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2017 | Buch

Definitions and Examples Kapitel lesen Erstes Kapitel lesen

Non-Convex Multi-Objective Optimization

verfasst von: Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Recent results on non-convex multi-objective optimization problems and methods are presented in this book, with particular attention to expensive black-box objective functions. Multi-objective optimization methods facilitate designers, engineers, and researchers to make decisions on appropriate trade-offs between various conflicting goals. A variety of deterministic and stochastic multi-objective optimization methods are developed in this book. Beginning with basic concepts and a review of non-convex single-objective optimization problems; this book moves on to cover multi-objective branch and bound algorithms, worst-case optimal algorithms (for Lipschitz functions and bi-objective problems), statistical models based algorithms, and probabilistic branch and bound approach. Detailed descriptions of new algorithms for non-convex multi-objective optimization, their theoretical substantiation, and examples for practical applications to the cell formation problem in manufacturing engineering, the process design in chemical engineering, and business process management are included to aide researchers and graduate students in mathematics, computer science, engineering, economics, and business management.

Inhaltsverzeichnis

Frontmatter

Basic Concepts

Frontmatter
Chapter 1. Definitions and Examples
Abstract
The problem of multi-objective optimization is considered:
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 2. Scalarization
Abstract
There were attempts to reduce multi-objective optimization problems to single-objective ones from the very beginning of their investigation [65, 99]. The reduction of a problem of multi-objective optimization to a single-objective optimization one normally is called scalarization. To find a discrete representation of the set of Pareto optimal solutions, a sequence of single-objective optimization problems should be solved, and they should hold the following theoretical properties:
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 3. Approximation and Complexity
Abstract
In computer science, the theory of computational complexity was developed to evaluate the hardness of problems, regarding the number of steps to obtain a solution; for the basic results of that theory we refer to [147]. The model of computations, used in that theory, is either the Turing machine or any other equivalent model. The input and output of the Turing machine should be encoded as finite strings of bits. Such an encoding can be used to represent the objects of discrete nature, and the theory of computational complexity is favorable, e.g., to the investigation of problems of single-objective combinatorial optimization [150, 151]. However, the Turing machine model is not suitable for the algorithms using real numbers. Therefore alternative complexity theories have been developed for the investigation of problems of continuous nature. For the fundamentals of the complexity of real number algorithms we refer to [19, 218], and for the complexity of problems of mathematical programming to [143, 150, 151].
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 4. A Brief Review of Non-convex Single-Objective Optimization
Abstract
Single-objective optimization methods are among the mathematical methods most widely used in various applications. The classical methods starting with linear programming have been proved very valuable tools for solving various economic and engineering problems. However, the growing complexity of the applied problems demanded the development of new ideas, methods, and algorithms. The classical mathematical optimization methods are based on the assumption that an objective function is convex. The convexity assumption, which is very fruitful for theoretical investigation, is hardly provable in many practical applications. Moreover, it is not truth very frequently. Therefore in the 1960s of the last century there begun active research in global optimization of non-convex problems. Naturally, single-objective problems foremost attracted attention of researchers. In this section we briefly review the approaches to single-objective global optimization, the multi-objective counterparts of which are considered in subsequent chapters. For the comprehensive presentation we refer to the representative monographs.
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas

Theory and Algorithms

Frontmatter
Chapter 5. Multi-Objective Branch and Bound
Abstract
Branch and bound approaches for optimization problems were developed in the 1960s [114, 121]. The main concept of a branch and bound algorithm is to detect and discard sets of feasible decisions which cannot contain optimal decisions. The search process can be illustrated as a search tree with the root corresponding to the search space and branches corresponding to its subsets. An iteration of the algorithm processes a node in the search tree that represents an unexplored subset of feasible decisions. The iteration has three main components: selection of the subset to be processed, branching corresponding to subdivision of the subset, and bound calculation.
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 6. Worst-Case Optimal Algorithms
Abstract
The question of the possibility to construct an optimal (in some sense) algorithm is of importance to the theory of multi-objective optimization similarly to any other theory of algorithmically solvable problems. In this chapter, we aim at finding a worst-case optimal approximation of the Pareto optimal set for multi-objective optimization problems, where the convexity of objective functions is not assumed. The class of Lipschitz functions is chosen as a model of objective functions since that model is one of the simplest and best researched models of global optimization [87]. Worst-case optimal algorithms are constructed for the cases of passive (non-adaptive) and sequential (adaptive) search in [249]. These results are the generalization to the multi-objective case of the results by Sukharev who investigated the worst-case optimal single-objective optimization algorithms in [210, 211].
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 7. Statistical Models Based Algorithms
Abstract
Multi-objective optimization problems with expensive objective functions are typical in engineering design, where time-consuming computations are involved for modeling technological processes. Frequently, the available software implements an algorithm to compute the values of objective functions, but neither details of implementation nor analytical properties of the objective functions are known. Nevertheless, the continuity of the objective functions can be normally assumed. The complexity of the computational model implies not only the expensiveness of the objective function but also the uncertainty in its properties, so that other analytical properties of f(x), besides the continuity, cannot be substantiated. Such unfavorable, from the optimization point of view, properties of f(x) as non-differentiability, non-convexity, and multimodality cannot be excluded. Difficulties of the black-box global optimization of expensive functions are well known from the experience gained in the single-objective case.
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 8. Probabilistic Bounds in Multi-Objective Optimization
Abstract
Randomization is one of the most important ideas used in the construction of heuristic methods for multi-objective optimization. Mathematically substantiated stochastic methods for non-convex multi-objective optimization attracted interest of researchers quite recently. A natural idea is to generalize single-objective optimization methods to the multi-objective case, and a prospective candidate is the well developed method based on the statistical methods of extremal values. A brief review of this approach, known as a Branch and Probability Bound (BPB) method is presented in Sect. 4.4. Some statistical procedures which are well known in single-objective optimization can be extended to multi-objective problems using scalarization. In this chapter, a version of multi-objective BPB method is described and discussed; this method is an extension of the BPB method developed for the case of a single-objective function in [237], see also [239], Sect. 2.6.1. The considered extension is based on the Tchebycheff scalarization method briefly discussed in Chap. 2
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas

Applications

Frontmatter
Chapter 9. Visualization of a Set of Pareto Optimal Decisions
Abstract
Applications of optimization methods for optimal design are most efficient when combined with the human engineer’s insight. Visualization is a technique which makes it easier for an engineer to understand the design space and interpret a design found by an optimization algorithm [255]. There are many techniques aimed at visualization of Pareto optimal solutions, especially for the bi-objective problems where a set of solutions, usually referred to as a Pareto front, is a curve in a two-dimensional solution space. The problem of visualization of the sets of optimal decisions is researched not so thoroughly. However, the visualization of a set of Pareto optimal decisions can significantly aid the choice of an appropriate engineering design. In this chapter, we present a case study of a process optimization where the application of a visualization technique was very helpful in the analysis of appropriate design variables.
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Chapter 10. Multi-Objective Optimization Aided Visualization of Business Process Diagrams
Abstract
Graphs are very popular models of many research subjects. Graphical presentation of a problem is advantageous for heuristic perception and understanding of relations between the objects considered. On the other hand, many efficient algorithmic techniques are available to attack mathematically stated graph problems. Therefore, graph models are especially useful where heuristic abilities of a human user in the formulation of a problem are combined with its algorithmic solution in the interactive mode. In the present chapter we consider the graph models of business processes which, in the literature on the management of business processes, are called business process diagrams (BPDs). To be more precise, a problem of drawing aesthetically pleasing layouts of BPDs is considered. The research of this problem was motivated by the fact that the aesthetic layouts are not only well readable but also most informative and practical [12].
Panos M. Pardalos, Antanas Žilinskas, Julius Žilinskas
Backmatter
Metadaten
Titel
Non-Convex Multi-Objective Optimization
verfasst von
Panos M. Pardalos
Antanas Žilinskas
Julius Žilinskas
Copyright-Jahr
2017
Electronic ISBN
978-3-319-61007-8
Print ISBN
978-3-319-61005-4
DOI
https://doi.org/10.1007/978-3-319-61007-8