learning rule sets and sugeno integrals for monotonic classification problems in some variants of the supervised classification setting the domains of the attributes and the set of classes are totally ordered sets the task of learning a classifier that is nondecreasing w r t each attribute is called monotonic classification several kinds of models can be used in this task in this paper we focus on decision rules we propose a method for learning a set of decision rules that optimally fits the training data while favoring short rules over long ones we give new results on the representation of sets of if then rules by extensions of sugeno integrals to distinct attribute domains where local utility functions are used to map attribute domains to a common totally ordered scale we study whether such qualitative extensions of sugeno integral provide compact representations of large sets of decision rules

properties of rule interestingness measures and alternative approaches to normalization of measures we are considering properties of interestingness measures of rules induced from data these are bayesian confirmation property two properties related to the case of entailment or refutation called ex 1 and logicality l and a group of symmetry properties we propose a modification of properties ex 1 and l called weak ex 1 and weak l that deploy the concept of confirmation in its larger sense we demonstrate that properties ex 1 and l do not fully reflect such understanding of the confirmation concept and thus we propose to substitute ex 1 by weak ex 1 and l by weak l moreover we introduce four new approaches to normalization of confirmation measures in order to transform measures so that they would obtain desired properties the analysis of the results of the normalizations of the confirmation measures takes into account all considered properties we advocate for two normalized confirmation measures measure z considered in the literature and newly proposed measure a finally we provide some ideas for combining them in a single measure keeping all desirable properties

invariant characterizations of fuzzy information systems under some homomorphisms based on data compression and related results abstract a fuzzy information system is an important mathematical model in the field of artificial intelligence and the notion of homomorphisms between fuzzy information systems is a kind of tool to study data compression in fuzzy information systems this paper investigates invariant characterizations of fuzzy information systems under some homomorphisms based on data compression upper reducts lower reducts and reducts in a fuzzy information system are first investigated then fuzzy sets in a fuzzy information system are divided into three categories according to their importance next measures of uncertainty for fuzzy information systems are investigated and in order to show the performance of the proposed measures a numerical experimental example on the ecoli data set that comes from uci repository of machine learning databases is given finally invariant and inverse invariant characterizations of a fuzzy information system under some homomorphisms are obtained

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

relationships among generalized rough sets in six coverings and pure reflexive neighborhood system rough set is a useful tool to deal with partition related uncertainty granularity and incompleteness of knowledge although the classical rough set is constructed on the basis of an indiscernibility relation it can also be generalized by using some weaker binary relations in this paper a systematic approach is used to study the generalized rough sets in six coverings and pure reflexive neighborhood system after two steps relationships among generalized rough sets in six coverings and pure reflexive neighborhood system are obtained the first step is to study the generalized rough sets in six coverings and to get relationships between every two covering rough set models the second step is to study the relationships between generalized rough sets in each covering and in pure reflexive neighborhood system the inclusion relations or equivalence relations among the seven upper lower approximations could be acquired finally the accuracy measures of generalized rough sets in six coverings and that in pure reflexive neighborhood system are compared the relationships among seven accuracy measures are also obtained some illustrative examples are employed to demonstrate our arguments

descent methods with linesearch in the presence of perturbations we consider the class of descent algorithms for unconstrained optimization with an armijo type stepsize rule in the case when the gradient of the objective function is computed inexactly an important novel feature in our theoretical analysis is that perturbations associated with the gradient are not assumed to be relatively small or to tend to zero in the limit as a practical matter we expect them to be reasonably small so that a meaningful approximate solution can be obtained this feature makes our analysis applicable to various difficult problems encounted in practice we propose a modified armijo type rule for computing the stepsize which guarantees that the algorithm obtains a reasonable approximate solution furthermore if perturbations are small relative to the size of the gradient then our algorithm retains all the standard convergence properties of descent methods

attribute reduction of data with error ranges and test costs in data mining applications we have a number of measurement methods to obtain a data item with different test costs and different error ranges test costs refer to time money or other resources spent in obtaining data items related to some object observational errors correspond to differences in measured and true value of a data item in supervised learning we need to decide which data items to obtain and which measurement methods to employ so as to minimize the total test cost and help in constructing classifiers this paper studies this problem in four steps first data models are built to address error ranges and test costs second error range based covering rough set is constructed to define lower and upper approximations positive regions and relative reducts a closely related theory deals with neighborhood rough set which has been successfully applied to heterogeneous attribute reduction the major difference between the two theories is the definition of neighborhood third the minimal test cost attribute reduction problem is redefined in the new theory fourth both backtrack and heuristic algorithms are proposed to deal with the new problem the algorithms are tested on ten uci university of california irvine datasets experimental results show that the backtrack algorithm is efficient on rational sized datasets the weighting mechanism for the heuristic information is effective and the competition approach can improve the quality of the result significantly this study suggests new research trends concerning attribute reduction and covering rough set

error estimates for approximate optimization by the extended ritz method an alternative to the classical ritz method for approximate optimization is investigated in the extended ritz method sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n tuples of functions from a given basis this alternative scheme called variable basis approximation includes functions computable by trigonometric polynomials with free frequencies free node splines neural networks and other nonlinear approximating families estimates of rates of approximate optimization by the extended ritz method are derived upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable basis functions the modulus of continuity of the functional to be minimized the modulus of tikhonov well posedness of the problem and certain norms tailored to the type of basis the results are applied to convex best approximation and to kernel methods in machine learning

the benders decomposition algorithm a literature review the benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems this paper presents a state of the art survey of this algorithm emphasizing its use in combinatorial optimization we discuss the classical algorithm the impact of the problem formulation on its convergence and the relationship to other decomposition methods we introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm the taxonomy provides the framework to synthesize the literature and to identify shortcomings trends and potential research directions we also discuss the use of the benders decomposition to develop efficient meta heuristics describe the limitations of the classical algorithm and present extensions enabling its application to a broader range of problems

random algorithms for convex minimization problems this paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems where the constraint set is specified as the intersection of possibly infinitely many constraint sets each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function the proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance but it is rather learned in time through observations also the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite we analyze the proposed algorithm for the case when the objective function is differentiable with lipschitz gradients and the case when the objective function is not necessarily differentiable the behavior of the algorithm is investigated both for diminishing and non diminishing stepsize values the almost sure convergence to an optimal solution is established for diminishing stepsize for non diminishing stepsize the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set as well as for the expected sub optimality of the function values along the weighted averages

