Continuous dynamical systems that realize discrete optimization on the hypercube

We study the problem of finding a local minimum of a multilinear function E over the discrete set {0,1}n. The search is achieved by a gradient-like system in [0,1]n with cost function E. Under mild restrictions on the metric, the stable attractors of the gradient-like system are shown to produce solutions of the problem, even when they are not in the vicinity of the discrete set {0,1}n. Moreover, the gradient-like system connects with interior point methods for linear programming and with the analog neural network studied by Vidyasagar (IEEE Trans. Automat. Control 40 (8) (1995) 1359), in the same context. © 2004 Elsevier B.V. All rights reserved.

Stochastic Algorithms for Discrete Parameter Simulation Optimization

We present two efficient discrete parameter simulation optimization (DPSO) algorithms for the long-run average cost objective. One of these algorithms uses the smoothed functional approximation (SFA) procedure, while the other is based on simultaneous perturbation stochastic approximation (SPSA). The use of SFA for DPSO had not been proposed previously in the literature. Further, both algorithms adopt an interesting technique of random projections that we present here for the first time. We give a proof of convergence of our algorithms. Next, we present detailed numerical experiments on a problem of admission control with dependent service times. We consider two different settings involving parameter sets that have moderate and large sizes, respectively. On the first setting, we also show performance comparisons with the well-studied optimal computing budget allocation (OCBA) algorithm and also the equal allocation algorithm. Note to Practitioners-Even though SPSA and SFA have been devised in the literature for continuous optimization problems, our results indicate that they can be powerful techniques even when they are adapted to discrete optimization settings. OCBA is widely recognized as one of the most powerful methods for discrete optimization when the parameter sets are of small or moderate size. On a setting involving a parameter set of size 100, we observe that when the computing budget is small, both SPSA and OCBA show similar performance and are better in comparison to SFA, however, as the computing budget is increased, SPSA and SFA show better performance than OCBA. Both our algorithms also show good performance when the parameter set has a size of 10(8). SFA is seen to show the best overall performance. Unlike most other DPSO algorithms in the literature, an advantage with our algorithms is that they are easily implementable regardless of the size of the parameter sets and show good performance in both scenarios.

Modified Discrete PSO to Increase the Delivered Energy Probability in Distribution energy Systems

This paper proposes a PSO based approach to increase the probability of delivering power to any load point by identifying new investments in distribution energy systems. The statistical failure and repair data of distribution components is the main basis of the proposed methodology that uses a fuzzyprobabilistic modeling for the components outage parameters. The fuzzy membership functions of the outage parameters of each component are based on statistical records. A Modified Discrete PSO optimization model is developed in order to identify the adequate investments in distribution energy system components which allow increasing the probability of delivering power to any customer in the distribution system at the minimum possible cost for the system operator. To illustrate the application of the proposed methodology, the paper includes a case study that considers a 180 bus distribution network.

Particle swarm optimization for two-connected networks with bounded rings

The Two-Connected Network with Bounded Ring (2CNBR) problem is a network design problem addressing the connection of servers to create a survivable network with limited redirections in the event of failures. Particle Swarm Optimization (PSO) is a stochastic population-based optimization technique modeled on the social behaviour of flocking birds or schooling fish. This thesis applies PSO to the 2CNBR problem. As PSO is originally designed to handle a continuous solution space, modification of the algorithm was necessary in order to adapt it for such a highly constrained discrete combinatorial optimization problem. Presented are an indirect transcription scheme for applying PSO to such discrete optimization problems and an oscillating mechanism for averting stagnation.

The cross-entropy method for continuous multi-extremal optimization

In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization. We demonstrate the effectiveness of the cross-entropy method for solving difficult continuous multi-extremal optimization problems, including those with non-linear constraints.

Optimal control of distributed generators and capacitors by hybrid DPSO

In this paper, a comprehensive planning methodology is proposed that can minimize the line loss, maximize the reliability and improve the voltage profile in a distribution network. The injected active and reactive power of Distributed Generators (DG) and the installed capacitor sizes at different buses and for different load levels are optimally controlled. The tap setting of HV/MV transformer along with the line and transformer upgrading is also included in the objective function. A hybrid optimization method, called Hybrid Discrete Particle Swarm Optimization (HDPSO), is introduced to solve this nonlinear and discrete optimization problem. The proposed HDPSO approach is a developed version of DPSO in which the diversity of the optimizing variables is increased using the genetic algorithm operators to avoid trapping in local minima. The objective function is composed of the investment cost of DGs, capacitors, distribution lines and HV/MV transformer, the line loss, and the reliability. All of these elements are converted into genuine dollars. Given this, a single-objective optimization method is sufficient. The bus voltage and the line current as constraints are satisfied during the optimization procedure. The IEEE 18-bus test system is modified and employed to evaluate the proposed algorithm. The results illustrate the unavoidable need for optimal control on the DG active and reactive power and capacitors in distribution networks.

