associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

solutions of fuzzy relation equations based on continuous t norms this study is concerned with fuzzy relation equations with continuous t norms in the form atr b where a and b are the fuzzy subsets of x and y respectively email protected xxy is a fuzzy relation and t is a continuous t norm the problem is how to determine a from r and b first an if and only if condition of being solvable is presented novel algorithms are then presented for determining minimal solutions when x and y are finite the proposed algorithms generate all minimal solutions for the equations making them efficient solving procedures

pseudomonotone and copositive star matrices abstract we study general and complementarity properties of matrices which are either pseudomonotone or copositive star on a closed convex cone in particular we show that if a matrix is pseudomonotone on the nonnegative orthant of r n then it belongs to p 0 q 0 we also show that if a matrix t is copositive star on a closed convex cone k then the linear complementarity problem lcp t k q is solvable for all q in r n iff zero is the only solution of the homogeneous problem lcp t k 0 we characterize pseudomonotonicity of a matrix t by the nonnegativity of the quadratic form tx x on a certain set

fuzzy equations max with conditionally cancellative operations fuzzy systems of equations email protected x b are studied where email protected 0 1 m x n email protected 0 1 m and stands for max product in the case of continuous strictly increasing triangular norm b s shieh information sciences 177 19 2007 4208 4215 showed that non zero coordinates of minimal solutions are coordinates of the greatest solution the goal of this paper is a presentation of the weakest assumptions about the operation which guarantee such result thus the fuzzy system of equations is considered for a left continuous increasing conditionally cancellative operation such that 1 0 0 minimal solutions of the system are described as orthogonal projections of the greatest solution and the number of minimal solutions of email protected x b is estimated moreover an efficient method is presented for the determination of the family of all minimal solutions

stability results for polyhedral complementarity problems in this work we study the multivalued complementarity problem for polyhedral multifunctions under homogeneity assumptions we employ an approach that consists in approximating the equivalent variational inequality formulation of the problem and studying the asymptotic behavior of sequences of solutions to these approximation problems to do this we employ results and the language of variational analysis the novelty of this approach lies in the fact that it allows us to obtain not only existence results but also stability ones we consider that our results can be used for developing numerical algorithms for solving multivalued complementarity problems

increasing continuous operations in fuzzy max equations and inequalities abstract in this paper solutions of fuzzy systems of equations a x b and systems of inequalities a x b a x b are considered where a 0 1 m n b 0 1 m stands for max product and x 0 1 n is an unknown vector in this paper the families of all solutions are described during the last thirty years this problem was considered by many authors our goal is to achieve the most general assumptions for which sets of solutions systems of equations and inequalities are the union of lattice intervals especially a method of determination of all minimal solutions in such systems is indicated as a result the families of all solutions of fuzzy systems of inequalities a x b and a x b are described as the main part of these results the method of determination of the family of all minimal solutions in max system of equations with an increasing operation continuous on the second argument is presented

system of fuzzy relation equations with inf composition complete set of solutions the problem of solvability of a system of equations with inf composition is considered on finite universes equations are expressed using operations of a bl algebra we study complete set of solutions of the respective system in the particular case one equation and in the general case in both cases various conditions of solvability are found and proved we characterize all maximal solutions and prove that under certain conditions each solution of the system an equation is less than or equal to a respective maximal one as a result we are able to characterize the complete set of solutions examples of single equations and their systems are considered and complete sets of solutions are found for each

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

stable monotone variational inequalities variational inequalities associated with monotone operators possibly nonlinear and multivalued and convex sets possibly unbounded are studied in reflexive banach spaces a variety of results are given which relate to a stability concept involving a natural parameter these include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity the monotone complementarity problem is covered as a special case and the results are sharpened for linear monotone complementarity and for generalized linear programming

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

some existence results for multivalued complementarity problems this paper presents some new results for the existence of a solution to a multivalued complementary problem these results are derived under a upper limiting homogeneity ulh property on the multifunction defining the complementarity problem and on a certain closedness interiority condition the closedness condition seems to be new in the context of existence proof for the multivalued complementarity problem we show that this closedness condition is satisfied by the class of polyhedral multifunctions

even more with the lemke complementarity algorithm this note presents a new termination result for the lemke linear complementarity algorithm that unifies two previous results special cases include linear complementarity problems satisfying the evers condition and those whosem matrix is copositive plus

data visualization by multimensional scaling a deterministic annealing approach abstract multidimensional scaling addresses the problem how proximity data can be faithfully visualized as points in a low dimensional euclidean space the quality of a data embedding is measured by a stress function which compares proximity values with euclidean distances of the respective points the corresponding minimization problem is non convex and sensitive to local minima we present a novel deterministic annealing algorithm for the frequently used objective sstress and for sammon mapping derived in the framework of maximum entropy estimation experimental results demonstrate the superiority of our optimization technique compared to conventional gradient descent methods

