Global formulation for interactive multiobjective optimization

Interactive methods are useful and realistic multiobjective optimization techniques and, thus, many such methods exist. However, they have two important drawbacks when using them in real applications. Firstly, the question of which method should be chosen is not trivial. Secondly, there are rather few practical implementations of the methods. We introduce a general formulation that can accommodate several interactive methods. This provides a comfortable implementation framework for a general interactive system. Besides, this implementation allows the decision maker to choose how to give preference information to the system, and enables changing it anytime during the solution process. This change-of-method option provides a very flexible framework for the decision maker.     

Pareto navigator for interactive nonlinear multiobjective optimization

We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.     

Trade-off analysis approach for interactive nonlinear multiobjective optimization

When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.     

A synchronous reference point-based interactive method for stochastic multiobjective programming

Stochastic multiobjective programming models are highly complex problems, due to the presence of random parameters, together with several conflicting criteria that have to be optimized simultaneously. Even the widely used concept of efficiency has to be redefined for these problems. The use of interactive procedures can somehow ease this complexity, allowing the decision maker to learn about the problem itself, and to look for his most preferred solution. Reference point schemes can be adapted to stochastic problem, by asking the decision maker to provide, not only desirable levels for the objectives, but also the desired probability to achieve these values. In this paper, we analyze the different kinds of achievement scalarizing functions that can be used in this environment, and we study the efficiency (in the stochastic sense) of the different solutions obtained. As a result, a synchronous interactive method is proposed for a class of stochastic multiobjective problems, where only the objective functions are random. Several solutions can be generated by this new method, making use of the same preferential information, using the different achievement scalarizing functions. The preferential information (levels and probabilities for the objectives) is incorporated into the achievement scalarizing functions in a novel way to generate the new solutions. The special case of linear normal problems is addressed separately. The performance of the algorithm is illustrated with a numerical example.     

An interactive multiple-response simulation optimization method

This study proposes a new interactive multicriteria method for determining the best levels of the decision variables needed to optimize a stochastic computer simulation with multiple response variables. The method, called the Pairwise Comparison Stochastic Cutting Plane (PCSCP) method, combines good features from interactive multiple objective mathematical programming and response surface methodology. The major characteristics of the PCSCP method are: (1) it interacts progressively with the decision-maker (DM) to obtain her preferences, (2) it uses experimental design to explore the decision space adequately while reducing the burden on the DM, and (3) it uses the preference information provided by the DM and the sampling error in the responses to reduce the decision space. The mechanics of the method are illustrated with a numerical example. Some computational studies evaluating the method are also reported.     

Interactive solution to multiobjective optimization problems

Sensitivity analysis is presented as a natural addition to interactive multiobjective optimization methods based on compromise programming. It is shown that useful information regarding trade-offs in the objectives can be generated effectively by means of an analysis of the sensitivity of solutions to variations in preference structures. An implementation based on sequential quadrative programming is provided and examples are given for illustration.     

Non-convex multiobjective optimization under uncertainty: a descent algorithm. Application to sandwich plate design and reliability

In this paper a novel algorithm for solving multiobjective design optimization problems with non-smooth objective functions and uncertain parameters is presented. The algorithm is based on the existence of a common descent vector for each sample of the random objective functions and on an extension of the stochastic gradient algorithm. The proposed algorithm is applied to the optimal design of sandwich material. Comparisons with the genetic algorithm NSGA-II and the DMS solver are given and show that it is numerically more efficient due to the fact that it does not necessitate the objective function expectation evaluation. It can moreover be entirely parallelizable. Another simple illustration highlights its potential for solving general reliability problems, replacing each probability constraint by a new objective written in terms of an expectation. Moreover, for this last application, the proposed algorithm does not necessitate the computation of the (small) probability of failure.     

A variant space search method and its application to the optimization design of compressor profiles

Optimization methods have been widely used in practical engineering, with search efficiency and global search ability being the main evaluation criteria. In this article, the Bezier curve equivalent recursion is used in a genetic algorithm (GA) to realize the variant space search to improve the search efficiency and global search ability. The parameters related to this method are investigated by an optimization test of the simple curve approximation, which is then used for optimization designs of supersonic and transonic profiles. The results show that the GA can be improved if the variant space search method is added.     

An interactive on-line multi-objective optimization approach with application to metal cutting turning operation

An on-line supervisory computer control model for optimization of multiple objective programming problems is developed and applied to machining problems. Specifically, the turning operation is formulated and solved by the approach. For the first time, the characteristics of on-line multi-objective problems are described and an approach for solving such problems is developed. The proposed system maintains feasibility of operations and optimizes objectives based on the machine performance and on-line interactions with the decision-maker (DM). The model is hierarchical and consists of two levets: level 1 is designed to protect automated machinery and the workpiece by monitoring the process outputs; and level 2 is designed to optimize the on-line multi-objective problem. The tasks of both levels are accomplished by on-line monitoring and control of changes in process outputs via real-time information provided by sensors. The recommended parameters, obtainable from the multi-objective optimization approach, are used to create and update a database for machinability problems for future use. An example is discussed.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

智能水凝胶的制备及其在生物医学中的应用

智能水凝胶对于外界微小的物理化学刺激,如温度、电场、磁场、光、pH、离子强度或压力等能够感知并在响应过程中有显著的溶胀行为或响应性。本文介绍了智能水凝胶的分类及其合成方法,同时综述了智能水凝胶在药物控释、医用敷料、组织工程、角膜接触镜材料以及作为组织充填材料方面的应用,并对其发展趋势及应用前景作了简单的评述。     

The method of moments and its optimization

The method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions. The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular-like domains. The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one-dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions. We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.     