Evolutionary design of bone scaffolds with reference to material selection

The favourable scaffold for bone tissue engineering should have desired characteristic features, such as adequate mechanical strength and three-dimensional open porosity, which guarantee a suitable environment for tissue regeneration. In fact, the design of such complex structures like bone scaffolds is a challenge for investigators. One of the aims is to achieve the best possible mechanical strength-degradation rate ratio. In this paper we attempt to use numerical modelling to evaluate material properties for designing bone tissue engineering scaffold fabricated via the fused deposition modelling technique. For our studies the standard genetic algorithm was used, which is an efficient method of discrete optimization. For the fused deposition modelling scaffold, each individual strut is scrutinized for its role in the architecture and structural support it provides for the scaffold, and its contribution to the overall scaffold was studied. The goal of the study was to create a numerical tool that could help to acquire the desired behaviour of tissue engineered scaffolds and our results showed that this could be achieved efficiently by using different materials for individual struts. To represent a great number of ways in which scaffold mechanical function loss could proceed, the exemplary set of different desirable scaffold stiffness loss function was chosen. © 2012 John Wiley & Sons, Ltd.

Optimal distribution network reinforcement considering load growth, line loss, and reliability

In this paper, a new comprehensive planning methodology is proposed for implementing distribution network reinforcement. The load growth, voltage profile, distribution line loss, and reliability are considered in this procedure. A time-segmentation technique is employed to reduce the computational load. Options considered range from supporting the load growth using the traditional approach of upgrading the conventional equipment in the distribution network, through to the use of dispatchable distributed generators (DDG). The objective function is composed of the construction cost, loss cost and reliability cost. As constraints, the bus voltages and the feeder currents should be maintained within the standard level. The DDG output power should not be less than a ratio of its rated power because of efficiency. A hybrid optimization method, called modified discrete particle swarm optimization, is employed to solve this nonlinear and discrete optimization problem. A comparison is performed between the optimized solution based on planning of capacitors along with tap-changing transformer and line upgrading and when DDGs are included in the optimization.

Optimal design of building structures using genetic algorithms

A general framework for multi-criteria optimal design is presented which is well-suited for automated design of structural systems. A systematic computer-aided optimal design decision process is developed which allows the designer to rapidly evaluate and improve a proposed design by taking into account the major factors of interest related to different aspects such as design, construction, and operation.

The proposed optimal design process requires the selection of the most promising choice of design parameters taken from a large design space, based on an evaluation using specified criteria. The design parameters specify a particular design, and so they relate to member sizes, structural configuration, etc. The evaluation of the design uses performance parameters which may include structural response parameters, risks due to uncertain loads and modeling errors, construction and operating costs, etc. Preference functions are used to implement the design criteria in a "soft" form. These preference functions give a measure of the degree of satisfaction of each design criterion. The overall evaluation measure for a design is built up from the individual measures for each criterion through a preference combination rule. The goal of the optimal design process is to obtain a design that has the highest overall evaluation measure - an optimization problem.

Genetic algorithms are stochastic optimization methods that are based on evolutionary theory. They provide the exploration power necessary to explore high-dimensional search spaces to seek these optimal solutions. Two special genetic algorithms, hGA and vGA, are presented here for continuous and discrete optimization problems, respectively.

The methodology is demonstrated with several examples involving the design of truss and frame systems. These examples are solved by using the proposed hGA and vGA.

