H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

A transfer algorithm for hierarchical clustering

Transfer algorithms are usually used to optimize an objective function that is defined on the set of partitions of a finite set X. In this paper we define an equivalence relation ? on the set of fuzzy equivalence relations on X and establish a bijection from the set of hierarchies on X to the set of equivalence classes with respect to ?. Thus, hierarchies can be identified with fuzzy equivalence relations and the transfer algorithm can be modified in order to optimize an objective function that is defined on the set of hierarchies on X.     

Economic design of variable parameters <Emphasis Type="Italic">X̄</Emphasis> control charts for processes with fuzzy mean shifts

Modern studies have shown that the X? control charts with variable parameters (VPs) detect process shifts faster than the traditional X? control charts. This article developed the economic design of the VP X? control chart to determine the values of the design parameters of the chart. However, this study, different from previous studies, was focused on the process that is subject to a disturbing cause, and the occurrence of the cause can result in a fuzzy mean shift (ie the magnitude of the mean shift could not be recognized exactly). The issue of economically selecting the design parameters for the chart was firstly formulated as a mathematical programming model with a fuzzy objective function that could cope with fuzzy number type of mean shift. A fuzzy-simulation-based genetic algorithm was then employed to search for the optimal values of the design parameters from the model. An industrial example was provided to illustrate the solution procedure, and was used for comparison between the VP and the traditional X? chart. Effects of model parameters on the solution of the economic design were also discussed.     

An extension to possibilistic fuzzy cluster analysis

We explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach, namely that the objective function is truly minimized only if all cluster centers are identical. Our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters, so that they are forced away from each other. We develop this approach for the possibilistic fuzzy c-means algorithm and the Gustafson–Kessel algorithm. In our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c-means algorithm with the advantages of a possibilistic approach w.r.t. the interpretation of the membership degrees.     

Fuzzy linear programming and applications

This paper presents a survey on methods for solving fuzzy linear programs. First LP models with soft constraints are discussed. Then LP problems in which coefficients of constraints and/or of the objective function may be fuzzy are outlined. Pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives. In addition to the commonly applied extended addition, based on the min-operator and used for the aggregation of the left-hand sides of fuzzy constraints and fuzzy objectives, a more flexible procedure, based on Yager's parametrized t-norm T_p, is presented. Finally practical applications of fuzzy linear programs are listed.     

Worst-case global optimization of black-box functions through Kriging and relaxation

A new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application.     

C-Programming: An outline

This paper presents the essentials of a method designed to solve optimization problems whose objective functions are of the form g(x)+ ψ(u(x)), where ψ is differentiable and either concave or convex. It is shown that solutions to such problems can be obtained through the solutions of the Lagrangian problem whose objective function is of the form g(x)+ λu(x).     

Competitive algorithms for the clustering of noisy data

In this paper, we consider the issue of clustering when outliers exist. The outlier set is defined as the complement of the data set. Following this concept, a specially designed fuzzy membership weighted objective function is proposed and the corresponding optimal membership is derived. Unlike the membership of fuzzy c-means, the derived fuzzy membership does not reduce with the increase of the cluster number. With the suitable redefinition of the distance metric, we demonstrate that the objective function could be used to extract c spherical shells. A hard clustering algorithm alleviating the prototype under-utilization problem is also derived. Artificially generated data are used for comparisons.     

Data-driven inverse optimization with imperfect information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent’s objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent’s true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the predicted decision (i.e., the decision implied by a particular candidate objective) differs from the agent’s actual response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches.     

Efficient computer experiment-based optimization through variable selection

A computer experiment-based optimization approach employs design of experiments and statistical modeling to represent a complex objective function that can only be evaluated pointwise by running a computer model. In large-scale applications, the number of variables is huge, and direct use of computer experiments would require an exceedingly large experimental design and, consequently, significant computational effort. If a large portion of the variables have little impact on the objective, then there is a need to eliminate these before performing the complete set of computer experiments. This is a variable selection task. The ideal variable selection method for this task should handle unknown nonlinear structure, should be computationally fast, and would be conducted after a small number of computer experiment runs, likely fewer runs (N) than the number of variables (P). Conventional variable selection techniques are based on assumed linear model forms and cannot be applied in this “large P and small N” problem. In this paper, we present a framework that adds a variable selection step prior to computer experiment-based optimization, and we consider data mining methods, using principal components analysis and multiple testing based on false discovery rate, that are appropriate for our variable selection task. An airline fleet assignment case study is used to illustrate our approach.     