inexact cuts in benders decomposition benders decomposition is a well known technique for solving large linear programs with a special structure in particular it is a popular technique for solving multistage stochastic linear programming problems early termination in the subproblems generated during benders decomposition assuming dual feasibility produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution we describe an inexact cut algorithm prove its convergence under easily verifiable assumptions and discuss a corresponding dantzig wolfe decomposition algorithm the paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling

partially monotonic decision trees abstract in multicriteria decision tasks certain features are linearly ordered according to the decision and are called criteria whereas others called regular attributes are not in practice regular attributes and criteria coexist in most classification tasks in this paper we propose a rank inconsistent rate that distinguishes attributes from criteria furthermore it represents the directions of the monotonic relationships between criteria and decisions we design a partially monotonic decision tree algorithm to extract decision rules for partially monotonic classification tasks experimental results show that the proposed algorithm is effective and efficient

dynamic programming approach to optimization of approximate decision rules this paper is devoted to the study of an extension of dynamic programming approach which allows sequential optimization of approximate decision rules relative to the length and coverage we introduce an uncertainty measure r t which is the number of unordered pairs of rows with different decisions in the decision table t for a nonnegative real number b we consider b decision rules that localize rows in subtables of t with uncertainty at most b our algorithm constructs a directed acyclic graph d b t which nodes are subtables of the decision table t given by systems of equations of the kind attribute value this algorithm finishes the partitioning of a subtable when its uncertainty is at most b the graph d b t allows us to describe the whole set of so called irredundant b decision rules we can describe all irredundant b decision rules with minimum length and after that among these rules describe all rules with maximum coverage we can also change the order of optimization the consideration of irredundant rules only does not change the results of optimization this paper contains also results of experiments with decision tables from uci machine learning repository

accelerated variance reduction stochastic admm for large scale machine learning recently many stochastic variance reduced alternating direction methods of multipliers admms e g sag admm and svrg admm have made exciting progress such as linear convergence rate for strongly convex sc problems however their best known convergence rate for non strongly convex non sc problems is o 1 t as opposed to o 1 t 2 of accelerated deterministic algorithms where t is the number of iterations thus there remains a gap in the convergence rates of existing stochastic admm and deterministic algorithms to bridge this gap we introduce a new momentum acceleration trick into stochastic variance reduced admm and propose a novel accelerated svrg admm method called asvrg admm for the machine learning problems with the constraint ax by c then we design a linearized proximal update rule and a simple proximal one for the two classes of admm style problems with b i and b i respectively where i is an identity matrix and is an arbitrary bounded constant note that our linearized proximal update rule can avoid solving sub problems iteratively moreover we prove that asvrg admm converges linearly for sc problems in particular asvrg admm improves the convergence rate from o 1 t to o 1 t 2 for non sc problems finally we apply asvrg admm to various machine learning problems and show that asvrg admm consistently converges faster than the state of the art methods

a novel method for attribute reduction of covering decision systems attribute reduction has become an important step in pattern recognition and machine learning tasks covering rough sets as a generalization of classical rough sets have attracted wide attention in both theory and application this paper provides a novel method for attribute reduction based on covering rough sets we review the concepts of consistent and inconsistent covering decision systems and their reducts and we develop a judgment theorem and a discernibility matrix for each type of covering decision system furthermore we present some basic structural properties of attribute reduction with covering rough sets based on a discernibility matrix we develop a heuristic algorithm to find a subset of attributes that approximate a minimal reduct finally the experimental results for uci data sets show that the proposed reduction approach is an effective technique for addressing numerical and categorical data and is more efficient than the method presented in the paper d g chen c z wang q h hu a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets information sciences 177 17 2007 3500 3518

electre tri nb a new multiple criteria ordinal classification method abstract this paper presents a new method for multiple criteria ordinal classification sorting problems in an ordinal classification problem the categories also called classes in related literature are pre defined and ordered from the best to the worst or from the worst to the best the actions not necessarily known a priori are assigned to the different ordered categories several electre type methods were designed to deal with this kind of problem however none of them proposes a characterization of the categories through a set of limiting profiles this is the novelty of the current method which may be considered as an extension of electre tri b it fulfills a set of structural requirements uniqueness of the assignments independence monotonicity homogeneity conformity and stability with respect to merging and splitting operations all these features will be presented in the current paper and illustrated through three examples

sequential covering rule induction algorithm for variable consistency rough set approaches we present a general rule induction algorithm based on sequential covering suitable for variable consistency rough set approaches this algorithm called vc domlem can be used for both ordered and non ordered data in the case of ordered data the rough set model employs dominance relation and in the case of non ordered data it employs indiscernibility relation vc domlem generates a minimal set of decision rules these rules are characterized by a satisfactory value of the chosen consistency measure we analyze properties of induced decision rules and discuss conditions of correct rule induction moreover we show how to improve rule induction efficiency due to application of consistency measures with desirable monotonicity properties

on projected stochastic gradient descent algorithm with weighted averaging for least squares regression the problem of least squares regression of a d dimensional unknown parameter is considered a stochastic gradient descent based algorithm with weighted iterate averaging that uses a single pass over the data is studied and its convergence rate is analyzed we first consider a bounded constraint set of the unknown parameter under some standard regularity assumptions we provide an explicit o 1 k upper bound on the convergence rate depending on the variance due to the additive noise in the measurements and the size of the constraint set we show that the variance term dominates the error and decreases with rate 1 k while the term that is related to the size of the constraint set decreases with rate log k k 2 we then compare the asymptotic ratio rho between the convergence rate of the proposed scheme and the empirical risk minimizer erm as the number of iterations approaches infinity we show that rho leq 4 for all d geq 1 when the random entries of the sensing vector are uncorrelated and identically distributed we further improve the upper bound by showing that rho leq 4 3 for the case of d 1 and unbounded parameter set when the random sensing entries are equal across time simulation results demonstrate strong performance of the algorithm as compared to existing methods and coincide with rho leq 4 3 even for large d in practice

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

an incremental gradient projection method with momentum term and adaptive stepsize rule we consider an incremental gradient method with momentum term for minimizing the sum of continuously differentiable functions this method uses a new adaptive stepsize rule that decreases the stepsize whenever sufficient progress is not made we show that if the gradients of the functions are bounded and lipschitz continuous over a certain level set then every cluster point of the iterates generated by the method is a stationary point in addition if the gradient of the functions have a certain growth property then the method is either linearly convergent in some sense or the stepsizes are bounded away from zero the new stepsize rule is much in the spirit of heuristic learning rules used in practice for training neural networks via backpropagation as such the new stepsize rule may suggest improvements on existing learning rules finally extension of the method and the convergence results to constrained minimization is discussed as are some implementation issues and numerical experience

rule learning for classification based on neighborhood covering reduction rough set theory has been extensively discussed in the domain of machine learning and data mining pawlak s rough set theory offers a formal theoretical framework for attribute reduction and rule learning from nominal data however this model is not applicable to numerical data which widely exist in real world applications in this work we extend this framework to numerical feature spaces by replacing partition of universe with neighborhood covering and derive a neighborhood covering reduction based approach to extracting rules from numerical data we first analyze the definition of covering reduction and point out its advantages and disadvantages then we introduce the definition of relative covering reduction and develop an algorithm to compute it given a feature space we compute the neighborhood of each sample and form a neighborhood covering of the universe and then employ the algorithm of relative covering reduction to the neighborhood covering thus derive a minimal covering rule set some numerical experiments are presented to show the effectiveness of the proposed technique

evaluation functions and decision conditions of three way decisions with game theoretic rough sets abstract three way decisions have been used over the years in many application areas the rough sets and its extensions provide useful approaches for three way decisions typically these approaches employ a single evaluation function or criterion to induce three way decisions when extending the rough set based three way decisions to multiple criteria decision making mcdm two issues are encountered the first issue is related to the construction and definition of aggregation mechanisms for dealing with differences in results of evaluation functions the second issue is related to the setting of choice structure for selecting the three types of decision choices in this article we consider the role and use of game theoretic rough set gtrs model to resolve and address these two issues the issue related to differences in evaluation functions is addressed with gtrs by implementing a game that considers multiple evaluation functions as game players the game theoretic analysis in the gtrs is employed to resolve the differences by determining a tradeoff between evaluation functions the issue related to choice structure is addressed by considering the conditions under which different game outcomes could constitute a game solution in particular the equilibrium analysis within games is used to construct the rules for three way decisions a demonstrative example is used to explain the use of the proposed approach the relationship between the proposed approach and the probabilistic rough sets is also discussed

some asymptotic results for learning in single hidden layer feedforward network models abstract we investigate the properties of a recursive estimation procedure the method of back propagation for a class of nonlinear regression models single hidden layer feedforward network models recently developed by cognitive scientists the results follow from more general results for a class of recursive m estimators obtained using theorems of ljung 1977 and walk 1977 for the method of stochastic approximation conditions are given ensuring that the back propagation estimator converges almost surely to a parameter value that locally minimizes expected squared error loss provided the estimator does not diverge and that the back propagation estimator is asymptotically normal when centered at this minimizer this estimator is shown to be statistically inefficient and a two step procedure that has efficiency equivalent to that of nonlinear least squares is proposed practical issues are illustrated by a numerical example involving approximation of the henon map

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac

combining ordered subsets and momentum for accelerated x ray ct image reconstruction statistical x ray computed tomography ct reconstruction can improve image quality from reduced dose scans but requires very long computation time ordered subsets os methods have been widely used for research in x ray ct statistical image reconstruction and are used in clinical pet and spect reconstruction in particular os methods based on separable quadratic surrogates os sqs are massively parallelizable and are well suited to modern computing architectures but the number of iterations required for convergence should be reduced for better practical use this paper introduces os sqs momentum algorithms that combine nesterov s momentum techniques with os sqs methods greatly improving convergence speed in early iterations if the number of subsets is too large the os sqs momentum methods can be unstable so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior experiments with simulated and real 3d ct scan data illustrate the performance of the proposed algorithms

attribute reduction in ordered decision tables via evidence theory rough set theory and dempster shafer theory of evidence are two distinct but closely related approaches to modeling and manipulating uncertain information it is quite natural to set up a hybrid model based on these two theories in this paper we investigate the problem of attribute reduction for ordered decision tables based on evidence theory belief and plausibility functions which are strongly connected with lower and upper approximation operators in dominance based rough set approach are proposed to define relative belief and plausibility reducts of ordered decision tables relationships among various types of relative reducts are thoroughly studied in consistent and inconsistent ordered decision tables a pair of numeric measures the inner and outer significance measures of a criterion is presented to search for a relative belief plausibility reduct which is meaningful for practical problems some real world tasks taken from the uci repository are employed to verify the feasibility and effectiveness of the proposed technique

a moment and sum of squares extension of dual dynamic programming with application to nonlinear energy storage problems abstract we present a finite horizon optimization algorithm that extends the established concept of dual dynamic programming ddp in two ways first in contrast to the linear costs dynamics and constraints of standard ddp we consider problems in which all of these can be polynomial functions second we allow the state trajectory to be described by probability distributions rather than point values and return approximate value functions fitted to these the algorithm is in part an adaptation of sum of squares techniques used in the approximate dynamic programming literature it alternates between a forward simulation through the horizon in which the moments of the state distribution are propagated through a succession of single stage problems and a backward recursion in which a new polynomial function is derived for each stage using the moments of the state as fixed data the value function approximation returned for a given stage is the point wise maximum of all polynomials derived for that stage this contrasts with the piecewise affine functions derived in conventional ddp we prove key convergence properties of the new algorithm and validate it in simulation on two case studies related to the optimal operation of energy storage devices with nonlinear characteristics the first is a small borehole storage problem for which multiple value function approximations can be compared the second is a larger problem for which conventional discretized dynamic programming is intractable

a multiple criteria sorting method where each category is characterized by several reference actions the electre tri nc method this paper presents electre tri nc a new sorting method which takes into account several reference actions for characterizing each category this new method gives a particular freedom to the decision maker in the co construction decision aiding process with the analyst to characterize the set of categories while there is no constraint for introducing only one reference action as typical of each category like in electre tri c almeida dias et al 2010 as in such a sorting method this new sorting method is composed of two joint rules electre tri nc also fulfills a certain number of natural requirements additional results on the behavior of the new method are also provided in this paper namely the ones with respect to the addition or removal of the reference actions used for characterizing a certain category a numerical example illustrates the manner in which electre tri nc can be used by a decision maker a comparison with some related sorting procedures is presented and it allows to conclude that the new method is appropriate to deal with sorting problems

robustness analysis for decision under uncertainty with rule based preference model we introduce robust ordinal regression to decision under uncertainty we propose an integrated framework for robustness analysis with the rule based preference model we formulate the procedures for deriving a univocal classification of acts we account for different types of indirect preference information we consider group decision under uncertainty with dominance based rough set approach we consider decision under uncertainty as a multi attribute classification problem where a set of acts is described by outcomes gained with given probabilities the decision maker dm provides desired classification for a small subset of reference acts such preference information is structured using dominance based rough set approach drsa and the resulting lower approximations of the quality class unions are used as an input for construction of an aggregate preference model we induce all minimal cover sets of rules being compatible with the non ambiguous assignment examples and satisfying some additional requirements that may be imposed by the dm applying such compatible instances of the preference model on a set of all acts we draw conclusions about the certainty of recommendation assured by different minimal cover sets of rules in particular we analyze the diversity of class assignments assignment based preference relations and class cardinalities then we solve an optimization problem to get a univocal precise classification for all acts taking into account the robustness concern this optimization problem admits incorporation of additional indirect and imprecise preferences in form of desired class cardinalities and assignment based pairwise comparisons finally we extend the proposed approach to group decision under uncertainty we present a set of indicators and outcomes giving an insight into the spaces of consensus and disagreement between the dms

