parameter dependent lmis in robust analysis characterization of homogeneous polynomially parameter dependent solutions via lmi relaxations this note investigates the robust stability of uncertain linear time invariant systems in polytopic domains by means of parameter dependent linear matrix inequality pd lmi conditions exploiting some algebraic properties provided by the uncertainty representation a systematic procedure to construct a family of finite dimensional lmi relaxations is provided the robust stability is assessed by means of the existence of a lyapunov function more specifically a homogeneous polynomially parameter dependent lyapunov hppdl function of arbitrary degree for a given degree if an hppdl solution exists a sequence of relaxations based on real algebraic properties provides sufficient lmi conditions of increasing precision and constant number of decision variables for the existence of an hppdl function which tend to the necessity alternatively if an hppdl solution of degree exists a sequence of relaxations which increases the number of variables and the number of lmis will provide an hppdl solution of larger degree the method proposed can be applied to determine homogeneous parameter dependent matrix solutions to a wide variety of pd lmis by transforming the infinite dimensional lmi problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite dimensional lmi conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter dependent solution of arbitrary degree illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature

affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

robust absolute stability and nonlinear state feedback stabilization based on polynomial lur e functions this paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter dependent lur e functions which ensure the stability of nonlinear systems with state dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex the proposed conditions are written as linear matrix inequalities parametrized in terms of the degree g of the parameter dependent solution and in terms of the relaxation level d of the inequality constraints based on the algebraic properties of positive matrix polynomials with parameters in the unit simplex as g and d increase progressively less conservative solutions are obtained the results in the paper include as special cases existing conditions for robust stability and for absolute stability analysis a convex solution suitable for the design of robust nonlinear state feedback stabilizing controllers is also provided numerical examples illustrate the efficiency of the proposed conditions c 2008 elsevier ltd all rights reserved

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

an efficient composite heuristic for the symmetric generalized traveling salesman problem the main purpose of this paper is to introduce a new composite heuristic for solving the generalized traveling salesman problem the proposed heuristic is composed of three phases the construction of an initial partial solution the insertion of a node from each non visited node subset and a solution improvement phase we show that the heuristic performs very well on 36 tsplib problems which have been solved to optimality by other researchers we also propose some simple heuristics that can be used as basic blocks to construct more efficient composite heuristics

lower bounding procedure for the asymmetric quadratic traveling salesman problem abstract in this paper we consider the asymmetric quadratic traveling salesman problem aqtsp given a directed graph and a function that maps every pair of consecutive arcs to a cost the problem consists in finding a cycle that visits every vertex exactly once and such that the sum of the costs is minimal we propose an extended linear programming formulation that has a variable for each cycle in the graph since the number of cycles is exponential in the graph size we propose a column generation approach moreover we apply a particular reformulation linearization technique on a compact representation of the problem and compute lower bounds based on lagrangian relaxation we compare our new bounds with those obtained by some linearization models proposed in the literature computational results on some set of benchmarks used in the literature show that our lower bounding procedures are very promising

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

polynomially parameter dependent lyapunov functions for robust stability of polytopic systems an lmi approach in this note robust stability of state space models with respect to real parametric uncertainty is considered specifically a new class of parameter dependent quadratic lyapunov functions for establishing stability of a polytope of matrices is introduced i e the homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs the choice of this class which contains parameter dependent quadratic lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form is motivated by the property that a polytope of matrices is stable if and only there exists an hpd qlf the main result of the note is a sufficient condition for determining the sought hpd qlf which amounts to solving linear matrix inequalities lmis derived via the complete square matricial representation csmr of homogeneous matricial forms and the lyapunov matrix equation numerical examples are provided to demonstrate the effectiveness of the proposed approach

an lmi condition for the robust stability of uncertain continuous time linear systems a new sufficient condition for the robust stability of continuous time uncertain linear systems with convex bounded uncertainties is proposed in this note the results are based on linear matrix inequalities lmis formulated at the vertices of the uncertainty polytope which provide a parameter dependent lyapunov function that assures the stability of any matrix inside the uncertainty domain with the aid of numerical procedures based on unidimensional search and the lmis feasibility tests a simple and constructive way to compute robust stability domains can be established

lin kernighan heuristic adaptations for the generalized traveling salesman problem abstract the lin kernighan heuristic is known to be one of the most successful heuristics for the traveling salesman problem tsp it has also proven its efficiency in application to some other problems in this paper we discuss possible adaptations of tsp heuristics for the generalized traveling salesman problem gtsp and focus on the case of the lin kernighan algorithm at first we provide an easy to understand description of the original lin kernighan heuristic then we propose several adaptations both trivial and complicated finally we conduct a fair competition between all the variations of the lin kernighan adaptation and some other gtsp heuristics it appears that our adaptation of the lin kernighan algorithm for the gtsp reproduces the success of the original heuristic different variations of our adaptation outperform all other heuristics in a wide range of trade offs between solution quality and running time making lin kernighan the state of the art gtsp local search