Setting targets for surrogate-based optimization

In the context of surrogate-based optimization (SBO), most designers have still very little guidance on when to stop and how to use infill measures with target requirements (e.g., one-stage approach for goal seeking and optimization); the reason: optimum estimates independent of the surrogate and optimization strategy are seldom available. Hence, optimization cycles are typically stopped when resources run out (e.g., number of objective function evaluations/time) or convergence is perceived, and targets are empirically set which may affect the effectiveness and efficiency of the SBO approach. This work presents an approach for estimating the minimum (target) of the objective function using concepts from extreme order statistics which relies only on the training data (sample) outputs. It is assumed that the sample inputs are randomly distributed so the outputs can be considered a random variable, whose density function is bounded (a, b), with the minimum (a) as its lower bound. Specifically, an estimate of the minimum (a) is obtained by: (i) computing the bounds (using training data and the moment matching method) of a selected set of analytical density functions (catalog), and (ii) identifying the density function in the catalog with the best match to the sample outputs distribution and corresponding minimum estimate (a). The proposed approach makes no assumption about the nature of the objective functions, and can be used with any surrogate, and optimization strategy even with high dimensional problems. The effectiveness of the proposed approach was evaluated using a compact catalog of Generalized Beta density functions and well-known analytical optimization test functions, i.e., F2, Hartmann 6D, and Griewangk 10D and in the optimization of a field scale alkali-surfactant-polymer enhanced oil recovery process. The results revealed that: (a) the density function (from a catalog) with the best match to a function outputs distribution, was the same for both large and reduced samples, (b) the true optimum value was always within a 95% confidence interval of the estimated minimum distribution, and (c) the estimated minimum represents a significant improvement over the present best solution and an excellent approximation of the true optimum value.     

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.     

Combination of fuzzy ranking and simulated annealing to improve discrete fracture inversion

Mathematical and computational modelling of discrete fracture networks is critical for the exploration and development of natural resource reservoirs. Utilizing the concept of fuzzy memberships, this paper advances the fundamental understanding in fracture network inversion and presents a systematic procedure to solve the most important problem in global optimization (simulated annealing): objective function formulation. First, a comprehensive field study identifies all potential components of an objective function. The components are statistical, geostatistical, mathematical and spatial measurements of fracture properties (location, orientation and size). The characteristic measurements can be input in parametric or non-parametric, discrete or continuum forms. Next, sensitivity analysis and fuzzy logic are combined to rank the candidate components based on their effects on the final objective function value and optimization convergence. The process negates guess works in objective function formulation by automatic selection of highly ranked components and their corresponding weighting factors. A case study is applied to a surface DFN in New York. The derived discrete fracture network is representative of the field data.     

Discrete-continuous structural optimization with distributed evolution strategies

This paper describes the development and application of distributed Evolution Strategies (ES) in the field of mixed-discrete structural optimization. ES are direct, probabilistic optimization methods based on the use of the three evolution operators recombination, mutation and selection in an evolving population of competing individuals in the design space. Advanced features of the employed ES include self-adaption of strategy parameters to achieve on-line tuning of the optimization process and a flexible selection scheme that allows scalable lifetimes of individuals

The property of using a population of coexisting but independent individuals allows the efficient realization in distributed computing environments. This approach results in a drastic reduction of response time and an improved applicability for large scale problems     

Monomial geometric programming with fuzzy relation equation constraints

Monomials are widely used. They are basic structural units of geometric programming. In the process of optimization, many objective functions can be denoted by monomials. We can often see them in resource allocation and structure optimization and technology management, etc. Fuzzy relation equations are important elements of fuzzy mathematics, and they have recently been widely applied in fuzzy comprehensive evaluation and cybernetics. In view of the importance of monomial functions and fuzzy relation equations, we present a fuzzy relation geometric programming model with a monomial objective function subject to the fuzzy relation equation constraints, and develop an algorithm to find an optimal solution based on the structure of the solution set of fuzzy relation equations. Two numerical examples are given to verify the developed algorithm. Our numerical results show that the algorithm is feasible and effective.     