a new approach to multi criteria sorting based on fuzzy outranking relations the theseus method abstract in this paper we propose the theseus method a new approach based on fuzzy outranking relations to multi criteria sorting problems compared with other outranking based methods theseus is inspired by another view of multi criteria classification problems it utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined this way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects the appropriate assignment is determined by solving a simple selection problem the capacity of a reference set for making appropriate assignments is related to a good characterization of the categories a single reference action characterizing a category may be insufficient to achieve well determined assignments in this paper the reference set capacity to perform appropriate assignments is characterized by some new concepts this capacity may be increased when more objects are added to the reference set theseus is a method for handling the preference information contained in such larger reference sets

variable consistency dominance based rough set approach to preference learning in multicriteria ranking abstract we present a methodology for non statistical preference learning in multicriteria ranking based on variable consistency dominance based rough set approach vc drsa a finite set of objects to be ranked is evaluated by a set of criteria which are real valued functions with ordinal or cardinal scales given the statement of a multicriteria ranking problem the only objective information one can get is the dominance relation over the set of objects the dominance relation is however too poor because it leaves many objects incomparable to enrich this relation and make the objects more comparable a decision maker dm must supply some preference information revealing her his value system with respect to multicriteria evaluations we are considering a frequent case when the preference information has the form of pairwise comparisons of some objects relatively well known to the dm called reference objects this information is thus composed of some decision examples on the reference objects it is the input data for a method that learns the preferences of the dm since this information is prone to inconsistencies we propose to structure it using vc drsa then the pairs of objects that are sufficiently consistent serve as a basis for induction of a preference model this model has the form of a set of if then decision rules application of these rules on the whole set of objects to be ranked yields a fuzzy preference structure directed weighted graph this preference structure is then exploited using a ranking method so as to work out a final recommendation we propose a list of properties that helps to choose a proper ranking method the methodology is illustrated by an example

rough data deduction based on the upper approximation this paper describes how to construct a structure called a rough deduction space it is an extension of an approximation space and incorporates a deduction relation related to data connections in the rough deduction space a notion of data deduction is introduced and is referred to as rough data deduction based on integrated information of both the upper approximation and the deduction relation rough data deduction accomplishes deductions from data to data which is different from any logical deduction in mathematical logic research on rough data deduction covers two activities the rough data deduction with respect to an equivalence relation and rough data deductions with respect to different equivalence relations the activities also concern the relationship between rough data deduction and rough relations that are rough representations of the deduction relation this leads to properties involving approximate features implied by rough data deduction and reflecting the characteristic that rough data deduction can describe rough data connections in particular since the research correlates closely to granules it may offer an avenue of research on granular computing as an example a specific problem is modeled by a rough deduction space the rough data connections in the problem are described by use of rough data deduction

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

fuzzy information systems and their homomorphisms abstract with the arrival of the information age information acquisition and communication have become more and more important in the field of information technology this paper uses the concept of homomorphism as a basic tool to study the communication between fuzzy information systems the concepts of consistent and compatible mappings with respect to fuzzy sets are firstly defined and their basic properties are studied then a pair of lower and upper rough fuzzy approximation operators is constructed by means of the concept of fuzzy mappings basic invariant properties of the approximation operators are investigated finally the concepts of fuzzy information system and its homomorphism are introduced and some invariant properties of fuzzy information systems under homomorphisms are examined it is proved that the attribute reductions of an original information system and its image system are equivalent to each other in the context of fuzzy attributes these results may have potential applications in attribute reduction and classification issues

induction of ordinal classification rules from decision tables with unknown monotonicity abstract we are considering induction of ordinal classification rules which assign objects to preference ordered decision classes within the dominance based rough set approach in order to extract such rules it is necessary to define dominance inconsistencies with respect to a set of condition attributes containing at least one ordinal condition attribute furthermore it is also assumed that we know if there exist increasing or decreasing monotonicity relationships between the values of ordinal condition and decision attributes very often however this information is unknown a priori one solution to this issue is to transform the ordinal condition attributes with unknown directions of preference to pairs of attributes with supposed inverse monotonic relationships both local and global monotonicity relationships can be represented by decision rules induced from transformed decision tables however in some cases transforming a decision table in this way is overcomplex in this paper we propose the inconsistency rates based on dominance and fuzzy preference relations that have the capacity of discovering monotonic relationships directly from data rather than induced decision rules moreover we propose a refined transformation method by introducing an additional monotonicity checking using these inconsistency rates to determine whether an ordinal condition attribute should be cloned or not experiments are also provided to evaluate the usefulness of the refined transformation method

dominance based rough set approach for group decisions the objective of this paper is to propose an approach to support group multicriteria classification the approach is composed of three phases the first phase exploits the knowledge provided by each decision maker to individually approximate the decision classes using rough approximation the second phase seeks to combine the outputs of individual approximation phase into a collective decision table by using an appropriate aggregation procedure the third phase uses the collective decision table in order to infer a set of collective decision rules which synthesize the judgements and perspectives of the different decision makers and to permit the classification of all decision objects the proposed approach relies on the dominance based rough set approach drsa which is used at two different levels first the drsa is used during the first phase to approximate the input data relative to each decision maker second the drsa is used during the third phase to approximate the collective decision table and generate the collective decision rules this paper presents the theoretical foundation of the proposed approach three case studies using real world data and a comparative study of recent similar proposals

rough approximation of a preference relation by dominance relations an original methodology for using rough sets to preference modeling in multi criteria decision problems is presented this methodology operates on a pairwise comparison table pct including pairs of actions described by graded preference relations on particular criteria and by a comprehensive preference relation it builds up a rough approximation of a preference relation by graded dominance relations decision rules derived from the rough approximation of a preference relation can be used to obtain a recommendation in multi criteria choice and ranking problems the methodology is illustrated by an example of multi criteria programming of water supply systems

