Utilitarian Mechanism Design for Multiobjective Optimization

In a classic optimization problem, the complete input data is assumed to be known to the algorithm. This assumption may not be true anymore in optimization problems motivated by the Internet where part of the input data is private knowledge of independent selfish agents. The goal of algorithmic mechanism design is to provide (in polynomial time) a solution to the optimization problem and a set of incentives for the agents such that disclosing the input data is a dominant strategy for the agents. In the case of NP-hard problems, the solution computed should also be a good approximation of the optimum. In this paper we focus on mechanism design for multiobjective optimization problems. In this setting we are given a main objective function and a set of secondary objectives which are modeled via budget constraints. Multiobjective optimization is a natural setting for mechanism design as many economical choices ask for a compromise between different, partially conflicting goals. The main contribution of this paper is showing that two of the main tools for the design of approximation algorithms for multiobjective optimization problems, namely, approximate Pareto sets and Lagrangian relaxation, can lead to truthful approximation schemes. By exploiting the method of approximate Pareto sets, we devise truthful deterministic and randomized multicriteria fully polynomial-time approximation schemes (FPTASs) for multiobjective optimization problems whose exact version admits a pseudopolynomial-time algorithm, as, for instance, the multibudgeted versions of minimum spanning tree, shortest path, maximum (perfect) matching, and matroid intersection. Our construction also applies to multidimensional knapsack and multiunit combinatorial auctions. Our FPTASs compute a $(1+\varepsilon)$-approximate solution violating each budget constraint by a factor $(1+\varepsilon)$. When feasible solutions induce an independence system, i.e., when subsets of feasible solutions are feasible as well, we present a PTAS (not violating any constraint), which combines the approach above with a novel monotone way to guess the heaviest elements in the optimum solution. Finally, we present a universally truthful Las Vegas PTAS for minimum spanning tree with a single budget constraint, where one wants to compute a minimum cost spanning tree whose length is at most a given value $L$. This result is based on the Lagrangian relaxation method, in combination with our monotone guessing step and with a random perturbation step (ensuring low expected running time). This result can be derandomized in the case of integral lengths. All the mentioned results match the best known approximation ratios, which are, however, obtained by nontruthful algorithms

A genetic algorithm for power distribution system planning

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.The planning of distribution systems consists in determining the optimum site and size of new substations and feeders in order to satisfy the future power demand with minimum investment and operational costs and an acceptable level of reliability. This problem is a combinatorial, non-linear and constrained optimization problem. Several solution methods based on genetic algorithms have been reported in the literature; however, some of these methods have been reported with applications to small systems while others have long solution time. In addition, the vast majority of the developed methods handle planning problems simplifying them as single-objective problems but, there are some planning aspects that can not be combined into a single scalar objective; therefore, they require to be treated separately. The cause of these shortcomings is the poor representation of the potential solutions and their genetic operators This thesis presents the design of a genetic algorithm using a direct representation technique and specialized genetic operators for power distribution system expansion planning problems. These operators effectively preserve and exploit critical configurations that contribute to the optimization of the objective function. The constraints of the problems are efficiently handle with new strategies. The genetic algorithm was tested on several theoretical and real large-scale power distribution systems. Problems of network reconfiguration for loss reduction were also included in order to show the potential of the algorithm to resolve operational problems. Both single-objective and multi-objective formulations were considered in the tests. The results were compared with results from other heuristic methods such as ant colony system algorithms, evolutionary programming, differential evolution and other genetic algorithms reported in the literature. From these comparisons it was concluded that the proposed genetic algorithm is suitable to resolve problems of largescale power distribution system planning. Moreover, the algorithm proved to be effective, efficient and robust with better performance than other previous methods.National Council for Science and Technology, Mexic