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

on the t partial order and properties for a t norm t on 0 1 a partial order t was recently defined and studied in this study for any fixed email protected 0 1 we define the set of incomparable elements according to t and this set is deeply investigated by means of the equivalence relation defined in 14 it is shown the set 0 1 denoting the set of all equivalence classes on t norms on 0 1 is uncountably infinite finally with the help of any t norm t on 0 1 it is obtained that the family t l l 0 1 of t norms on 0 1 if t is a divisible t norm then it is obtained that 0 1 t l is a lattice thus we give an answer to an open problem in 10

parallel and deterministic algorithms from mrfs surface reconstruction deterministic approximations to markov random field mrf models are derived one of the models is shown to give in a natural way the graduated nonconvexity gnc algorithm proposed by a blake and a zisserman 1987 this model can be applied to smooth a field preserving its discontinuities a class of more complex models is then proposed in order to deal with a variety of vision problems all the theoretical results are obtained in the framework of statistical mechanics and mean field techniques a parallel iterative algorithm to solve the deterministic equations of the two models is presented together with some experiments on synthetic and real images

on multivalued complementarity problems in banach spaces in this paper we prove a new existence result for multivalued complementarity problems in banach spaces our result represents a refinement and improvement of the previously known results

convex potentials and their conjugates in analog mean field optimization this paper deals with the problem of mapping hybrid i e both discrete and continuous constrained optimization problems onto analog networks the saddle point paradigm of mean field methods in statistical physics provides a systematic procedure for finding such a mapping via the notion of effective energy specifically it is shown that within this paradigm to each closed bounded constraint set is associated a smooth convex potential function using the conjugate or the legendre fenchel transform of the convex potential the effective energy can be transformed to yield a cost function that is a natural generalization of the analog hopfield energy descent dynamics and deterministic annealing can then be used to find the global minimum of the original minimization problem when the conjugate is hard to compute explicitly it is shown that a minimax dynamics similar to that of arrow and hurwicz in lagrangian optimization can be used to find the saddle points of the effective energy as an illustration of its wide applicability the effective energy framework is used to derive hopfield like energy functions and descent dynamics for two classes of networks previously considered in the literature winner take all networks and rotor networks even when the cost function of the original optimization problem is not quadratic

a novel optimizing network architecture with applications we present a novel optimizing network architecture with applications in vision learning pattern recognition and combinatorial optimization this architecture is constructed by combining the following techniques 1 deterministic annealing 2 self amplification 3 algebraic transformations 4 clocked objectives and 5 softassign deterministic annealing in conjunction with self amplification avoids poor local minima and ensures that a vertex of the hypercube is reached algebraic transformations and clocked objectives help partition the relaxation into distinct phases the problems considered have doubly stochastic matrix constraints or minor variations thereof we introduce a new technique softassign which is used to satisfy this constraint experimental results on different problems are presented and discussed

fuzzy relation equations and inequalities abstract this paper generalizes a method of solution of fuzzy relation equations introduced by sanchez 1976 and developed by pedrycz 1982 we find the greatest or the least solution of a composite l fuzzy relation equation with arbitrary monotone composition

constrained nets for graph matching and other quadratic assignment problems some time ago durbin and willshaw proposed an interesting parallel algorithm the elastic net for approximately solving some geometric optimization problems such as the traveling salesman problem recently it has been shown that their algorithm is related to neural networks of hopfield and tank and that they both can be understood as the semiclassical approximation to statistical mechanics of related physical models the main point of the elastic net algorithm is seen to be in the way one deals with the constraints when evaluating the effective cost function free energy in the thermodynamic analogy and not in its geometric foundation emphasized originally by durbin and willshaw as a consequence the elastic net algorithm is a special case of the more general physically based computations and can be generalized to a large class of nongeometric problems in this paper we further elaborate on this observation and generalize the elastic net to the quadratic assignment problem we work out in detail its special case the graph matching problem because it is an important problem with many applications in computational vision and neural modeling simulation results on random graphs and on structured hand designed graphs of moderate size 20 100 nodes are discussed