a convergent incremental gradient method with a constant step size an incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented the method requires a single gradient evaluation per iteration and uses a constant step size for the case that the gradient is bounded and lipschitz continuous we show that the method visits infinitely often regions in which the gradient is small under certain unimodality assumptions global convergence is established in the quadratic case a global linear rate of convergence is shown the method is applied to distributed optimization problems arising in wireless sensor networks and numerical experiments compare the new method with other incremental gradient methods

rough sets methodology for sorting problems in presence of multiple attributes and criteria abstract we consider a sorting classification problem in the presence of multiple attributes and criteria called the mac however the domains of attributes are not preference ordered while the domains of criteria scales are totally ordered by preference relations among the attributes we distinguish between qualitative attributes and quantitative attributes in order to construct a comprehensive preference model that could be used to support the sorting task we consider preferential information of the decision maker dm in the form of assignment examples i e exemplary assignments of some reference actions to the decision classes the preference model inferred from these examples is a set of if then decision rules the rules are derived from rough approximations of decision classes made up of reference actions they satisfy conditions of completeness and dominance and manage with possible ambiguity inconsistencies in the set of examples our idea of rough approximations involves three relations together indiscernibility similarity and dominance defined on qualitative and quantitative attributes and on criteria respectively the usefulness of this approach is illustrated by an example

a rough margin based support vector machine by introducing the rough set theory into the support vector machine svm a rough margin based svm rmsvm is proposed to deal with the overfitting problem due to outliers similar to the classical svm the rmsvm searches for the separating hyper plane that maximizes the rough margin defined by the lower and upper margin in this way more data points are adaptively considered rather than the few extreme value points used in the classical svm in addition different support vectors may have different effects on the learning of the separating hyper plane depending on their positions in the rough margin points in the lower margin have more effects than those in the boundary of the rough margin from experimental results on six benchmark datasets the classification accuracy of this algorithm is improved without additional computational expense compared with the classical n svm

new variants of pairwise classification abstract pairwise classification is the technique that deals with multi class problems by converting them into a series of binary problems one for each pair of classes typically k class classification rules tend to be easier to learn for k 2 than for k 2 only one decision boundary requires attention this paper presents new methods for obtaining class membership probability estimates for multi class classification problems by coupling the probability estimates created by binary classifiers classifiers used include linear bayes normal classifier parzen density based classifier naive bayes classifier binary decision tree classifier and random neural net classifier the accuracy of new pairwise classifiers is examined on some real data sets the classification errors were estimated by stratified version of 10 fold cross validation technique i e the training examples were partitioned into 10 equal sized blocks with similar class distributions as in the original set the validation technique was repeated 10 times for each data set

multicriteria decision support using rules that represent rough graded preference relations the approach described in this paper aims to support multicriteria choice and ranking of actions when the input preference information acquired from the decision maker is a graded comprehensive pairwise comparison or ranking of reference actions it is based on decision rule preference model induced from a rough approximation of the graded comprehensive preference relation among the reference actions the set of decision rules applied to a new set of actions provides a graded fuzzy preference relation which can be exploited by weighted fuzzy net flow score or lexicographic fuzzy net flow score procedure to obtain a final recommendation in terms of the best choice or of the ranking

mining incomplete data with singleton subset and concept probabilistic approximations abstract rough set theory provides a very useful idea of lower and upper approximations for inconsistent data for incomplete data these approximations are not unique in this paper we investigate properties of three well known generalizations of approximations singleton subset and concept these approximations were recently further generalized as to include an additional parameter interpreted as a probability in this paper we report novel properties of singleton subset and concept probabilistic approximations additionally we validated such approximations experimentally our main objective was to test which of the singleton subset and concept probabilistic approximations are the most useful for data mining our conclusion is that for a given incomplete data set all three approaches should be applied and the best approach should be selected as a result of ten fold cross validation finally we conducted experiments on complexity of rule sets and the total number of singleton subset and concept approximations

related family a new method for attribute reduction of covering information systems in terms of attribute reduction of covering based rough sets the discernibility matrix is used as a conventional method to compute all attribute reducts however it is inapplicable to attribute reduction in certain circumstances in this article a new method referred to as the related family is introduced to compute all attribute reducts and relative attribute reducts for covering rough sets its core idea is to remove superfluous attributes while keeping the approximation space of covering information system unchanged the related family method is more powerful than the discernibility matrix method since the former can handle complicated cases that could not be handled by the latter in addition a simplified version of the related family and its corresponding heuristic algorithm are also presented

multicriteria sorting using a valued indifference relation under a preference disaggregation paradigm abstract most multicriteria decision methods need the definition of a significant amount of preferential information from a decision agent the preference disaggregation analysis paradigm infers the model s parameter values from holistic judgments provided by a decision agent here a new method for inferring the parameters of a fuzzy outranking model for multicriteria sorting is proposed this approach allows us to use most of the preferential information contained in a reference set the central idea is to characterize the quality of the model by measuring discrepancies and concordances amongst i the preference relations derived from the outranking model and ii the preferential information contained in the reference set the model s parameters are inferred from a multiobjective optimization problem according to some additional preferential information from a decision agent once the model has been fitted sorting decisions about new objects are performed by using a fuzzy indifference relation this proposal performs very well in some examples

on the convergence of algorithms with implications for stochastic and nondifferentiable optimization studies of the convergence of algorithms often revolve around the existence of a function with respect to which monotonic descent is required in this paper we show that under relatively lenient conditions stage dependent descent not necessarily monotonic is sufficient to guarantee convergence this development also provides the impetus to examine optimization algorithms we show that one of the important avenues in the study of convergence namely the theory of epi convergence imposes stronger conditions than are necessary to establish the convergence of an optimization algorithm working from a relaxation of epi convergence we introduce the notion of compatibility and prove several results that permit relaxations of conditions imposed by previous approaches to algorithmic convergence finally to illustrate the usefulness of the concepts we combine stage dependent descent with results derivable from compatibility to provide a basis for the convergence of a general algorithmic statement that might be used for stochastic and nondifferentiable optimization

stochastic primal dual hybrid gradient algorithm with arbitrary sampling and imaging applications we propose a stochastic extension of the primal dual hybrid gradient algorithm studied by chambolle and pock in 2011 to solve saddle point problems that are separable in the dual variable the anal

