Showing papers on "Multi-objective optimization published in 1998"

Efficient Global Optimization of Expensive Black-Box Functions

Abstract: In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.

Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study

Abstract: Since 1985 various evolutionary approaches to multiobjective optimization have been developed, capable of searching for multiple solutions concurrently in a single run. But the few comparative studies of different methods available to date are mostly qualitative and restricted to two approaches. In this paper an extensive, quantitative comparison is presented, applying four multiobjective evolutionary algorithms to an extended 0/1 knapsack problem.

Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems

Abstract: This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives.

Multiobjective optimization and multiple constraint handling with evolutionary algorithms. I. A unified formulation

Abstract: In optimization, multiple objectives and constraints cannot be handled independently of the underlying optimizer. Requirements such as continuity and differentiability of the cost surface add yet another conflicting element to the decision process. While "better" solutions should be rated higher than "worse" ones, the resulting cost landscape must also comply with such requirements. Evolutionary algorithms (EAs), which have found application in many areas not amenable to optimization by other methods, possess many characteristics desirable in a multiobjective optimizer, most notably the concerted handling of multiple candidate solutions. However, EAs are essentially unconstrained search techniques which require the assignment of a scalar measure of quality, or fitness, to such candidate solutions. After reviewing current revolutionary approaches to multiobjective and constrained optimization, the paper proposes that fitness assignment be interpreted as, or at least related to, a multicriterion decision process. A suitable decision making framework based on goals and priorities is subsequently formulated in terms of a relational operator, characterized, and shown to encompass a number of simpler decision strategies. Finally, the ranking of an arbitrary number of candidates is considered. The effect of preference changes on the cost surface seen by an EA is illustrated graphically for a simple problem. The paper concludes with the formulation of a multiobjective genetic algorithm based on the proposed decision strategy. Niche formation techniques are used to promote diversity among preferable candidates, and progressive articulation of preferences is shown to be possible as long as the genetic algorithm can recover from abrupt changes in the cost landscape.

A Rigorous Framework for Optimization of Expensive Functions by Surrogates

Abstract: The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which design application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations to the objective function and managing the use of these approximations as surrogates for optimization. The result is to obtain convergence to a minimizer of an expensive objective function subject to simple constraints. The approach is widely applicable because it does not require, or even explicitly approximate, derivatives of the objective. Numerical results are presented for a 31-variable helicopter rotor blade design example and for a standard optimization test example.

A multi-objective genetic local search algorithm and its application to flowshop scheduling

Abstract: We propose a hybrid algorithm for finding a set of nondominated solutions of a multi objective optimization problem. In the proposed algorithm, a local search procedure is applied to each solution (i.e., each individual) generated by genetic operations. Our algorithm uses a weighted sum of multiple objectives as a fitness function. The fitness function is utilized when a pair of parent solutions are selected for generating a new solution by crossover and mutation operations. A local search procedure is applied to the new solution to maximize its fitness value. One characteristic feature of our algorithm is to randomly specify weight values whenever a pair of parent solutions are selected. That is, each selection (i.e., the selection of two parent solutions) is performed by a different weight vector. Another characteristic feature of our algorithm is not to examine all neighborhood solutions of a current solution in the local search procedure. Only a small number of neighborhood solutions are examined to prevent the local search procedure from spending almost all available computation time in our algorithm. High performance of our algorithm is demonstrated by applying it to multi objective flowshop scheduling problems.

An evolutionary algorithm for multiobjective optimization: the strength Pareto approach

Abstract: Evolutionary algorithms EA have proved to be well suited for optimization prob lems with multiple objectives Due to their inherent parallelism they are able to capture a number of solutions concurrently in a single run In this report we propose a new evolutionary approach to multicriteria optimization the Strength Pareto Evolutionary Algorithm SPEA It combines various features of previous multiobjective EAs in a unique manner and is characterized as follows a besides the population a set of individuals is maintained which contains the Pareto optimal solutions generated so far b this set is used to evaluate the tness of an individual according to the Pareto dominance relationship c unlike the commonly used tness sharing population diversity is preserved on basis of Pareto dominance rather than distance d a clustering method is incorporated to reduce the Pareto set without destroying its characteristics The proof of principle results on two problems suggest that SPEA is very e ective in sampling from along the entire Pareto optimal front and distributing the generated solutions over the tradeo surface Moreover we compare SPEA with four other multiobjective EAs as well as a single objective EA and a random search method in application to an extended knapsack problem Regarding two complementary quantitative measures SPEA outperforms the other approaches at a wide margin on this test problem Finally a number of suggestions for extension and application of the new algorithm are discussed

Stochastic Dynamic Programming and the Control of Queueing Systems

Abstract: Optimization Criteria. Finite Horizon Optimization. Infinite Horizon Discounted Cost Optimization. An Inventory Model. Average Cost Optimization for Finite State Spaces. Average Cost Optimization Theory for Countable State Spaces. Computation of Average Cost Optimal Policies for Infinite State Spaces. Optimization Under Actions at Selected Epochs. Average Cost Optimization of Continuous Time Processes. Appendices. Bibliography. Index.

Multiobjective optimization and multiple constraint handling with evolutionary algorithms. II. Application example

Abstract: For part I see ibid, 26-37 The evolutionary approach to multiple function optimization formulated in the first part of the paper is applied to the optimization of the low-pressure spool speed governor of a Pegasus gas turbine engine This study illustrates how a technique such as the multiobjective genetic algorithm can be applied, and exemplifies how design requirements can be refined as the algorithm runs Several objective functions and associated goals express design concerns in direct form, ie, as the designer would state them While such a designer-oriented formulation is very attractive, its practical usefulness depends heavily on the ability to search and optimize cost surfaces in a class much broader than usual, as already provided to a large extent by the genetic algorithm (GA) The two instances of the problem studied demonstrate the need for preference articulation in cases where many and highly competing objectives lead to a nondominated set too large for a finite population to sample effectively It is shown that only a very small portion of the nondominated set is of practical relevance, which further substantiates the need to supply preference information to the GA The paper concludes with a discussion of the results

Policy optimization for dynamic power management

Abstract: Dynamic power management schemes (also called policies) can be used to control the power consumption levels of electronic systems, by setting their components in different states, each characterized by a performance level and a power consumption. In this paper, we describe power-managed systems using a finite-state, stochastic model. Furthermore, we show that the fundamental problem of finding an optimal policy which maximizes the average performance level of a system, subject to a constraint on the power consumption, can be formulated as a stochastic optimization problem called policy optimization. Policy optimization can be solved exactly in polynomial time (in the number of states of the model). We implemented a policy optimization tool and tested the quality of the optimal policies on a realistic case study.

Finding Acceptable Solutions in the Pareto-Optimal Range using Multiobjective Genetic Algorithms

Abstract: This paper investigates the problem of using a genetic algorithm to converge on a small, user-defined subset of acceptable solutions to multiobjective problems, in the Pareto-optimal (P-O) range. The paper initially explores exactly why separate objectives can cause problems in a genetic algorithm (GA). A technique to guide the GA to converge on the subset of acceptable solutions is then introduced.

MOGAC: a multiobjective genetic algorithm for hardware-software cosynthesis of distributed embedded systems

Abstract: In this paper, we present a hardware-software cosynthesis system, called MOGAC, that partitions and schedules embedded system specifications consisting of multiple periodic task graphs. MOGAC synthesizes real-time heterogeneous distributed architectures using an adaptive multiobjective genetic algorithm that can escape local minima. Price and power consumption are optimized while hard real-time constraints are met. MOGAC places no limit on the number of hardware or software processing elements in the architectures it synthesizes. Our general model for bus and point-to-point communication links allows a number of link types to be used in an architecture. Application-specific integrated circuits consisting of multiple processing elements are modeled. Heuristics are used to tackle multirate systems, as well as systems containing task graphs whose hyperperiods are large relative to their periods. The application of a multiobjective optimization strategy allows a single cosynthesis run to produce multiple designs that trade off different architectural features. Experimental results indicate that MOGAC has advantages over previous work in terms of solution quality and running time.

A Spatial Predator-Prey Approach to Multi-objective Optimization: A Preliminary Study

Abstract: This paper presents a novel evolutionary approach of approximating the shape of the Pareto-optimal set of multi-objective optimization problems. The evolutionary algorithm (EA) uses the predator-prey model from ecology. The prey are the usual individuals of an EA that represent possible solutions to the optimization task. They are placed at vertices of a graph, remain stationary, reproduce, and are chased by predators that traverse the graph. The predators chase the prey only within its current neighborhood and according to one of the optimization criteria. Because there are several predators with different selection criteria, those prey individuals, which perform best with respect to all objectives, are able to produce more descendants than inferior ones. As soon as a vertex for the prey becomes free, it is refilled by descendants from alive parents in the usual way of EA, i.e., by inheriting slightly altered attributes. After a while, the prey concentrate at Pareto-optimal positions. The main objective of this preliminary study is the answer to the question whether the predator-prey approach to multi-objective optimization works at all. The performance of this evolutionary algorithm is examined under several step-size adaptation rules.

On a multi-objective evolutionary algorithm and its convergence to the Pareto set

Abstract: Although there are many versions of evolutionary algorithms that are tailored to multi-criteria optimization, theoretical results are apparently not yet available. In this paper, it is shown that results known from the theory of evolutionary algorithms in case of single-criterion optimization do not carry over to the multi-criterion case. At first, three different step size rules are investigated numerically for a selected problem with two conflicting objectives. The empirical results obtained by these experiments lead to the observation that only one of these step size rules may have the property to ensure convergence to the Pareto set. A theoretical analysis finally shows that a special version of an evolutionary algorithm with this step size rule converges with probability one to the Pareto set for the test problem under consideration.

Using approximations to accelerate engineering design optimization

Abstract: Optimization problems that arise in engineering design are often characterized by several features that hinder the use of standard nonlinear optimization techniques. Foremost among these features is that the functions used to define the engineering optimization problem often are computationally intensive. Within a standard nonlinear optimization algorithm, the computational expense of evaluating the functions that define the problem would necessarily be incurred for each iteration of the optimization algorithm. Faced with such prohibitive computational costs, an attractive alternative is to make use of surrogates within an optimization context since surrogates can be chosen or constructed so that they are typically much less expensive to compute. For the purposes of this paper, we will focus on the use of algebraic approximations as surrogates for the objective. In this paper we introduce the use of so-called merit functions that explicitly recognize the desirability of improving the current approximation to the objective during the course of the optimization. We define and experiment with the use of merit functions chosen to simultaneously improve both the solution to the optimization problem (the objective) and the quality of the approximation. Our goal is to further improve the effectiveness of our general approach without sacrificing any of its rigor.

A neural network approach to multiobjective optimization for water quality management in a river basin

Abstract: A new neural network-based multiobjective optimization of water quality management for water pollution control and river basin planning is presented. Past research on water quality management problems has shown that traditional multiobjective decision making does not provide an adequate solution since it depends directly on the decision maker's (DM's) preferences, which may not be clearly defined. In order to overcome the DM's preferences problem this study uses a neural network algorithm to form a model for the solution of the multiobjective problems of water quality management in a river basin. Using the backpropagation algorithm of feedforward neural networks, a multiobjective programming model can simulate the DM's preferences, providing direct help to the analysts involved in real applications. Before describing the details of the multiobjective optimization problem, neural networks are proposed which can be used to predict the DM's preference structures. To demonstrate the procedures and the performance of the neural network-based approach, the case of the Tou-Chen River Basin in Taiwan is selected for analysis and discussion.

Dynamic Optimization in a Discontinuous World

Abstract: Many engineering tasks can be formulated as dynamic optimization or open-loop optimal control problems, where we search a priori for the input profiles to a dynamic system that optimize a given performance measure over a certain time period. Further, many systems of interest in the chemical processing industries experience significant discontinuities during transients of interest in process design and operation. This paper discusses three classes of dynamic optimization problems with discontinuities: path-constrained problems, hybrid discrete/continuous problems, and mixed-integer dynamic optimization problems. In particular, progress toward a general numerical technology for the solution of large-scale discontinuous dynamic optimization problems is discussed.

