58 research outputs found

    Set optimization - a rather short introduction

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    Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

    Modelo de computación evolutivo para redes sostenibles, eficientes y resistentes.

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    We present a new approach to adapt the differential evolution (DE) algorithm so that it can be applied in combinatorial optimization problems. The differential evolution algorithm has been proposed as an optimization algorithm for the continuous domain, using real numbers to encode the solutions, and its main operator, the mutation, uses a arithmetic operations to create a mutant using three different random solutions. This mutation operator cannot be used in combinatorial optimization problems, which have a domain of a discrete and finite set of objects. Based on this concept, we present an idea of representing each solution as a set, and replace the arithmetic operators in the classic DE genetic operators by set operators. Using a well known NP-hard problem, the traveling salesman problem (TSP), as an example of a combinatorial optimization problem, we study different possibilities for the mutation operator, presenting the advantages and disadvantages of each, before setting with the best one. We also explain the modifications made to adapt the algorithm for a multiobjective optimization algorithm. Some of these modifications are inherent to the different type of problems, other modification are proposed to improve the algorithm. Amongst the later modification are using more than one population in the evolution process. We also present a new self-adaptive variation of the multiobjective optimization algorithm, although this is not limited to the multi-objective case, and can be used also in the single-objective

    Evaluating sets of multi-attribute alternatives with uncertain preferences

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    In a decision-making problem, there can be uncertainty regarding the user preferences concerning the available alternatives. Thus, for a decision support system, it is essential to analyse the user preferences to make personalised recommendations. In this thesis we focus on Multiattribute Utility Theory (MAUT) which aims to define user preference models and elicitation procedures for alternatives evaluated with a vector of a fixed number of conflicting criteria. In this context, a preference model is usually represented with a real value function over the criteria used to evaluate alternatives, and an elicitation procedure is a process of defining such value function. The most preferred alternative will then be the one that maximises the value function. With MAUT models, it is common to represent the uncertainty of the user preferences with a parameterised value function. Each instantiation of this parameterisation then represents a user preference model compatible with the preference information collected so far. For example, a common linear value function is the weighted sum of the criteria evaluating an alternative, which is parameterised with respect to the set of weights. We focus on this type of preference models and in particular on value functions evaluating sets of alternatives rather single alternatives. These value functions can be used for example to define if a set of alternatives is preferred to another one, or which is the worst-case loss in terms of utility units of recommending a set of alternatives. We define the concept of setwise minimal equivalent subset (SME) and algorithms for its computation. Briefly, SME is the subset of an input set of alternatives with equivalent value function and minimum cardinality. We generalise standard preference relations used to compare single alternatives with the purpose of comparing sets of alternatives. We provide computational procedures to compute SME and evaluate preference relations with particular focus on linear value functions. We make extensive use of the Minimax Regret criterion, which is a common method to evaluate alternatives for potential questions and recommendations with uncertain value functions. It prescribes an outcome that minimises the worst-case loss with respect to all the possible parameterisation of the value function. In particular, we focus on its setwise generalisation, namely \textit{Setwise Minimax Regret} (SMR), which is the worst-case loss of recommending a set of alternatives. We provide a novel and efficient procedure for the computation of the SMR when supposing a linear value function. We also present a novel incremental preference elicitation framework for a supplier selection process, where a realistic medium-size factory inspires constraints and objectives of the underlying optimization problem. This preference elicitation framework applies for generic multiattribute combinatorial problems based on a linear preference model, and it is particularly useful when the computation of the set of Pareto optimal alternatives is practically unfeasible

    Computing Immutable Regions for Subspace Top-k Queries

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    National Research Foundation (NRF) Singapore under International Research Centre @ Singapore Funding Initiativ