incremental rough set approach for hierarchical multicriteria classification abstract multicriteria classification refers to classify objects evaluated by a set of criteria to preference ordered decision classes dominance based rough set approach has been successfully introduced to express and reason inconsistencies with a dominance principle in multicriteria classification problems hierarchical attribute values exist extensively within many real world applications which provide a hierarchical form to organize view and analyze data from different perspectives for accommodating the preference variability in this study we consider an extension of dominance based rough set approach by applying an incremental learning technique for hierarchical multicriteria classification while attribute values dynamically vary across different levels of granulations we formalize the dynamic characteristics of knowledge granules with the cut refinement and coarsening through attribute value taxonomies in the hierarchical multicriteria decision systems in consequence incremental algorithms for computing dominance based rough approximations of preference ordered decision classes are developed by applying the resulted prior knowledge as the input and only recomputing those outputs which depend on the changed attribute values this paper presents the theoretical foundation of the proposed approach example analysis and experimental evaluation are also provided for illustration of the feasibility and efficiency

robust ordinal regression for dominance based rough set approach to multiple criteria sorting we present a new multiple criteria sorting method deriving from dominance based rough set approach drsa the preference information supplied by the decision maker dm is a set of possibly imprecise and inconsistent assignment examples on a subset of reference alternatives relatively well known to the dm to structure the data we use drsa and subsequently represent the assignment examples by all minimal sets of rules covering all alternatives from the lower approximations of class unions such a set of rules is called minimal cover set it is one of the instances of the preference model compatible with dm s preference information in this way we implement the principle of robust ordinal regression ror to decision rule preference model for each alternative we derive the necessary and possible assignments specifying the range of classes to which the alternative is assigned by all or at least one compatible set of rules respectively as well as class acceptability indices we also introduce the notion of a representative compatible minimal cover set of rules whose selection builds on the results of ror addressing the robustness concern application of the approach is demonstrated by classifying 69 land zones in 4 classes representing different risk levels

axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough set decision rules abstract utility or value functions play an important role of preference models in multiple criteria decision making we investigate the relationships between these models and the decision rule preference model obtained from the dominance based rough set approach the relationships are established by means of special cancellation properties used in conjoint measurement as axioms for representation of aggregation procedures we are considering a general utility function and three of its important special cases associative operator sugeno integral and ordered weighted maximum for each of these aggregation functions we give a representation theorem establishing equivalence between a very weak cancellation property the specific utility function and a set of rough set decision rules each result is illustrated by a simple example of multiple criteria decision making the results show that the decision rule model we propose has clear advantages over a general utility function and its particular cases

a cutting plane method from analytic centers for stochastic programming the stochastic linear programming problem with recourse has a dual block angular structure it can thus be handled by benders decomposition or by kelley s method of cutting planes equivalently the dual problem has a primal block angular structure and can be handled by dantzig wolfe decomposition the two approaches are in fact identical by duality here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations the only significant difference form the aforementioned methods is that new cutting planes or columns by duality will be generated not from the optimum of the linear programming relaxation but from the analytic center of the set of localization

the decomposition principle and algorithms for linear programming abstract the computational difficulties that continue to plague decomposition algorithms namely long tail convergence and numerical instabilities have served to dampen enthusiasm about their computational effectiveness the use of interior points of subproblems in decomposition procedures may have a significant role to play in alleviating such computational difficulties indeed dantzig wolfe decomposition provides the arena within which simplex techniques for master problems and interior point techniques for subproblems complement one another in a useful way in combination they could lead to more effective decomposition algorithms than we have today we formulate a particular algorithm along these lines and illustrate its convergence and numerical characteristics through numerical experiments we make these experiments the basis for a discussion of the merits of using interior points in decomposition

an aggregated fuzzy naive bayes data classifier in this study an aggregated fuzzy naive bayes classifier is proposed for decision making problems where both linguistic and numerical information are available in the solution process of such problems all attributes are considered as fuzzy numbers and a procedure based on 2 tuple fuzzy linguistic representation model is generated for combining them this procedure and subsequent fuzzy naive bayes classification are performed based on arithmetic operations defined by chen s function principle the proposed method was demonstrated on 2 well known examples from the literature in which both numerical and linguistic attributes were considered the results show that the proposed aggregated fuzzy naive bayes classifier is notably efficient in decision making where the attributes are in more realistic forms

decomposition in general mathematical programming in this paper a unifying framework is presented for the generalization of the decomposition methods originally developed by benders 1962 and dantzig and wolfe 1960 these generalizations calledvariable decomposition andconstraint decomposition respectively are based on the general duality theory developed by tind and wolsey the framework presented is of a general nature since there are no restrictive conditions imposed on problem structure moreover inaccuracies and duality gaps that are encountered during computations are accounted for the two decomposition methods are proven not to cycle if certain fairly general conditions are met furthermore finite convergence can be ensured under the traditional finiteness conditions and asymptotic convergence can be guaranteed once certain continuity conditions are met the obvious symmetry between both types of decomposition methods is explained by establishing a duality relation between the two which extends a similar result in linear programming a remaining asymmetry in the asymptotic convergence results is argued to be a direct consequence of a fundamental asymmetry that resides in the tind wolsey duality theory it can be shown that in case the latter asymmetry disappears the former does too other decomposition techniques such as lagrangean decomposition and cross decomposition turn out to be captured by the general framework presented here as well

incremental least squares methods and the extended kalman filter in this paper we propose and analyze nonlinear least squares methods which process the data incrementally one data block at a time such methods are well suited for large data sets and real time operation and have received much attention in the context of neural network training problems we focus on the extended kalman filter which may be viewed as an incremental version of the gauss newton method we provide a nonstochastic analysis of its convergence properties and we discuss variants aimed at accelerating its convergence

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

robust inversion dimensionality reduction and randomized sampling we consider a class of inverse problems in which the forward model is the solution operator to linear odes or pdes this class admits several dimensionality reduction techniques based on data averaging or sampling which are especially useful for large scale problems we survey these approaches and their connection to stochastic optimization the data averaging approach is only viable however for a least squares misfit which is sensitive to outliers in the data and artifacts unexplained by the forward model this motivates us to propose a robust formulation based on the student s t distribution of the error we demonstrate how the corresponding penalty function together with the sampling approach can obtain good results for a large scale seismic inverse problem with 50 corrupted data

multi criteria classification a new scheme for application of dominance based decision rules abstract we are considering the problem of multi criteria classification in this problem a set of if then decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes given a sample of classification examples called learning data set the rules are induced from dominance based rough approximations of preference ordered decision classes according to the variable consistency dominance based rough set approach vc drsa the main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by i no rule ii exactly one rule iii several rules the proposed classification scheme can be applied to both learning data set to restore the classification known from examples and testing data set to predict classification of new objects a hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals

incremental proximal methods for large scale convex optimization we consider the minimization of a sum sum i 1 mf i x consisting of a large number of convex component functions f i for this problem incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective we propose new incremental methods consisting of proximal iterations applied to single components as well as combinations of gradient subgradient and proximal iterations we provide a convergence and rate of convergence analysis of a variety of such methods including some that involve randomization in the selection of components we also discuss applications in a few contexts including signal processing and inference machine learning