Multiobjective dynamic optimization of an industrial nylon 6 semibatch reactor using genetic algorithm

Abstract: The nondominated sorting genetic algorithm (NSGA) is adapted and used to obtain multiobjective Pareto optimal solutions for three grades of nylon 6 being produced in an industrial semibatch reactor. The total reaction time and the concentra- tion of an undesirable cyclic dimer in the product are taken as two individual objectives for minimization, while simultaneously requiring the attainment of design values of the final monomer conversion and for the number-average chain length. Substantial improvements in the operation of the nylon 6 reactor are indicated by this study. The technique used is very general in nature and can be used for multiobjective optimization of other reactors. Good mathematical models accounting for all the physicochemical aspects operative in a reactor (and which have been preferably tested on industrial data) are a prerequisite for such optimization studies. q 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 69-87, 1998

Conditions for Global Optimality 2

Abstract: In this paper bearing the same title as our earlier survey-paper [11] we pursue the goal of characterizing the global solutions of an optimization problem, i.e. getting at necessary and sufficient conditions for a feasible point to be a global minimizer (or maximizer) of the objective function. We emphasize nonconvex optimization problems presenting some specific structures like ’convex-anticonvex‘ ones or quadratic ones.

Selective Breeding in a Multiobjective Genetic Algorithm

Abstract: This paper describes an investigation of the efficacy of various elitist selection strategies in a multiobjective Genetic Algorithm implementation, with parents being selected both from the current population and from the archive record of nondominated solutions encountered during search. It is concluded that, because the multiobjective optimization process naturally maintains diversity in the population, it is possible to improve the performance of the algorithm through the use of strong elitism and high selection pressures without suffering the disadvantages of genetic convergence which such strategies would bring in single objective optimization.

Genetic Algorithm Development for Multiobjective Optimization of Structures

Abstract: Genetic algorithms (GAs) have the characteristic of maintaining a population of solutions and can search in a parallel manner for many nondominated solutions. These features coincide with the requirement of seeking a Pareto optimal set in a multiobjective (multicriteria, vector) optimization problem. The rationale for multiobjective optimization via GAs is that at each generation, the fitness of each individual is defined according to its nondominated property. Because nondominated individuals are assigned the highest fitness values, the convergence of a population will go to the nondominated zone: the Pareto optimal set. Based on this concept, a Pareto GA, whose goal is to locate the Pareto optimal set of a multiobjective optimization problem, is developed. In this GA, to avoid missing Pareto optimal points during evolutionary processes, a new concept called Pareto-set filter is adopted. At each generation, the points of rank 1 are put into the filter and undergo a nondominated check. In addition, a niche technique is provided to prevent genetic drift in population evolution. This technique sets a replacement rule for reproduction procedures. For a constrained optimization problem, a revised penalty function method is introduced to transfer a constrained problem into a nonconstralned one. The transferred function of a point contains information on a point's status (feasible or infeasible), position in a search region, and distance to the Pareto optimal set Two multiobjective optimization examples, a 25-bar space truss optimal design (objectives: structural weight and virtual work, constraints: stresses) and a four-bar pyramid truss with control system (objectives: structural weight and control effort, constraints: closed-loop frequencies) are provided to demonstrate analysis procedures of the proposed Pareto GA.

Design of reliable systems using static and dynamic fault trees

Abstract: The design process of a reliable system is, by nature, iterative. Traditional approaches to the design of a reliable system follow the requirement determination, preliminary design, analysis, evaluation, and redesign stages until what is regarded as an acceptable design is achieved. The system requirements typically consist of requirements on reliability, cost, weight, power consumption, physical size, etc. Within available resources, there can exist numerous approaches that completely satisfy all the design requirements. However as modern systems are becoming more and more complex, it is difficult to enumerate all the acceptable designs to find the optimal design configuration. A design optimization tool is greatly needed. This paper embeds a genetic algorithm (GA) into a fault tree method to determine the heuristic optimal design configuration of a reliable system. For optimization, a fault tree which can represent the failure causes of potential designs is used. Two new gates (CHO & RED) are introduced in this research. GA are developed and integrated into a fault-tree solver to find the optimal design. Improvement techniques to accelerate GA convergence and to avoid the GA-premature problem are implemented. Multi-objective optimization is discussed and methods for it are developed. Several techniques to accelerate the optimization process are implemented which appreciably reduce the calculation time. Simulation results show that the integration of GA optimization capabilities with fault-tree reliability-analysis provides a robust, powerful system-design tool. The methodology is applied to an example of a cardiac-assist system.

