fuzzy inclusion and fuzzy equality of two fuzzy subsets fuzzy operations for fuzzy subsets in this paper we introduce a new definition of fuzzy inclusion and equality between fuzzy subsets the main idea is that we evaluate a fuzzy equality not only by one value as was done up to the present but by two ones i e by a pair of numbers from the closed unit interval basic properties of the proposed fuzzy inclusion and equality are given using the new notion of fuzzy equality we next define fuzzy operations for fuzzy subsets these operations are also characterized by a pair of numbers all fundamental laws for fuzzy operations are listed some connections between llukasiewicz logic and introduced definitions of fuzzy inclusion and equality are also described

primal and dual linear decision rules in stochastic and robust optimization linear stochastic programming provides a flexible toolbox for analyzing real life decision situations but it can become computationally cumbersome when recourse decisions are involved the latter are usually modeled as decision rules i e functions of the uncertain problem data it has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence in this paper we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program by employing techniques that are commonly used in modern robust optimization we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes the gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation our method remains applicable if the stochastic program has random recourse and multiple decision stages it also extends to cases involving ambiguous probability distributions

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

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

arithmetic operations of non normal fuzzy sets using gradual numbers abstract this paper will study the arithmetic operations of non normal fuzzy sets using the concept of gradual numbers that can be regarded as the elements of gradual sets we shall present the idea in which the fuzzy sets can be formulated as consisting of gradual elements like the usual set consisting of usual elements when the universal set is taken to be the real number system the gradual element is also called a gradual number in this case the arithmetic operations of non normal fuzzy sets can be defined by using the gradual numbers

operations on fuzzy sets an axiomatic approach abstract we introduce intersection and union of fuzzy sets constructed by means of operations on the unit interval which may depend on the considered element of the universe of discourse a similar approach for complementation was given by lowen 7 the main result is that element dependent operations are not compatible with the preservation of the intersection and union under the inverse image of a function finally we consider fuzzy algebras built with these set theoretic operations

on the performance of affine policies for two stage adaptive optimization a geometric perspective we consider two stage adjustable robust linear optimization problems with uncertain right hand side mathbf b b belonging to a convex and compact uncertainty set mathcal u u we provide an a priori approximation bound on the ratio of the optimal affine in mathbf b b solution to the optimal adjustable solution that depends on two fundamental geometric properties of mathcal u u a the symmetry and b the simplex dilation factor of the uncertainty set mathcal u u and provides deeper insight on the power of affine policies for this class of problems the bound improves upon a priori bounds obtained for robust and affine policies proposed in the literature we also find that the proposed a priori bound is quite close to a posteriori bounds computed in specific instances of an inventory control problem illustrating that the proposed bound is informative

nonuniform sampling and antialiasing in image representation a unified approach to the representation and processing of a class of images which are not bandlimited but belong to the space of locally bandlimited signals is presented a nonuniform sampling theorem clark et al 1985 for functions belonging to this space is extended and a class of nonstationary stochastic processes is considered the space of locally bandlimited signals is shown to be a reproducing kernel space a generalized projection theorem can therefore be applied yielding either a continuous or a discrete projection filter the former can be used for image conditioning prior to nonuniform sampling while the latter provides a general tool for image representation by nonuniform sampling schemes the problem of finding the local bandwidth of a given signal in order to generate an optimal sampling scheme is addressed in the context of signal representation in the combined position frequency space the stochastic estimation of parameters which characterize the local bandwidth is discussed bounds on the error resulting from the utilization of nonexact position varying signal parameters are derived

frequency domain analysis and synthesis of image pyramid generating kernels construction of image pyramids is described as a two di mensional decimation process frequently employed generating kernels are compared to the optimal kernel that assures minimal information loss after the resolution reduction i e the one corresponding to an ideal low pass filter physically realizable optimal generating kernels are presented the amount of computation required for generation of the image pyramid can be reduced significantly by employing half band filters as components of the optimal kernel image pyramids generated by the optimal kernel show a better command of details than the ones generated by a simple 4 4 averaging or a computationally equivalent kernel