    An Algorithm for Biobjective Mixed Integer Quadratic Programs

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    Multiobjective quadratic programs (MOQPs) are appealing since convex quadratic programs have elegant mathematical properties and model important applications. Adding mixed-integer variables extends their applicability while the resulting programs become global optimization problems. Thus, in this work, we develop a branch and bound (BB) algorithm for solving biobjective mixed-integer quadratic programs (BOMIQPs). An algorithm of this type does not exist in the literature. The algorithm relies on five fundamental components of the BB scheme: calculating an initial set of efficient solutions with associated Pareto points, solving node problems, fathoming, branching, and set dominance. Considering the properties of the Pareto set of BOMIQPs, two new fathoming rules are proposed. An extended branching module is suggested to cooperate with the node problem solver. A procedure to make the dominance decision between two Pareto sets with limited information is proposed. This set dominance procedure can eliminate the dominated points and eventually produce the Pareto set of the BOMIQP. Numerical examples are provided. Solving multiobjective quadratic programs (MOQPs) is fundamental to our research. Therefore, we examine the algorithms for this class of problems with different perspectives. The scalarization techniques for (strictly) convex MOPs are reviewed and the available algorithms for computing efficient solutions for MOQPs are discussed. These algorithms are compared with respect to four properties of MOQPs. In addition, methods for solving parametric multiobjective quadratic programs are studied. Computational studies are provided with synthetic instances, and examples in statistics and portfolio optimization. The real-life context reveals the interplay between the scalarizations and provides an additional insight into the obtained parametric solution sets

    Districting Problems - New Geometrically Motivated Approaches

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    This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures

    Localization in urban environments. A hybrid interval-probabilistic method

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    Ensuring safety has become a paramount concern with the increasing autonomy of vehicles and the advent of autonomous driving. One of the most fundamental tasks of increased autonomy is localization, which is essential for safe operation. To quantify safety requirements, the concept of integrity has been introduced in aviation, based on the ability of the system to provide timely and correct alerts when the safe operation of the systems can no longer be guaranteed. Therefore, it is necessary to assess the localization's uncertainty to determine the system's operability. In the literature, probability and set-membership theory are two predominant approaches that provide mathematical tools to assess uncertainty. Probabilistic approaches often provide accurate point-valued results but tend to underestimate the uncertainty. Set-membership approaches reliably estimate the uncertainty but can be overly pessimistic, producing inappropriately large uncertainties and no point-valued results. While underestimating the uncertainty can lead to misleading information and dangerous system failure without warnings, overly pessimistic uncertainty estimates render the system inoperative for practical purposes as warnings are fired more often. This doctoral thesis aims to study the symbiotic relationship between set-membership-based and probabilistic localization approaches and combine them into a unified hybrid localization approach. This approach enables safe operation while not being overly pessimistic regarding the uncertainty estimation. In the scope of this work, a novel Hybrid Probabilistic- and Set-Membership-based Coarse and Refined (HyPaSCoRe) Localization method is introduced. This method localizes a robot in a building map in real-time and considers two types of hybridizations. On the one hand, set-membership approaches are used to robustify and control probabilistic approaches. On the other hand, probabilistic approaches are used to reduce the pessimism of set-membership approaches by augmenting them with further probabilistic constraints. The method consists of three modules - visual odometry, coarse localization, and refined localization. The HyPaSCoRe Localization uses a stereo camera system, a LiDAR sensor, and GNSS data, focusing on localization in urban canyons where GNSS data can be inaccurate. The visual odometry module computes the relative motion of the vehicle. In contrast, the coarse localization module uses set-membership approaches to narrow down the feasible set of poses and provides the set of most likely poses inside the feasible set using a probabilistic approach. The refined localization module further refines the coarse localization result by reducing the pessimism of the uncertainty estimate by incorporating probabilistic constraints into the set-membership approach. The experimental evaluation of the HyPaSCoRe shows that it maintains the integrity of the uncertainty estimation while providing accurate, most likely point-valued solutions in real-time. Introducing this new hybrid localization approach contributes to developing safe and reliable algorithms in the context of autonomous driving
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