled discrete representation of the Pareto manifold or the Pareto front corresponding to the equilibrium points of a multi-objective optimization problem. We show how the introduction of simple additional constraints into a continuation procedure produces equispaced points in either of those two sets. Moreover, we describe in detail a novel algorithm for global continuation that requires two orders of magnitude less function evaluations than evolutionary algorithms commonly used to solve this problem. The performance of the methods is demonstrated on problems from the current literature

Safety and Reliability - Safe Societies in a Changing World

The contributions cover a wide range of methodologies and application areas for safety and reliability that contribute to safe societies in a changing world. These methodologies and applications include: - foundations of risk and reliability assessment and management - mathematical methods in reliability and safety - risk assessment - risk management - system reliability - uncertainty analysis - digitalization and big data - prognostics and system health management - occupational safety - accident and incident modeling - maintenance modeling and applications - simulation for safety and reliability analysis - dynamic risk and barrier management - organizational factors and safety culture - human factors and human reliability - resilience engineering - structural reliability - natural hazards - security - economic analysis in risk managemen