construction of fuzzy σ algebras using triangular norms abstract generalizing the definitions given by the author fuzzy sets and systems4 1980 83 93 we introduce and study t fuzzy algebras t being any triangular norm the main result is that for a large class of triangular norms each t fuzzy algebra is generated i e consists of all functions x 0 1 being measurable with respect to some algebra on x

multistage robust mixed integer optimization with adaptive partitions we present a new partition and bound method for multistage adaptive mixed integer optimization amio problems that extends previous work on finite adaptability the approach analyzes the optimal solution to a static nonadaptive version of an amio problem to gain insight into which regions of the uncertainty set are restricting the objective function value we use this information to construct partitions in the uncertainty set leading to a finitely adaptable formulation of the problem we use the same information to determine a lower bound on the fully adaptive solution the method repeats this process iteratively to further improve the objective until a desired gap is reached we provide theoretical motivation for this method and characterize its convergence properties and the growth in the number of partitions using these insights we propose and evaluate enhancements to the method such as warm starts and smarter partition creation we describe in detail how to apply finite adaptability to multista

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

ordinal sums of representable uninorms abstract we investigate properties of an ordinal sum of uninorms in the case that the summands are proper representable uninorms we show sufficient and necessary conditions for a uninorm to be an ordinal sum of representable uninorms an example is also included

connective generators for archimedian triangular operators abstract triangular norms form the kernel of fuzzy logic and fuzzy systems here in this paper my main motivation is to provide three new connective generators that can generate different classes of fuzzy connectives in this study i have tried to show that there are many possible fuzzy connective generators that can be associative and yet monotonic the study contemplates mainly on three different connective generators these generators use simple monotonic and continuous functions that are both bounded in the domain 0 1 and the range 0 1

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

piecewise static policies for two stage adjustable robust linear optimization in this paper we consider two stage adjustable robust linear optimization problems under uncertain constraints and study the performance of piecewise static policies these are a generalization of static policies where we divide the uncertainty set into several pieces and specify a static solution for each piece we show that in the worst case there is no piecewise static policy with a polynomial number of pieces that has a significantly better performance than an optimal static policy this is quite surprising as piecewise static policies are significantly more general than static policies the proof is based on a combinatorial argument and the structure of piecewise static policies

the distributivity property of increasing binary operations abstract this paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory in particular some assumptions namely associativity and commutativity and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified moreover some other properties e g componentwise convexity concavity for special operations from the class of quasi copulas is considered

a nonlinear semidefinite optimization relaxation for the worst case linear optimization under uncertainties in this paper we consider the so called worst case linear optimization wclo with uncertainties in the right hand side of the constraints such a problem arises from numerous applications such as systemic risk estimate in finance and stochastic optimization we first show the problem is np hard and present a coarse semidefinite relaxation sdr for wclo an iterative procedure is introduced to sequentially refine the relaxation model based on the solution of the current relaxation model by simply changing some parameters in the coarse sdr we show that the sequence of the proposed sdrs will converge to some nonlinear semidefinite optimization problem sdo a bi section search algorithm is proposed to solve the resulting nonlinear sdo our preliminary experimental results illustrate that the nonlinear sdr can provide very tight bounds for the original wclo and is able to locate the exact global solution in most cases within a few iterations on average

design of near optimal decision rules in multistage adaptive mixed integer optimization in recent years decision rules have been established as the preferred solution method for addressing computationally demanding multistage adaptive optimization problems despite their success existing decision rules a are typically constrained by their a priori design and b do not incorporate in their modeling adaptive binary decisions to address these problems we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions respectively we then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed integer optimization we demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems we provide global lower bounds and show that our approach is i practically tractable and ii provides high quality solutions that outperform alternative methods

on locally internal monotonic operations this paper deals with monotonic binary operations f 0 1 2 0 1 with the property called locally internal property that the value at any point x y is always one of its arguments x y after stating a theorem that characterizes this kind of operations some special cases are studied in detail by considering additional properties of the operation commutativity existence of a neutral element and associativity in case of locally internal associative monotonic operations with neutral element a characterization theorem gives an improvement of a well known theorem of czogala and drewniak on idempotent associative and increasing operations with neutral element as well as an improvement of a characterization theorem for left and right continuous idempotent uninorms