inexact bundle methods for two stage stochastic programming stochastic programming problems arise in many practical situations in general the deterministic equivalents of these problems can be very large and may not be solvable directly by general purpose optimization approaches for the particular case of two stage stochastic programs we consider decomposition approaches akin to a regularized l shaped method that can handle inexactness in the subproblem solution from a nonsmooth optimization perspective these variants amount to applying a proximal bundle method to an oracle that gives inaccurate values for the objective function and a subgradient rather than forcing early termination of the subproblem optimization to define inexact oracles we select a small subset of scenarios for which the subproblem solution is exact and we replace the information for the remaining scenarios by a fast procedure that does not involve solving an optimization problem the inaccurate oracle information creates inexact cuts in the master program which are well handled by th

handling multicriteria preferences in cluster analysis in the framework of multicriteria decision aid a lot of interest has been devoted to sorting problems in which the set of categories is pre defined besides preference oriented multicriteria clustering has received little attention usual geometric and related metrics are not well suited for this problem here we propose a clustering method based on a valued indifference relation inspired by outranking methods we suggest a method based on comparing cluster centers and an average net flow score of clusters to build a complete ranking of the set of clusters that is a way of defining a set of ordered categories for sorting purposes the new approach performs very well in some examples

a cost sensitive decision tree algorithm based on weighted class distribution with batch deleting attribute mechanism minimal cost classification is an important issue in data mining and machine learning recently many enhanced algorithms based on the c4 5 algorithm have been proposed to tackle this issue one disadvantage in these methods is that they are inefficient for medium or large data sets to overcome this problem we present a cost sensitive decision tree algorithm based on weighted class distribution with a batch deleting attribute mechanism bdadt in the bdadt algorithm a heuristic function is designed for evaluating attributes in node selection this contains a weighted information gain ratio a test cost and a user specified non positive parameter for adjusting the effect of the test cost meanwhile a batch deleting attribute mechanism is incorporated into our algorithm this mechanism deletes redundant attributes according to the values of the heuristic function in the process of assigning nodes to improve the efficiency of decision tree construction experiments are conducted on 20 uci data sets with representative test cost normal distribution to evaluate the proposed bdadt algorithm the experimental results show that the average total costs obtained by the proposed algorithm are smaller than the existing cs c4 5 and cs gainratio algorithms furthermore the proposed algorithm significantly increases the efficiency of cost sensitive decision tree construction

extended rough set based attribute reduction in inconsistent incomplete decision systems a systematic study of attribute reduction in inconsistent incomplete decision systems iidss has not yet been performed and no complete methodology of attribute reduction has been developed for iidss to date in an iids there are various ways to handle missing values in this paper a missing attribute value may be replaced with any known value of a corresponding attribute such a missing attribute value is called a do not care condition in this way this paper establishes reduction concepts specifically for iidss mainly by extending related reduction concepts from other types of decision systems into iidss and then derives their relationships and properties with these derived properties the extended reducts are divided into two distinct types heritable reducts and nonheritable reducts and algorithms for computing them are presented using the relationships derived here the eight types of extended reducts established for iidss can be converted to five equivalent types then five discernibility function based approaches are proposed each for a particular kind of reduct each approach can find all reducts of its associated type the theoretical analysis of the proposed approaches is described in detail finally numerical experiments have shown that the proposed approaches are effective and suitable for handling both numerical and categorical attributes but that they have different application conditions the proposed approaches can provide a solution to the reduction problem for iidss

reducts and constructs in classic and dominance based rough sets approach abstract the idea of the reduct as defined in the classic rough sets approach crsa has proven to be inspiring enough to get into closely related theories including the dominance based rough sets approach drsa the procedure of reduction is generally similar to that of feature selection but narrower as it is the descriptive rather than the predictive aspect of data exploration that constitutes its principal goal crsa reducts are thus defined as minimal subsets of attributes that retain sufficiently high quality of object description developed within crsa the crsa reducts have given rise to the generalized notion of crsa constructs which have turned out to be superior to reducts in numerous practical experiments with real life data sets the generalization process is continued in this paper in which a definition of constructs in the context of drsa is introduced the definition fully analogous to that of crsa constructs differs only in that it is context based in drsa while context free in crsa consequently the presented drsa constructs are expected to have analogous properties to that of crsa constructs including superiority to drsa reducts in experiments with real life data sets

on the relation between rough set reducts and jumping emerging patterns abstract this paper presents the relations between rough set reducts and jumping emerging patterns observations are introduced formally and supported by brief examples furthermore we propose practical applications of these observations to the minimal reduct problem and to testing whether a given attribute set is differentiating we believe that our study can be expanded so as to include other types of reducts and emerging patterns

dominance based rough fuzzy set approach and its application to rule induction abstract theories of fuzzy sets and rough sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness respectively marrying both leads to the hybrid notion of rough fuzzy sets in order to get a more accurate account of imperfect information in this paper our attention is paid to ordered fuzzy decision systems where condition criteria are preference ordered and decision classes are not only ordered but also fuzzy first the dominance based rough fuzzy approximations of an upward or downward cumulated fuzzy set are introduced in ordered fuzzy decision systems second lower and upper reducts relative to a certain cumulated fuzzy set are proposed to eliminate redundant criteria in the system then two approaches to attribute reduction are presented based on the discernibility matrix and the heuristic strategy respectively also decision rules are extracted directly from these approximations and some applicable and simplified decision rules are obtained according to requirements of decision makers finally a case study in bankruptcy risk analysis is used to illustrate the mechanism of the proposed methods

dominance based rough set approach to incomplete ordered information systems dominance based rough set approach has attracted much attention in practical applications ever since its inception this theory has greatly promoted the research of multi criteria decision making problems involving preferential information this paper mainly deals with approaches to attribute reduction in incomplete ordered information systems in which some attribute values may be lost or absent by introducing a new kind of dominance relation named the characteristic based dominance relation to incomplete ordered information systems we expand the potential applications of dominance based rough set approach to eliminate information that is not essential attribute reduction in the sense of reducing attributes is needed an approach on the basis of the discernibility matrix and the discernibility function to computing all relative reducts is investigated in incomplete ordered information systems consistent incomplete ordered decision tables to reduce the computational burden a heuristic algorithm with polynomial time complexity for finding a unique relative reduct is designed by using the inner and outer significance measures of each criterion candidate moreover some numerical experiments are employed to verify the feasibility and effectiveness of the proposed algorithms