基于PAR的高可信装备保障算法推演

算法被誉为计算的“灵魂”（spirit of computing），算法程序的可靠性和正确性对软件系统的可信度起着至关重要的作用。动态规划、贪心、分支限界等传统的算法设计策略缺乏有效的选择标准，一些算法形式化开发方法和工具的应用水平和范围也很有限，因此算法的可靠性已成为高可信软件系统的一个主要瓶颈。 作为军事运筹学的一个重要分支，装备保障计算主要研究军事活动中装备领域的决策优化问题，保障计算软件特别是算法的可靠性在很大程度上决定了装备保障的效率乃至军事行动的成败。利用形式化软件工程的研究成果、特别是引入算法程序的形式化开发方法，对我军通用装备保障领域中的大量算法类问题进行有效求解，有利于优化保障结构、统筹保障资源、整合保障力量、提高保障时效，促进装备整体战斗力水平的提升。 本文以高可信软件开发方法PAR为基础，面向装备保障计算的领域需求，提出了一类离散最优化问题（discrete optimization problem, DOP）的结构模型和算法推演技术，并成功推演了一系列典型的装备保障算法。本论文的主要创新性贡献如下： （1）定义了DOP的结构模型，提出了基于单点结构（Singleton）的问题分划递推策略，进而通过PAR算法推演生成问题的高效求解算法，涵盖了多种传统算法设计策略，显著提高了算法程序设计的机械化水平。 （2）提出了描述DOP分划递推过程的问题简约图（problem reduction graph, PRG）模型，并针对典型DOP结构推演得到了一组PRG构造算法模式，其中每个算法模式都涵盖了满足特定代数结构性质的一大类具体问题。 （3）阐述了PAR算法推演的范畴模型，为算法设计和重用提供了抽象而有效的方法指导。 （4）对我军通用装备保障领域具有代表性的48个问题进行了形式化的算法推演，构建了领域算法库。 （5）在PAR平台的基础上设计了装备保障算法开发平台的原型COPALM，支持DOP算法推演和重用，进而提高了相关应用软件的开发效率。

Implementing and testing branch-and-bound algorithm for the quadratic 0-1 problem on transputers

Discrete optimization problems are very difficult to solve, even if the dimention is small. For most of them the problem of finding an ε-approximate solution is already NP-hard. The branch-and-bound algorithms are the most used algorithms for solving exactly this sort of problems.

A computational study of a parallel Branch and Bound algorithm for the quadratic 0-1 programming problem on transputers

Discrete optimization problems are very difficult to solve, even if the dimantion is small. For most of them the problem of finding an ε-approximate solution is already NP-hard.

« Resolution Search » et problèmes d’optimisation discrète

Thèse réalisée en cotutelle avec l'Université d'Avignon.

Heuristic solution methods for multi-attribute vehicle routing problems

Le Problème de Tournées de Véhicules (PTV) est une clé importante pour gérér efficacement des systèmes logistiques, ce qui peut entraîner une amélioration du niveau de satisfaction de la clientèle. Ceci est fait en servant plus de clients dans un temps plus court. En terme général, il implique la planification des tournées d'une flotte de véhicules de capacité donnée basée à un ou plusieurs dépôts. Le but est de livrer ou collecter une certain quantité de marchandises à un ensemble des clients géographiquement dispersés, tout en respectant les contraintes de capacité des véhicules. Le PTV, comme classe de problèmes d'optimisation discrète et de grande complexité, a été étudié par de nombreux au cours des dernières décennies. Étant donné son importance pratique, des chercheurs dans les domaines de l'informatique, de la recherche opérationnelle et du génie industrielle ont mis au point des algorithmes très efficaces, de nature exacte ou heuristique, pour faire face aux différents types du PTV. Toutefois, les approches proposées pour le PTV ont souvent été accusées d'être trop concentrées sur des versions simplistes des problèmes de tournées de véhicules rencontrés dans des applications réelles. Par conséquent, les chercheurs sont récemment tournés vers des variantes du PTV qui auparavant étaient considérées trop difficiles à résoudre. Ces variantes incluent les attributs et les contraintes complexes observés dans les cas réels et fournissent des solutions qui sont exécutables dans la pratique. Ces extensions du PTV s'appellent Problème de Tournées de Véhicules Multi-Attributs (PTVMA). Le but principal de cette thèse est d'étudier les différents aspects pratiques de trois types de problèmes de tournées de véhicules multi-attributs qui seront modélisés dans celle-ci. En plus, puisque pour le PTV, comme pour la plupart des problèmes NP-complets, il est difficile de résoudre des instances de grande taille de façon optimale et dans un temps d'exécution raisonnable, nous nous tournons vers des méthodes approcheés à base d’heuristiques.

Equal opportunity networks, distance-balanced graphs, and Wiener game

Given a graph G and a set X ⊆ V(G), the relative Wiener index of X in G is defined as WX (G) = {u,v}∈X 2  dG(u, v) . The graphs G (of even order) in which for every partition V(G) = V1 +V2 of the vertex set V(G) such that |V1| = |V2| we haveWV1 (G) = WV2 (G) are called equal opportunity graphs. In this note we prove that a graph G of even order is an equal opportunity graph if and only if it is a distance-balanced graph. The latter graphs are known by several characteristic properties, for instance, they are precisely the graphs G in which all vertices u ∈ V(G) have the same total distance DG(u) = v∈V(G) dG(u, v). Some related problems are posed along the way, and the so-called Wiener game is introduced.