design of affine controllers via convex optimization we consider a discrete time time varying linear dynamical system perturbed by process noise with linear noise corrupted measurements over a finite horizon we address the problem of designing a general affine causal controller in which the control input is an affine function of all previous measurements in order to minimize a convex objective in either a stochastic or worst case setting this controller design problem is not convex in its natural form but can be transformed to an equivalent convex optimization problem by a nonlinear change of variables which allows us to efficiently solve the problem our method is related to the classical design procedure for time invariant infinite horizon linear controller design and the more recent purified output control method we illustrate the method with applications to supply chain optimization and dynamic portfolio optimization and show the method can be combined with model predictive control techniques when perfect state information is available

extending scope of robust optimization comprehensive robust counterparts of uncertain problems in this paper we propose a new methodology for handling optimization problems with uncertain data with the usual robust optimization paradigm one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set whereas with the proposed approach we require also a controlled deterioration in performance when the data is outside the uncertainty set the extension of robust optimization methodology developed in this paper opens up new possibilities to solve efficiently multi stage finite horizon uncertain optimization problems in particular to analyze and to synthesize linear controllers for discrete time dynamical systems

the aggregation operations for decision making in probabilistic fuzzy environment this paper deals with some important classes of aggregation operations on various kinds of sets applied to decision making problems these operations are mainly based on general concepts such as triangular norms t and s norms in this paper we focus particularly on operations on probabilistic sets and their distribution function representation the considerations are illustrated by means of numerical examples

generalized medians distance aggregation operators are introduced in the case of singleton based distance a well known class of k medians is recovered for interval based distance characterization of p r distance aggregation operators is given in general as well as for idempotent operators based on the triangle of median factors medians are introduced owa operators are shown to be a linear convex combination of special j medians

generalised fuzzy aggregation in estimating the risk of desertification of a burned forest this paper investigates two aspects of using fuzzy logic with a gis the use of non conventional aggregation operators for conjunctive and disjunctive reasoning and the use of weights when combining aggregates of different significance to the classification two methods of weighting are used weights as powers of the membership functions and the novel concept of membership functions that take values greater than 1 the two approaches are investigated in conjunction with 121 possible combinations of aggregation operators that are used to reason in the disjunctive and conjunctive level of the problem of assessing the risk of desertification after a forest fire

optimality of affine policies in multistage robust optimization in this paper we prove the optimality of disturbance affine control policies in the context of one dimensional constrained multistage robust optimization our results cover the finite horizon case with minimax worst case objective and convex state costs plus linear control costs we develop a new proof methodology which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function apart from providing an elegant and conceptually simple proof technique the approach also entails very fast algorithms for the case of piecewise affine state costs which we explore in connection with a classical inventory management application

on the adaptivity gap in two stage robust linear optimization under uncertain packing constraints in this paper we study the performance of static solutions in two stage adjustable robust packing linear optimization problem with uncertain constraint coefficients such problems arise in many important applications such as revenue management and resource allocation problems where demand requests have uncertain resource requirements the goal is to find a two stage solution that maximizes the worst case objective value over all possible realizations of the second stage constraints from a given uncertainty set we consider the case where the uncertainty set is column wise and constraint wise any constraint describing the set involve entries of only a single column or a single row this is a fairly general class of uncertainty sets to model constraint coefficient uncertainty we show that the two stage adjustable robust problem is varomega log n hard to approximate on the positive side we show that a static solution is an o big log n cdot min log vargamma log m n big approximation for the two stage adjustable robust problem where m and n denote the numbers of rows and columns of the constraint matrix and vargamma is the maximum possible ratio of upper bounds of the uncertain constraint coefficients therefore for constant vargamma surprisingly the performance bound for static solutions and therefore the adaptivity gap matches the hardness of approximation for the adjustable problems furthermore in general the static solution provides nearly the best efficient approximation for the two stage adjustable robust problem