Optimisation of piping network design for district cooling system

A district cooling system (DeS) is a.scheme for centralised cooling energy distribution which takes advantage of economies of scale and load diversity. . A cooling medium (chilled water) is generated at a central refrigeration plant and then supplied to a district area, comprising multiple buildings, through a closed-loop piping circuit. Because of the substantial capital investment involved, an optimal design of the distribution piping . configuration is one of the crucial factors for successful implementation of a district 1'. cooling scheme. Since there. exists an enormous number of different combinations of the piping configuration, it is not feasible to evaluate each individual case using an exhaustive approach. This thesis exammes the problem of determining an optimal distribution piping configuration using a genetic algorithm (GA). In order to estimate the spatial and temporal distribution of cooling loads; the climatic conditions of Hong Kong were investigated and a weather database in the form of a typical meteorological year (TMY) was developed. Detailed thermal modelling of a number of prototypical buildings was carried out to determine benchmark cooling loads. A novel Local Search/Looped Local Search algorithm was developed for finding optimal/near-optimal distribution piping configurations. By means of computational . experiments, it was demonstrated that there is a promising improvement to GA performance by including the Local Search/Looped Local Search algorithm, in terms of both solution quality and computational efficiency. The effects on the search performance of a number of parameters were systematically investigated to establish the most effective settings. In order to illustrate the effectiveness of the Local Search/Looped Local Search algorithm, a benchmark problem - the optimal communication,spanning tree (OCST) was used for comparison. The results showed that the Looped Local Search method developed in this work was an effective tool for optimal network design of the distribution piping system in DCS, as well as for optimising the OCST problem.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

OPTIMISATION TECHNIQUES FOR TELECOMMUNICATION NETWORKS

This thesis deals with various facets of the optimisation problem for telecommunication networks and proposes a number of new techniques for their solution. The necessary essentials, Graph Theory, Complexity Theory and Telecommunication Principles, are investigated. The relevant graphs are enumerated and the requirements of suitable optimisation algorithms for certain graphical problems are established. The Private Automatic Branch Exchange (PABX) is introduced. the variety of telecommunications traffic as well as the practical requirements of a connection topology are discussed. The fundamental Network Optimisation Problem (NJP) is defined and analysed. Simple exhaustive methods of solution are considered together with partial solution algorithms and simplification methods. Centralised networks with and without concentrators are introduced. Extensions and modifications are proposed for some techniques and existing practical methods of dealing with the NOP are investigated. A number of new ideas are proposed for the practical solution of the NOP. Reduction methods are presented for replacing large unmanageable networks with smaller ones, on which optimisation can take place. Fixed topology techniques are introduced for initial tandem switch selection purposes and perturbation methods are considered which can be applied to such an initial solution. Lookahead methods of link removal are introduced for the purposes of determining the tandem interconnection network together with the traffic routeing strategy. A composite method is proposed incorporating all of these concepts and the results of a number of numerical experiments upon actual network problem; are presented. the extension of the proposed techniques to other areas of problem solving and optimisation is considered. In particular, a new method for the solution of the Euclidean Travelling Salesman Problem (ETSP) is presented. A brief discussion is undertaken, in conclusion, concerning the practical difficulties of the NOP and The restrictions this placed upon solution algorithms of various types.Brit1sh Telecom, Ta1lis Consultancy, Londo

The role of Walsh structure and ordinal linkage in the optimisation of pseudo-Boolean functions under monotonicity invariance.

Optimisation heuristics rely on implicit or explicit assumptions about the structure of the black-box fitness function they optimise. A review of the literature shows that understanding of structure and linkage is helpful to the design and analysis of heuristics. The aim of this thesis is to investigate the role that problem structure plays in heuristic optimisation. Many heuristics use ordinal operators; which are those that are invariant under monotonic transformations of the fitness function. In this thesis we develop a classification of pseudo-Boolean functions based on rank-invariance. This approach classifies functions which are monotonic transformations of one another as equivalent, and so partitions an infinite set of functions into a finite set of classes. Reasoning about heuristics composed of ordinal operators is, by construction, invariant over these classes. We perform a complete analysis of 2-bit and 3-bit pseudo-Boolean functions. We use Walsh analysis to define concepts of necessary, unnecessary, and conditionally necessary interactions, and of Walsh families. This helps to make precise some existing ideas in the literature such as benign interactions. Many algorithms are invariant under the classes we define, which allows us to examine the difficulty of pseudo-Boolean functions in terms of function classes. We analyse a range of ordinal selection operators for an EDA. Using a concept of directed ordinal linkage, we define precedence networks and precedence profiles to represent key algorithmic steps and their interdependency in terms of problem structure. The precedence profiles provide a measure of problem difficulty. This corresponds to problem difficulty and algorithmic steps for optimisation. This work develops insight into the relationship between function structure and problem difficulty for optimisation, which may be used to direct the development of novel algorithms. Concepts of structure are also used to construct easy and hard problems for a hill-climber

Structure-Based Evolutionary Design Applied to Wire Antennas

A new design technique for antennas, namely the Structure-based Evolutionary Design (SED), is introduced and described in detail. SED is a new global random search method derived by the “genetic programming”, a strategy proposed by Koza. The proposed technique will be compared with the genetic algorithms (GA), a widely used design technique, showing the numerous advantages of our approach with respect to standard ones. SED assumes no “a priori” structure, but it builds up the structure of the individuals as the procedure evolves. Therefore SED is able to determine both the structure shape and dimensions as an outcome of the procedure (infinite-dimensional solution space), acting on subparts of the whole structure, and allowing to explore effectively the far more vast solution space. We thoroughly discuss both the general features of SED and its application to wire antenna design. The antenna internal representation, which is a key to the successful implementation of SED, and the construction of fitness functions from the antenna specifications will be described in detail. The proposed approach has been assessed with many different cases, using as design requirements both Gain and VSWR in a frequency band as wide as possible, and with the smallest size. The results obtained with SED are finally compared with other popular algorithms like Particle Swarm Optimization (PSO) and Differential Evolution (DE), showing that both the computational cost and the complexity are of the same order of magnitude, but the performances obtained by SED are significantly higher

Approximation of Multiobjective Optimization Problems

We study optimization problems with multiple objectives. Such problems are pervasive across many diverse disciplines -- in economics, engineering, healthcare, biology, to name but a few -- and heuristic approaches to solve them have already been deployed in several areas, in both academia and industry. Hence, there is a real need for a rigorous investigation of the relevant questions. In such problems we are interested not in a single optimal solution, but in the tradeoff between the different objectives. This is captured by the tradeoff or Pareto curve, the set of all feasible solutions whose vector of the various objectives is not dominated by any other solution. Typically, we have a small number of objectives and we wish to plot the tradeoff curve to get a sense of the design space. Unfortunately, typically the tradeoff curve has exponential size for discrete optimization problems even for two objectives (and is typically infinite for continuous problems). Hence, a natural goal in this setting is, given an instance of a multiobjective problem, to efficiently obtain a ``good'' approximation to the entire solution space with ``few'' solutions. This has been the underlying goal in much of the research in the multiobjective area, with many heuristics proposed for this purpose, typically however without any performance guarantees or complexity analysis. We develop efficient algorithms for the succinct approximation of the Pareto set for a large class of multiobjective problems. First, we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy the Pareto curve of a multiobjective optimization problem. We provide approximation algorithms with tight performance guarantees for bi-objective problems and make progress for the more challenging case of three and more objectives. Subsequently, we propose and study the notion of the approximate convex Pareto set; a novel notion of approximation to the Pareto set, as the appropriate one for the convex setting. We characterize when such an approximation can be efficiently constructed and investigate the problem of computing minimum size approximate convex Pareto sets, both for discrete and convex problems. Next, we turn to the problem of approximating the Pareto set as efficiently as possible. To this end, we analyze the Chord algorithm, a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as, multiobjective and parametric optimization, computational geometry, and graphics