rough sets and intelligent data analysis rough set based data analysis starts from a data table called an information system the information system contains data about objects of interest characterized in terms of some attributes often we distinguish in the information system condition and decision attributes such information system is called a decision table the decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table with every decision table a set of decision rules called a decision algorithm can be associated it is shown that every decision algorithm reveals some well known probabilistic properties in particular it satisfies the total probability theorem and the bayes theorem these properties give a new method of drawing conclusions from data without referring to prior and posterior probabilities inherently associated with bayesian reasoning

convergence property of gradient type methods with non monotone line search in the presence of perturbations in this paper two new kinds of methods which are called gradient type method and hybrid projection method with perturbations are proposed and non monotone line search technique is employed at the same time global convergence of these methods is proved only in the case where the gradient function is uniformly continuous on an open convex set containing the iteration sequence numerical examples are given at the end of this paper

a stochastic successive minimization method for nonsmooth nonconvex optimization with applications to transceiver design in wireless communication networks consider the problem of minimizing the expected value of a cost function parameterized by a random variable the classical sample average approximation method for solving this problem requires minimization of an ensemble average of the objective at each step which can be expensive in this paper we propose a stochastic successive upper bound minimization method ssum which minimizes an approximate ensemble average at each iteration to ensure convergence and to facilitate computation we require the approximate ensemble average to be a locally tight upper bound of the expected cost function and be easily optimized the main contributions of this work include the development and analysis of the ssum method as well as its applications in linear transceiver design for wireless communication networks and online dictionary learning moreover using the ssum framework we extend the classical stochastic sub gradient method to the case of minimizing a nonsmooth nonconvex objective function and establish its convergence

a cutting plane approach for large scale capacitated multi period facility location using a specialized interior point method we propose a cutting plane approach namely benders decomposition for a class of capacitated multi period facility location problems the novelty of this approach lies on the use of a specialized interior point method for solving the benders subproblems the primal block angular structure of the resulting linear optimization problems is exploited by the interior point method allowing the either exact or inexact efficient solution of large instances the consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting plane method the methodology proposed allowed the solution of instances of up to 200 potential locations one million customers and three periods resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables those problems were solved by the specialized approach in less than one hour and a half outperforming other state of the art methods which exhausted the 144 gb of available memory in the largest instances

on rough set and fuzzy sublattice let l be a lattice with the least element 0 and the greatest element 1 and let be a full congruence relation on l in this paper the notion of upper and lower approximations of a fuzzy subset of l is introduced and some important properties will be studied

warm start of the primal dual method applied in the cutting plane scheme a practical warm start procedure is described for the infeasible primal dual interior point method ipm employed to solve the restricted master problem within the cutting plane method in contrast to the theoretical developments in this field the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow moreover it treats systematically the case when a large number of cuts are added at one time the technique proposed in this paper has been implemented in the context of hopdm the state of the art yet public domain interior point code numerical results confirm a high degree of efficiency of this approach regardless of the number of cuts added at one time can be thousands in the largest examples and regardless of the depth of the new cuts reoptimizations are usually done with a few additional iterations

fuzzy preference based rough sets preference analysis is an important task in multi criteria decision making the rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations the existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria in this paper we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria fuzzy preference relations are incorporated into a fuzzy rough set model which leads to a fuzzy preference based rough set model the measure of attribute dependency of the pawlak s rough set model is generalized to compute the relevance between criteria and decisions the definitions of upward dependency downward dependency and global dependency are introduced algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets

divide to conquer decomposition methods for energy optimization modern electricity systems provide a plethora of challenging issues in optimization the increasing penetration of low carbon renewable sources of energy introduces uncertainty in problems traditionally modeled in a deterministic setting the liberalization of the electricity sector brought the need of designing sound markets ensuring capacity investments while properly reflecting strategic interactions in all these problems hedging risk possibly in a dynamic manner is also a concern the fact of representing uncertainty and or competition of different companies in a multi settlement power market considerably increases the number of variables and constraints for this reason usually a trade off needs to be found between modeling and numerical tractability the more details are brought into the model the harder becomes the optimization problem for structured optimization and generalized equilibrium problems we explore some variants of solution methods based on lagrangian relaxation and on benders decomposition throughout we keep as a leading thread the actual practical value of such techniques in terms of their efficiency to solve energy related problems

multicoordination methods for solving convex block angular programs several decomposition methods are considered for solving block angular programs baps in parallel we present a computational comparison of synchronous multicoordination methods the most efficient of these approaches is shown to involve an intermediate number of blocks in the coordination phase

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

inexact subgradient methods with applications in stochastic programming in many instances the exact evaluation of an objective function and its subgradients can be computationally demanding by way of example we cite problems that arise within the context of stochastic optimization where the objective function is typically defined via multi dimensional integration in this paper we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods we refer to this new class of methods as inexact subgradient algorithms with relatively mild conditions imposed on the approximations we show that the inexact subgradient algorithms inherit properties associated with their traditional i e exact counterparts within the context of stochastic optimization the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi gradient methods additionally we study methods in which steplengths may be defined adaptively in a manner that reflects the improvement in the objective function approximations as the iterations proceed we illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs

approximate distribution reducts in inconsistent interval valued ordered decision tables abstract many methods based on the rough set theory to deal with information systems have been proposed in recent decades in practice some information systems are based on dominance relations and may be inconsistent because of various factors moreover taking the imprecise evaluations and assignments in the description of objects into account single valued information systems have been generalized to interval valued information systems in this paper by introducing a dominance relation to interval valued ordered information systems we establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation to extract the minimal decision rules approximate distribution reducts are proposed in inconsistent interval valued ordered decision tables this paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct and two equivalent definitions of approximate distribution reducts are also introduced in addition numerical examples are employed to examine the validity of the approaches proposed in this paper

approximations and uncertainty measures in incomplete information systems there are mainly two methodologies dealing with uncertainty measurement issue in rough set theory pure rough set approach and information theory approach pure rough set approach is based on the concepts of accuracy roughness and approximation accuracy proposed by pawlak information theory approach is based on shannon s entropy or its variants several authors have extended the information theory approach into incomplete information systems however there are few studies on extending the pure rough set approach to incomplete information systems this paper focuses on constructing uncertainty measures in incomplete information systems by pure rough set approach three types of definitions of lower and upper approximations and corresponding uncertainty measurement concepts including accuracy roughness and approximation accuracy are investigated theoretical analysis indicates that two of the three types can be used to evaluate the uncertainty in incomplete information systems experiments on incomplete real life data sets have been conducted to test the two selected types the first type and the third type of uncertainty measures results show that the two types of uncertainty measures are effective