multiple boolean algebras and their application to fuzzy sets abstract a cycle of semilattices is a set e together with p semilattice operations which can be linearly ordered so that each operation is distributive to the following and the last is distributive to the first a multiple boolean algebra is defined to be a cycle of semilattices satisfying certain additional requirements 1 i each operation has an absorbing and an identity element 2 ii there is a bijection e e carrying each operation into the next and the last into the first 3 iii an axiom is satisfied that generalizes the fact that in boolean algebras the meet and the join of an element with its complement are equal to the zero and the identity elements respectively it is shown that for every n 1 and p 2 there exist multiple boolean algebras of order p and cardinality p n the new theoretical construct can be applied to the theory of fuzzy sets if a suitable finite subset of the unit interval 0 1 is taken as the set of the grades of membership the set of the fuzzy subsets in a universe can be equipped with additional binary operations in such manner that it becomes isomorphic to a pair of multiple boolean algebras one conjugate to the other

fuzzy cardinals based on the generalized equality of fuzzy subsets the purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality the proposed idea will be compared with several approaches which were earlier offered in fuzzy literature using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals i e we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has respectively at most least and less than more than k elements various properties of different fuzzy cardinals and semicardinals as well as some interdependences between them are also given

on the associativity functional equation abstract let a b be any bounded closed real interval the class of all continuous nondecreasing associative functions m a b 2 a b fulfilling the boundary conditions m a a a and m b b b is described

a small set of axioms for residuated logic we show that a set of three axioms together with modus ponens suffices to build up a formal system for the class r of all quadruples l t j in which l is a partially ordered set with a top element and t is a triangular norm on l that has a residuated implication j we show that this r is not a variety

infinite horizon performance bounds for uncertain constrained systems we present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances we take as a performance metric either an expected value or minimax discounted cost over an infinite horizon and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances and we show that the two bounds are closely related in particular the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set we show that our performance bounds can be computed via solution of a finite dimensional convex optimization problem and provide numerical examples to illustrate the efficacy of our method

a tractable approach for designing piecewise affine policies in two stage adjustable robust optimization we consider the problem of designing piecewise affine policies for two stage adjustable robust linear optimization problems under right hand side uncertainty it is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large a significant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set here we address this challenge by introducing a new framework in which the uncertainty set is approximated by a dominating simplex the corresponding policy is then based on a mapping from the uncertainty set to the simplex although our piecewise affine policy has exponentially many pieces it can be computed efficiently by solving a compact linear program given the dominating simplex furthermore we can find the dominating simplex in a closed form if the uncertainty set satisfies some symmetries and can be computed using a mip in general we would like to remark that our policy is an approximate piecewise affine policy and is not necessarily a generalization of the class of affine policies nevertheless the performance of our policy is significantly better than the affine policy for many important uncertainty sets such as ellipsoids and norm balls both theoretically and numerically for instance for hypersphere uncertainty set our piecewise affine policy can be computed by an lp and gives a o m 1 4 approximation whereas the affine policy requires us to solve a second order cone program and has a worst case performance bound of o sqrt m

fuzzy sets and their operations ii among the basic operations which can be performed on fuzzy sets are the operations of union intersection complement algebraic product and algebraic sum in addition to these operations new operations called bounded sum and bounded difference were defined by l a zadeh to investigate the fuzzy reasoning which provides a way of dealing with the reasoning problems which are too complex for precise solution this paper investigates the algebraic properties of fuzzy sets under these nex operations of bounded sum and bounded difference and the properties of fuzzy sets in the case where these new operations are combined with the well known operations of union intersection algebraic product and algebraic sum

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

on the power and limitations of affine policies in two stage adaptive optimization we consider a two stage adaptive linear optimization problem under right hand side uncertainty with a min max objective and give a sharp characterization of the power and limitations of affine policies where the second stage solution is an affine function of the right hand side uncertainty in particular we show that the worst case cost of an optimal affine policy can be omega m 1 2 delta times the worst case cost of an optimal fully adaptable solution for any 0 where m is the number of linear constraints we also show that the worst case cost of the best affine policy is o sqrt m times the optimal cost when the first stage constraint matrix has non negative coefficients moreover if there are only k m uncertain parameters we generalize the performance bound for affine policies to o sqrt k which is particularly useful if only a few parameters are uncertain we also provide an o sqrt k approximation algorithm for the general case without any restriction on the constraint matrix but the solution is not an affine function of the uncertain parameters we also give a tight characterization of the conditions under which an affine policy is optimal for the above model in particular we show that if the uncertainty set mathcal u subseteq mathbb r m is a simplex then an affine policy is optimal however an affine policy is suboptimal even if mathcal u is a convex combination of only m 3 extreme points only two more extreme points than a simplex and the worst case cost of an optimal affine policy can be a factor 2 worse than the worst case cost of an optimal fully adaptable solution for any 0