Properties of certain zero column-sum matrices with applications to the optimization of chemical reactors

This work is motivated by linear chemical reactor systems. The mathematical model of these systems employs a finite dimensional concentration vector which yields the properties of a discrete probability distribution. Central in the response of the system is a rate matrix. The properties of these matrices are analyzed in terms of the theories of Markoff and M-matrices. A linear objective function is selected and the optimization of a cascade system relative to changes of the sizes of the tanks is pursued. This amounts to the optimization of the objective function on R₊^m. The global optimum is shown to lie on the diagonal of the domain. Hence, the search for optimum can be simplified to a single dimension. Other related topics such as the effect of the number of tanks in the cascade on the optimum, conditions for off-diagonal stationary points and the constrained optimization are also considered.     

Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization

This paper investigates vector optimization problems with objective and the constraints are multifunctions. By using a special scalarization function introduced in optimization by Hiriart-Urruty, we establish optimality conditions in terms of Lagrange-Fritz-John and Lagrange-Kuhn-Tucker multipliers. When all the data of the problem are subconvexlike we derive the results by Li, and hence those of Lin and Corley. We also show how the generalized Moreau-Rockafellar type theorem to multifunctions obtained recently by Lin can be derived from the well-known results in scalar optimization. In the last, vector optimization problem in which objective and the constraints are defined by multifunctions and depends on a parameter u, and the resulting value multifunction M(u) are considered. With the help of the generalized Moreau-Rockafellar type theorem we establish the weak subdifferential of M in terms of the weak subdifferential of objective and constraint multifunctions.     

On the calculation of a membership function for the solution of a fuzzy linear optimization problem

In the present paper the fuzzy linear optimization problem (with fuzzy coefficients in the objective function) is considered. Recent concepts of fuzzy solution to the fuzzy optimization problem based on the level-cut and the set of Pareto optimal solutions of a multiobjective optimization problem are applied. Chanas and Kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem, which is based on calculating the sum of lengths of certain intervals. The purpose of this paper is to determine a method for realizing this idea. We derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered.     

Linear scaling and the DIRECT algorithm

In this paper, we discuss the performance of the DIRECT global optimization algorithm on problems with linear scaling. We show with computations that the performance of DIRECT can be affected by linear scaling of the objective function. We also provide a theoretical result which shows that DIRECT does not perform well when the absolute value of the objective function is large enough. Then we present DIRECT-a, a modification of DIRECT, to eliminate the sensitivity to linear scaling of the objective function. We prove theoretically that linear scaling of the objective function does not affect the performance of DIRECT-a. Similarly, we prove that some modifications of DIRECT are also unaffected by linear scaling of the objective function, while the original DIRECT algorithm is sensitive to linear scaling. Numerical results in this paper show that DIRECT-a is more robust than the original DIRECT algorithm, which support the theoretical results. Numerical results also show that careful choices of the parameter ε can help DIRECT perform well when the objective function is poorly linearly scaled.     

An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters

Real engineering design problems are generally characterized by the presence of many often conflicting and incommensurable objectives. Naturally, these objectives involve many parameters whose possible values may be assigned by the experts. The aim of this paper is to introduce a hybrid approach combining three optimization techniques, dynamic programming (DP), genetic algorithms and particle swarm optimization (PSO). Our approach integrates the merits of both DP and artificial optimization techniques and it has two characteristic features. Firstly, the proposed algorithm converts fuzzy multiobjective optimization problem to a sequence of a crisp nonlinear programming problems. Secondly, the proposed algorithm uses H-SOA for solving nonlinear programming problem. In which, any complex problem under certain structure can be solved and there is no need for the existence of some properties rather than traditional methods that need some features of the problem such as differentiability and continuity. Finally, with different degree of α we get different α-Pareto optimal solution of the problem. A numerical example is given to illustrate the results developed in this paper.