Single- and multiobjective optimization problems in robust parameter design

This paper reviews the evolution of off-line quality engineering methods with respect to one or more quality criteria, and presents some recent results. The fundamental premises that justify the use of robust product/process design are established with an illustrative example. The use of designed experiments to model quality criteria and their optimization is briefly reviewed. The fact that most design-for-quality problems involve multiple quality criteria motivates the development of multiobjective optimization techniques for robust parameter design. Two situations are considered: one in which response surface models for the quality characteristics can be obtained using regression and considered over a continuous factor space, and one in which the problem scenario and the experiment permit only discrete parameter settings for the design factors. In the former scenario, a multiobjective optimization technique based on the reference-point method is presented; this technique also incorporates an inference mechanism to deal with uncertainty in the response surface models caused by finite, noisy data. In the discrete-factors scenario, an efficient method to reduce computational complexity for a class of models is presented.     

A survey and annotated bibliography of multiobjective combinatorial optimization

This paper provides a survey of the research in and an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe the main characteristics of MOCO problems, and review the main properties and theoretical results for these problems. The main parts of the paper are a section on the review of the available solution methodology, both exact and heuristic, and a section on the annotation of the existing literature in the field organized problem by problem. We conclude the paper by stating open questions and areas of future research. Received: February 7, 2000 / Accepted: April 14, 2000     

An optimization method for learning statistical classifiers in structural reliability

Monte Carlo simulation is a general and robust method for structural reliability analysis, affected by the serious efficiency problem consisting in the need of computing the limit state function a very large number of times. In order to reduce this computational effort the use of several kinds of solver surrogates has been proposed in the recent past. Proposals include the Response Surface Method (RSM), Neural Networks (NN), Support Vector Machines (SVM) and several other methods developed in the burgeoning field of Statistical Learning (SL). Many of these techniques can be employed either for function approximation (regression approach) or for pattern recognition (classification approach). This paper concerns the use of these devices for discriminating samples into safe and failure classes using the classification approach, because it constitutes the core of Monte Carlo simulation as applied to reliability analysis as such. Due to the flexibility of most SL methods, a critical step in their use is the generation of the learning population, as it affects the generalization capacity of the surrogate. To this end it is first demonstrated that the optimal population from the information viewpoint lies around in the vicinity of the limit state function. Next, an optimization method assuring a small as well as highly informative learning population is proposed on this basis. It consists in generating a small initial quasi-random population using Sobol sequence for triggering a Particle Swarm Optimization (PSO) performed over an iteration-dependent cost function defined in terms of the limit state function. The method is evaluated using SVM classifiers, but it can be readily applied also to other statistical classification techniques because the distinctive feature of the SVM, i.e. the margin band, is not actively used in the algorithm. The results show that the method yields results for the probability of failure that are in very close agreement with Monte Carlo simulation performed on the original limit state function and requiring a small number of learning samples.     

Anisotropic mesh optimization and its application in adaptivity

The construction of solution-adapted meshes is addressed within an optimization framework. An approximation of the second spatial derivative of the solution is used to get a suitable metric in the computational domain. A mesh quality is proposed and optimized under this metric, accounting for both the shape and the size of the elements. For this purpose, a topological and geometrical mesh improvement method of high generality is introduced. It is shown that the adaptive algorithm that results recovers optimal convergence rates in singular problems, and that it captures boundary and internal layers in convection-dominated problems. Several important implementation issues are discussed. © 1997 John Wiley & Sons, Ltd.     

Linear multiobjective maxmin optimization and some Pareto and lexmaxmin extensions

Summary This is a lexmaxmin extension of some results on linear maxmin programming recently obtained by Posner and Wu, Kaplan, Gupta and Arora, and Kabe. The findings of these authors will be combined with previous results by the author of the present paper to give further insight into the interrelation between optimization with respect to maxmin, Pareto, and lexmaxmin.

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Zusammenfassung Diese Arbeit enthält Lexmaxmin-Erweiterungen einiger jüngerer Ergebnisse von Posner und Wu, von Kaplan, von Gupta und Arora und von Kabe über lineare Maxmin-Probleme. Die Beobachtungen der genannten Autoren werden mit früheren Ergebnissen des Autors der vorliegenden Arbeit mit dem Ziel vereint, weitere Einsichten in den Zusammenhang zwischen Maxmin-, Pareto- und Lexmaxmin-Optimierung zu vermitteln.

     

A single-loop optimization method for reliability analysis with second order uncertainty

Reliability analysis may involve random variables and interval variables. In addition, some of the random variables may have interval distribution parameters owing to limited information. This kind of uncertainty is called second order uncertainty. This article develops an efficient reliability method for problems involving the three aforementioned types of uncertain input variables. The analysis produces the maximum and minimum reliability and is computationally demanding because two loops are needed: a reliability analysis loop with respect to random variables and an interval analysis loop for extreme responses with respect to interval variables. The first order reliability method and nonlinear optimization are used for the two loops, respectively. For computational efficiency, the two loops are combined into a single loop by treating the Karush–Kuhn–Tucker (KKT) optimal conditions of the interval analysis as constraints. Three examples are presented to demonstrate the proposed method.     

Software reliability measurement in imperfect debugging environment and its application

In practice, debugging operations during the testing phase of software development are not always performed perfectly. In other words, not all the software faults detected are corrected and removed. Generally, this is called imperfect debugging. In this paper, we discuss a software reliability growth model considering imperfect debugging. Defining a random variable representing the cumulative number of faults corrected up to a specified testing time, this model is described by a semi-Markov process. Then, several quantitative measures are derived for software reliability assessment in an imperfect debugging environment. The application of this model to optimal software release problems is also discussed. Finally, numerical illustrations for software reliability measurement and optimal software release policies are presented.