real coextensions as a tool for constructing triangular norms we present in this paper a universal method of constructing left continuous triangular norms l c t norms the starting point is an arbitrary possibly finite totally ordered monoid fulfilling the conditions that are characteristic for l c t norms commutativity negativity and quanticity we show that under suitable conditions we can extend this structure by substituting each element for a real interval the process can be iterated and if the final structure obtained in this way is order isomorphic to a closed real interval its monoidal operation can up to isomorphism be identified with a l c t norm we specify the constituents needed for the construction in an explicit way we furthermore illustrate the method on the basis of a number of examples

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

when are static and adjustable robust optimization problems with constraint wise uncertainty equivalent adjustable robust optimization aro generally produces better worst case solutions than static robust optimization ro however aro is computationally more difficult than ro in this paper we provide conditions under which the worst case objective values of aro and ro problems are equal we prove that when the uncertainty is constraint wise the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact then robust solutions are also optimal for the corresponding aro problem furthermore we prove that if some of the uncertain parameters are constraint wise and the rest are not then under a similar set of assumptions there is an optimal decision rule for the aro problem that does not depend on the constraint wise uncertain parameters also we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non constraint wise uncertain parameters finally we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

on the power of robust solutions in two stage stochastic and adaptive optimization problems we consider a two stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution in particular we show that if the right hand side of the constraints is uncertain and belongs to a symmetric uncertainty set such as hypercube ellipsoid or norm ball and the probability measure is also symmetric then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two stage stochastic problem furthermore we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric we refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two stage stochastic problem as the stochasticity gap we also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive if both the objective coefficients and right hand side are uncertain we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric however we prove that the adaptability gap ratio of optimal cost of the robust problem and the optimal cost of a two stage fully adaptable problem is at most four even if both the objective coefficients and the right hand side of the constraints are uncertain and belong to a symmetric uncertainty set the bound holds for the class of positive uncertainty sets as well moreover if the uncertainty set is a hypercube special case of a symmetric set the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain

selected topics in robust convex optimization robust optimization is a rapidly developing methodology for handling optimization problems affected by non stochastic uncertain but bounded data perturbations in this paper we overview several selected topics in this popular area specifically 1 recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data 2 tractability of robust counterparts 3 links between ro and traditional chance constrained settings of problems with stochastic data and 4 a novel generic application of the ro methodology in robust linear control

on the order of triangular norms comments on a triangular norm hierarchy by e cretu in cretu fuzzy sets and systems 120 2001 371 the members of the families of frank dubois prade yager and hamacher t norms respectively are compared in a pointwise way with the minimum the product and the lukasiewicz t norm all these results are well known and trivial moreover these families of t norms cannot only be compared with the three basic t norms above but all these families are monotone with respect to their index a fact which is also well known and straightforward to prove with the exception of the family of frank t norms whose monotonicity has been first proven in butnariu and klement triangular norm based measures and games with fuzzy coalitions kluwer dordrecht 1993

a characterization of idempotent nullnorms nullnorms are generalization of triangular norms and conorms with a zero element anywhere in the unit interval nullnorms are built up from triangular norms and triangular conorms if nullnorm is idempotent then it is composed of a minimum and a maximum the idea of this paper is to find minimal assumptions on an operation in order to get an idempotent nullnorm

nonuniform image representation in area of interest systems this paper is concerned with image representation by data distributed nonuniformly and in particular with a representation scheme suitable for area of interest imaging the class of signals under consideration whose information density varies with position can be represented according to a nonuniform sampling scheme position varying projection operators are presented as simple low pass filtering operations in a fourier like domain sequential projections are used for pyramidal representation of nonuniformly sampled images it is shown that irregular random sampling prevents under certain mild restrictions aliasing effects