A Multiple objective Ant--Q algorithm for the design of water distribution irrigation networks

Abstract: The diiculty in solving multiple objective optimization problems with traditional techniques, has urge researchers to use alternative approaches. Ant-Q algorithms have shown good results in the solution of combinatorial optimization problems, however little work has been done for multiple objective problems. This paper describes an Ant-Q algorithm called MOAQ, that can solve multiple objective optimization problems. MOAQ considers a family of agents for each objective function involved. Each family nds solutions that depend on solutions found by the rest of the families, creating a negotiation mechanism and nding compromise solutions for all the objectives involved. The compromise solutions are evaluated in the Pareto sense, assigning rewards to the non-dominated solutions tting all problems constraints, and punishments to the solutions violating any of them. 1 We compare and contrast the solutions obtained with MOAQ with the solutions obtained with two recently developed genetic algorithm approaches in two artiicial problems, demonstrating the ability of MOAQ to nd and maintain the Pareto frontier. Furthermore, it is shown how MOAQ is applied to a complex real-world problem; the design of irrigation water distribution networks, with very promising results.

On the robustness of population-based versus point-based optimization in the presence of noise

Abstract: Practical optimization problems often require the evaluation of solutions through experimentation, stochastic simulation, sampling, or even interaction with the user. Thus, most practical problems involve noise. We address the robustness of population-based versus point-based optimization on a range of parameter optimization problems when noise is added to the deterministic objective function values. Population-based optimization is realized by a genetic algorithm and an evolution strategy. Point-based optimization is implemented as the classical Hooke-Jeeves pattern search strategy and threshold accepting as a modern local search technique. We investigate the performance of these optimization methods for varying levels of additive normally distributed fitness-independent noise and different sample sizes for evaluating individual solutions. Our results strongly favour population-based optimization, and the evolution strategy in particular.

Design and Analysis of Parametric Query Optimization Algorithms

Abstract: Query optimizers normally compile queries into one optimal plan by assuming complete knowledge of all cost parameters such as selectivity and resource availability. The execution of such plans could be sub-optimal when cost parameters are either unknown at compile time or change significantly between compile time and runtime [Loh89, GrW89]. Parametric query optimization [INS+92, CG94, GK94] optimizes a query into a number of candidate plans, each optimal for some region of the parameter space. In this paper, we present parametric query optimization algorithms. Our approach is based on the property that for linear cost functions, each parametric optimal plan is optimal in a convex polyhedral region of the parameter space. This property is used to optimize linear and non-linear cost functions. We also analyze the expected sizes of the parametric optimal set of plans and the number of plans produced by the Cole and Graefe algorithm [CG94].

Global and multi-objective optimization for lens design by real-coded genetic algorithms

Abstract: This paper presents new lens optimization methods based on real-coded genetic algorithms (GAs). We take advantage of GA's capability of global optimization and multi-objective optimization against two serious problems in conventional lens optimization techniques: (1) choosing a starting point by trial and error, and (2) combining plural criteria to a single criterion. In this paper, two criteria for lenses, the resolution and the distortion, are considered. First, we propose a real-coded GA that optimizes a single criterion, a weighted sum of the resolution and the distortion. To overcome a problem of the difficulty in generating feasible lenses especially in large-scale problems, we introduce an enforcement operator to modify an infeasible solution into a feasible one. By applying the proposed method to some small- scale problems, we show that the proposed method can find empirically optimal and suboptimal lenses. We also apply the proposed method to some relatively large-scale problems and show that the proposed method can effectively work under large-scale problems. Next, regarding the lens design problem as a multi-objective optimization problem, we propose a real-coded multi-objective GA that explicitly optimizes the two criteria. We show that effectiveness of the proposed method in multi-objective lens optimization by applying it to a three-element lens design problem.