asymptotic stability of continuous time systems with saturation nonlinearities abstract a new criterion is established for global asymptotic stability of second order systems modeled by equations of the type dot x ax where is the saturation function the derivation is based on the bendixon s theorem on limit cycles and a closer study of the trajectories of the systems applications to stabilization of more general cascade nonlinear systems are also discussed

a random key genetic algorithm for the generalized traveling salesman problem the generalized traveling salesman problem is a variation of the well known traveling salesman problem in which the set of nodes is divided into clusters the objective is to find a minimum cost tour passing through one node from each cluster we present an effective heuristic for this problem the method combines a genetic algorithm ga with a local tour improvement heuristic solutions are encoded using random keys which circumvent the feasibility problems encountered when using traditional ga encodings on a set of 41 standard test problems with symmetric distances and up to 442 nodes the heuristic found solutions that were optimal in most cases and were within 1 of optimality in all but the largest problems with computation times generally within 10 seconds the heuristic is competitive with other heuristics published to date in both solution quality and computation time

detecting homogeneous groups in clustering using the euclidean distance the use of the euclidean distance as dissimilarity measure to detect groups in a sample implies that the features space is homogeneous the clustering process can be realised under this condition taking the sample as one group but when the groups are detected we must assure that each one is homogeneous in order to apply the euclidean distance to detect homogeneous groups we propose a criterion which treats each group as a fuzzy set in the same universe of discourse of the sample

technical communique absolute stability criteria with prescribed decay rate for finite dimensional and delay systems we provide here an extension of popov criterion permitting to check exponential stability with prescribed decay rate otherwise called a stability of nonlinear delay systems with sector bounded nonlinearities as for the celebrated result the main hypothesis is expressed under a frequency form for the delay free case the latter is equivalent to a linear matrix inequality whose solution may be found by widespread algorithms

convex set based fuzzy clustering prototype based methods are commonly used in cluster analysis and the results may be highly dependent on the prototype used we propose a two level fuzzy clustering method that involves adaptively expanding and merging convex polytopes where the convex polytopes are considered as a flexible prototype therefore the dependency on the use of a specified prototype can be eliminated also the proposed method makes it possible to effectively represent an arbitrarily distributed data set without a priori knowledge of the number of clusters in the data set in the first level of our proposed method each cluster is represented by a convex polytope which is described by its set of vertices specifically nonlinear membership functions are utilized to determine whether an input pattern creates a new cluster or whether an existing cluster should be modified in the second level the expandable clusters that are selected by an intercluster distance measure are merged to improve clustering efficiency and to reduce the order dependency of the incoming input patterns several experimental results are given to show the validity of our method

stability of polytopes of matrices via affine parameter dependent lyapunov functions asymptotically exact lmi conditions abstract this paper investigates necessary and sufficient conditions for the existence of an affine parameter dependent lyapunov function assuring the hurwitz or schur stability of a polytope of matrices a systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given at each step a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter dependent lyapunov function necessity is asymptotically attained through a relaxation based on a generalization of polya s theorem to the case of matrix valued functions numerical experiments illustrate the results

an ensemble of discrete differential evolution algorithms for solving the generalized traveling salesman problem in this paper an ensemble of discrete differential evolution algorithms with parallel populations is presented in a single populated discrete differential evolution dde algorithm the destruction and construction dc procedure is employed to generate the mutant population whereas the trial population is obtained through a crossover operator the performance of the dde algorithm is substantially affected by the parameters of dc procedure as well as the choice of crossover operator in order to enable the dde algorithm to make use of different parameter values and crossover operators simultaneously we propose an ensemble of dde edde algorithms where each parameter set and crossover operator is assigned to one of the parallel populations each parallel parent population does not only compete with offspring population generated by its own population but also the offspring populations generated by all other parallel populations which use different parameter settings and crossover operators as an application area the well known generalized traveling salesman problem gtsp is chosen where the set of nodes is divided into clusters so that the objective is to find a tour with minimum cost passing through exactly one node from each cluster the experimental results show that none of the single populated variants was effective in solving all the gtsp instances whereas the edde performed substantially better than the single populated variants on a set of problem instances furthermore through the experimental analysis of results the performance of the edde algorithm is also compared against the best performing algorithms from the literature ultimately all of the best known averaged solutions for larger instances are further improved by the edde algorithm

quantifying vagueness in decision models abstract when modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints whereas qualitative statements are easily given in the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information though it is quite advanced formally the practical determination of its core elements i e membership functions and operators has only been explored to a very limited extent this paper presents results of empirical research which focused on the problem of modelling vagueness i e determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as young man long sticks high profits etc

efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem abstract the generalized traveling salesman problem gtsp is a well known combinatorial optimization problem with a host of applications it is an extension of the traveling salesman problem tsp where the set of cities is partitioned into so called clusters and the salesman has to visit every cluster exactly once while the gtsp is a very important combinatorial optimization problem and is well studied in many aspects the local search algorithms used in the literature are mostly basic adaptations of simple tsp heuristics hence a thorough and deep research of the neighborhoods and local search algorithms specific to the gtsp is required we formalize the procedure of adaptation of a tsp neighborhood for the gtsp and classify all other existing and some new gtsp neighborhoods for every neighborhood we provide efficient exploration algorithms that are often significantly faster than the ones known from the literature finally we compare different local search implementations empirically

lmi conditions for robust stability analysis based on polynomially parameter dependent lyapunov functions abstract the robust stability of uncertain linear systems in polytopic domains is investigated in this paper the main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter dependent lyapunov matrix functions of arbitrary degree on the uncertain parameters the conditions exploit the positivity of the uncertain parameters being constructed in such a way that as the degree of the polynomial increases the number of linear matrix inequalities and free variables increases and the test becomes less conservative if a feasible solution exists for a certain degree the conditions will also be verified for larger degrees for any given degree the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability both continuous and discrete time uncertain systems are addressed as illustrated by numerical examples

brief circle and popov criteria as tools for nonlinear feedback design we convert classical absolute stability criteria into nonlinear design procedures which employ efficient numerical tools such as lmi s we first analyze an earlier circle criterion design and show that its feasibility is limited by conditions on the unstable part of the zero dynamics and the relative degree then we develop an extended circle criterion design which eliminates the relative degree obstacle the restrictions on the zero dynamics are relaxed by using the popov multiplier which also reduces controller complexity the results are illustrated with several design examples

a lagrangian based approach for the asymmetric generalized traveling salesman problem this paper presents an optimal approach for the asymmetric generalized traveling salesman problem gtsp the gtsp is defined on a directed graph in which the nodes are grouped into m predefined mutually exclusive and exhaustive sets with the arc set containing no intraset arcs the problem is to find a minimum cost m arc directed cycle which includes exactly one node from each set our approach employs a lagrangian relaxation to compute a lower bound on the total cost of an optimal solution the lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution finally we use an efficient branch and bound procedure which exploits the multiple choice structure of the node sets we present computational results for the optimal approach tested on a series of randomly generated problems the results show success on a range of problems with up to 104 nodes

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

a double loop hybrid algorithm for the traveling salesman problem with arbitrary neighbourhoods abstract this paper addresses the traveling salesman problem with arbitrary neighbourhoods which is an np hard problem in combinatorial optimization and is important in operations research and theoretical computer science existing methods based on neighbourhood predigestion and discretization are impractical in cases in which the neighbourhoods are arbitrary in this paper the neighbourhoods of the traveling salesman problem are generalized to arbitrarily connected regions in planar euclidean space a novel approach to solving this problem is proposed including a boundary based encoding scheme and a double loop hybrid algorithm based on particle swarm optimization and genetic algorithm in the hybrid algorithm linear descending inertia weight particle swarm optimization is adopted to search continuous visiting positions in the outer loop and the genetic algorithm is used to optimize the discrete visiting sequence in the inner loop the boundary based encoding scheme can reduce the search space significantly without degrading the solution quality and the hybrid algorithm can find a high quality solution in a reasonable time the computational results on both small and large instances demonstrate that the proposed approach can guarantee a high quality solution in a reasonable time compared with three other state of the art algorithms iterative algorithm branch and bound algorithm and upper and lower bound algorithm moreover the proposed approach works efficiently in a real world application that cannot be solved by existing algorithms

global asymptotic stabilization of systems satisfying two different sector conditions global asymptotic stabilization for a class of nonlinear systems is addressed the dynamics of these systems are composed of a linear part to which is added some nonlinearities which satisfy two different sector bound conditions depending on whether the state is near or far from the origin the proposed approach is based on the uniting of control lyapunov functions in this framework the stabilization problem may be recast as an lmi optimization problem for which powerful semidefinite programming softwares exist this is illustrated by means of three numerical examples

aggregating membership values by a choquet fuzzy integral based operator a new type of aggregator the choquet fuzzy integral based operator is introduced and applied in aggregation of membership values in this paper we generalized the definition of the fuzzy integral as a tool for membership aggregation the generalization involves replacing the max or min operator in information integration with a choquet fuzzy integral based aggregator resulting in increased flexibility the characteristics of the choquet fuzzy integral are studied and a two phase aggregating process is proposed also we demonstrate a good classification result using the proposed aggregator to integrate the memberships of several clusters in a complex unconstrained handwritten digit recognition domain

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

an existence result for polynomial solutions of parameter dependent lmis we show in this paper that any system of linear matrix inequalities depending continuously upon scalar parameters and solvable for any value of the latter in a fixed compact set admits a branch of solutions polynomial with respect to the parameters this result is useful for studying e g parametric robustness or gain scheduling issues

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported