a newton cg augmented lagrangian method for semidefinite programming we consider a newton cg augmented lagrangian method for solving semidefinite programming sdp problems from the perspective of approximate semismooth newton methods in order to analyze the rate of convergence of our proposed method we characterize the lipschitz continuity of the corresponding solution mapping at the origin for the inner problems we show that the positive definiteness of the generalized hessian of the objective function in these inner problems a key property for ensuring the efficiency of using an inexact semismooth newton cg method to solve the inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems numerical experiments on a variety of large scale sdp problems with the matrix dimension n up to 4 110 and the number of equality constraints m up to 2 156 544 show that the proposed method is very efficient we are also able to solve the sdp problem fap36 with n 4 110 and m 1 154 467 in the seventh dimacs implementation challenge much more accurately than in previous attempts

bounds for the quadratic assignment problem using the bundle method semidefinite programming sdp has recently turned out to be a very powerful tool for approximating some np hard problems the nature of the quadratic assignment problem qap suggests sdp as a way to derive tractable relaxations we recall some sdp relaxations of qap and solve them approximately using a dynamic version of the bundle method the computational results demonstrate the efficiency of the approach our bounds are currently among the strongest ones available for qap we investigate their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree

complexity study for the robust stable marriage problem abstract the robust stable marriage problem rsm is a variant of the classic stable marriage problem in which the robustness of a given stable matching is measured by the number of modifications required to find an alternative stable matching should some pairings break due to an unforeseen event we focus on the complexity of finding an a b supermatch an a b supermatch is defined as a stable matching in which if any a non fixed men women break up it is possible to find another stable matching by changing the partners of those a men women and the partners of at most b others we first discuss a model based on independent sets for finding 1 1 supermatches secondly in order to show that deciding whether or not there exists a 1 b supermatch is np complete we first introduce a sat formulation for which the decision problem is np complete by using schaefer s dichotomy theorem we then show the equivalence between this sat formulation and finding a 1 1 supermatch on a specific family of instances we also focus on studying the threshold between the cases in p and np complete for this problem

a proximal projection bundle method for lagrangian relaxation including semidefinite programming we give a proximal bundle method for minimizing a convex function f over a convex set c it requires evaluating f and its subgradients with a fixed but possibly unknown accuracy epsilon 0 each iteration involves solving an unconstrained proximal subproblem and projecting a certain point onto c the method asymptotically finds points that are epsilon optimal in lagrangian relaxation of convex programs it allows for e accurate solutions of lagrangian subproblems and finds e optimal primal solutions for semidefinite programming problems it extends the highly successful spectral bundle method to the case of inexact eigenvalue computations

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

type 2 fuzzistics for nonsymmetric interval type 2 fuzzy sets forward problems interval type 2 fuzzy sets it2 fs play a central role in fuzzy sets as models for words and in engineering applications of t2 fss these fuzzy sets are characterized by their footprints of uncertainty fou which in turn are characterized by their boundaries upper and lower membership functions mf the centroid of an it2 fs which is an it1 fs provides a measure of the uncertainty in the it2 fs the main purpose of this paper is to quantify the centroid of a non symmetric it2 fs with respect to geometric properties of its fou this is very important because interval data collected from subjects about words suggests that the fous of most words are non symmetrical using the results in this paper it is possible to formulate and solve forward problems i e to go from parametric non symmetric it2 fs models to data with associated uncertainty bounds we provide some solutions to such problems for non symmetrical triangular trapezoidal gaussian and shoulder fous

computational experience with a bundle approach for semidefinite cutting plane relaxations of max cut and equipartition we propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities our approach is based on lagrangian duality where the inequalities are dualized and only a basic set of constraints is maintained explicitly this leads to function evaluations requiring to solve a relatively simple semidefinite program our approach provides accurate solutions to semidefinite relaxations of the max cut and the equipartition problem which are not achievable by direct approaches based only on interior point methods

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

lattices of convex normal functions the algebra of truth values of type 2 fuzzy sets is the set of all functions from the unit interval into itself with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min this algebra has been studied rather extensively both from a theoretical and from a practical point of view it has a number of interesting subalgebras and this paper is about the subalgebra of all convex normal functions and closely related ones these particular algebras are de morgan algebras and our concern is principally with their completeness as lattices a special feature of our treatment is a representation of these algebras as monotone functions with pointwise order making the operations more intuitive

the complexity of counting stable marriages in an instance of size n of the stable marriage problem each of n men and n women ranks the members of the opposite sex in order of preference a stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching it is well known that at least one stable matching exists for every stable marriage instance so that the decision version of the problem always has a yes answer furthermore efficient algorithms are known for the determination of such a stable matching so that the search version of the problem is polynomially solvable however by exploring the structure of the set of stable matchings for any particular instance of the problem and exploiting its relationship with the set of antichains of an associated partially ordered set we prove that the enumeration version of the problem determining the number of stable matchings is p complete and therefore cannot be solved in polynomial time if

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

a cholesky dual method for proximal piecewise linear programming a quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term subject to simple bounds on variables it may be used for search direction finding in nondifferentiable optimization algorithms an efficient implementation is described that updates a cholesky factorization of active constraints and provides good accuracy via iterative refinement numerical experience is reported for some large problems

robustness and regularization of support vector machines we consider regularized support vector machines svms and show that they are precisely equivalent to a new robust optimization formulation we show that this equivalence of robust optimization and regularization has implications for both algorithms and analysis in terms of algorithms the equivalence suggests more general svm like algorithms for classification that explicitly build in protection to noise and at the same time control overfitting on the analysis front the equivalence of robustness and regularization provides a robust optimization interpretation for the success of regularized svms we use this new robustness interpretation of svms to give a new proof of consistency of kernelized svms thus establishing robustness as the reason regularized svms generalize well

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

convex normal functions revisited the lattice l u of upper semicontinuous convex normal functions with convolution ordering arises in studies of type 2 fuzzy sets in 2002 kawaguchi and miyakoshi extended t norms as logical connectives of fuzzy truth values multiple valued logic 8 1 2002 53 69 showed that this lattice is a complete heyting algebra later harding et al lattices of convex normal functions fuzzy sets and systems 159 2008 1061 1071 gave an improved description of this lattice and showed it was a continuous lattice in the sense of gierz et al a compendium of continuous lattices springer berlin 1980 in this note we show the lattice l u is isomorphic to the lattice of decreasing functions from the real unit interval 0 1 to the interval 0 2 under pointwise ordering modulo equivalence almost everywhere this allows development of further properties of l u it is shown that l u is completely distributive is a compact hausdorff topological lattice whose topology is induced by a metric and is self dual via a period two antiautomorphism we also show the lattice l u has another realization of natural interest in studies of type 2 fuzzy sets it is isomorphic to a quotient of the lattice l of all convex normal functions under the convolution ordering this quotient identifies two convex normal functions if they agree almost everywhere and their intervals of increase and decrease agree almost everywhere

an alternating linearization bundle method for convex optimization and nonlinear multicommodity flow problems we give a bundle method for minimizing the sum of two convex functions one of them being known only via an oracle of arbitrary accuracy each iteration involves solving two subproblems in which the functions are alternately represented by their linearizations our approach is motivated by applications to nonlinear multicommodity flow problems encouraging numerical experience on large scale problems is reported

answering an open question in fuzzy metric spaces abstract this paper answers affirmatively a problem posted by gregori et al in 2012 proving that the parametric distance involved in a stationary fuzzy metric space is uniformly continuous

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

characterizing idempotent nullnorms on a special class of bounded lattices abstract nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms this paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing only two distinct elements incomparable with a furthermore some basic properties for the bounded lattice containing only two distinct elements incomparable with a are presented

a proximal bundle method with approximate subgradient linearizations we give a proximal bundle method for minimizing a convex function f over a closed convex set it only requires evaluating f and its subgradients with an accuracy epsilon 0 which is fixed but possibly unknown it asymptotically finds points that are epsilon optimal when applied to lagrangian relaxation it allows for epsilon accurate solutions of lagrangian subproblems and finds epsilon optimal solutions of convex programs

numerical evaluation of sbmethod we report numerical results for sbmethod a publically available implementation of the spectral bundle method applied to the 7 th dimacs challenge test sets that are semidefinite relaxations of combinatorial optimization problems the performance of the code is heavily influenced by parameters that control bundle update and eigenvalue computation unfortunately no mathematically sound guidelines for setting them are known based on our experience with sbmethod we propose heuristics for dynamically updating the parameters as well as a heuristc for improving the starting point these are now the default settings of sbmethod version 1 1 we compare their performance on the dimacs instances to our previous best choices for version 1 0 sbmethod version 1 1 is also part of the independent dimacs benchmark by h mittelmann based on these results we try to analyze strengths and weaknesses of our approach in comparison to other codes for large scale semidefinite programming

join and meet operations for type 2 fuzzy sets with nonconvex secondary memberships in this paper we will present two theorems for the join and meet operations for general type 2 fuzzy sets with arbitrary secondary memberships which can be nonconvex and or nonnormal type 1 fuzzy sets these results will be used to derive the join and meet operations of the more general descriptions of interval type 2 fuzzy sets presented in a paper by bustince sola et al interval type 2 fuzzy sets are generalization of interval valued fuzzy sets towards a wider view on their relationship ieee trans fuzzy syst vol 23 pp 1876 1882 2015 where the secondary grades can be nonconvex hence this study will help to explore the potential of type 2 fuzzy logic systems which use the general forms of interval type 2 fuzzy sets which are not equivalent to interval valued fuzzy sets several examples for both general type 2 and the more general forms of interval type 2 fuzzy sets are presented

optimizing condition numbers in this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite n times n matrices we show that the condition number is a clarke regular strongly pseudoconvex function we prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program this asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

an inexact bundle variant suited to column generation we give a bundle method for constrained convex optimization instead of using penalty functions it shifts iterates towards feasibility by way of a slater point assumed to be known besides the method accepts an oracle delivering function and subgradient values with unknown accuracy our approach is motivated by a number of applications in column generation in which constraints are positively homogeneous so that zero is a natural slater point and an exact oracle may be time consuming finally our convergence analysis employs arguments which have been little used so far in the bundle community the method is illustrated on a number of cutting stock problems

the algebra of fuzzy truth values the purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type 2 fuzzy sets

piecewise quadratic approximations in convex numerical optimization we present a bundle method for convex nondifferentiable minimization where the model is a piecewise quadratic convex approximation of the objective function unlike standard bundle approaches the model only needs to support the objective function from below at a properly chosen small subset of points as opposed to everywhere we provide the convergence analysis for the algorithm with a general form of master problem which combines features of trust region stabilization and proximal stabilization taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information numerical results are also reported

an algorithm for nonsmooth convex minimization with errors a readily implementable algorithm is given for minimizing any convex not necessarily differentiable function f of several variables at each iteration the method requires only one approximate evaluation of f and its e subgradient and finds a search direction by solving a small quadratic programming problem the algorithm generates a minimizing sequence of points which converges to a solution wheneverf has any minimizers

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

a new look at type 2 fuzzy sets and type 2 fuzzy logic systems we propose the concept of a conditional fuzzy set and prove that a type 2 fuzzy set is equivalent to a conditional fuzzy set we show that both the conditional fuzzy sets and the type 2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other whereas in the type 2 fuzzy set framework the secondary variable depends on the primary variable by definition it is this dependence between the primary and secondary variables that makes the type 2 fuzzy sets a complex subject while the conditional fuzzy sets do not have this built in dependence and thus are much easier to analyze with the fuzzy relation formulation powerful tools in fuzzy set theory such as zadeh s compositional rule of inference can be used to obtain the marginal fuzzy sets of the type 2 and conditional fuzzy sets transforming the type 2 problems back to the conventional type 1 domain with the help of the marginal fuzzy set concept we show that a type 2 fuzzy logic system can be designed in the same way as designing a type 1 fuzzy logic system

weighted fuzzy interpolative reasoning systems based on interval type 2 fuzzy sets abstract in this paper we present a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on interval type 2 fuzzy sets we also apply the proposed weighted fuzzy interpolative reasoning method to deal with the truck backer upper control problem the proposed method satisfies the seven evaluation indices for fuzzy interpolative reasoning the experimental results show that the proposed method outperforms the existing methods it provides us with a useful way for dealing with fuzzy interpolative reasoning in sparse fuzzy rule based systems

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

solving a class of semidefinite programs via nonlinear programming in this paper we introduce a transformation that converts a class of linear and nonlinear semidefinite programming sdp problems into nonlinear optimization problems for those problems of interest the transformation replaces matrix valued constraints by vector valued ones hence reducing the number of constraints by an order of magnitude the class of transformable problems includes instances of sdp relaxations of combinatorial optimization problems with binary variables as well as other important sdp problems we also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data this transformation together with the efficient gradient formulas enables the solution of very large scale sdp problems by gradient based nonlinear optimization techniques in particular we propose a first order log barrier method designed for solving a class of large scale linear sdp problems this algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original sdp problem global convergence of the algorithm is established under mild and reasonable assumptions

proximity control in bundle methods for convex proximal bundle methods for minimizing a convex functionf generate a sequence x k by takingxk 1 to be the minimizer of hat f k x u k x x k 2 2 where hat f k is a sufficiently accurate polyhedral approximation tof andu k 0 the usual choice ofu k 1 may yield very slow convergence a technique is given for choosing u k adaptively that eliminates sensitivity to objective scaling some encouraging numerical experience is reported

proximal decomposition via alternating linearization a new approximate proximal point method for minimizing the sum of two convex functions is introduced it replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models the method updates the linear models and the prox center as well as the prox coefficient it is monotone in terms of the objective values and converges to a solution of the problem if any a dual version of the method is derived and analyzed applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented

a spectral bundle method with bounds semidefinite relaxations of quadratic 0 1 programming or graph partitioning problems are well known to be of high quality however solving them by primal dual interior point methods can take much time even for problems of moderate size the recent spectral bundle method of helmberg and rendl can solve quite efficiently large structured equality constrained semidefinite programs if the trace of the primal matrix variable is fixed as happens in many applications we extend the method so that it can handle inequality constraints without seriously increasing computation time in addition we introduce inexact null steps this abolishes the need of computing exact eigenvectors for subgradients which brings along significant advantages in theory and in practice encouraging preliminary computational results are reported

answering an open problem on t norms for type 2 fuzzy sets abstract this paper proves that a binary operation on 0 1 ensuring that the binary operation is a t norm or is a t conorm is a t norm where and are special convolution operations defined by f g x sup f y g z y z x f g x sup f y g z y z x for any f g map 0 1 0 1 where and are a continuous t norm and a continuous t conorm on 0 1 answering negatively an open problem posed in 8 besides some characteristics of t norm and t conorm are obtained in terms of the binary operations and

uninorms in fuzzy systems modeling abstract fuzzy systems modeling provides a framework in which input output relationships can be modeled by partitioning the input into fuzzy regions for which we can describe the output the fuzzy modeling inference process the procedure used to obtain the output of a fuzzy systems model for a given input is shown to involve an aggregation step in which the contributions of the different components of the fuzzy systems model are combined it is shown that the uninorm operator provides a general class of operators to implement this step the uninorm it is recalled is an operator that provides a generalization of the t norm and t conorm by allowing the identity element to be any value in the unit interval another step in the inference process is the determination of the contribution of each component of the model based on its relevancy to the current input the performance of this step requires the use of an operator related to the implication operator which we call a relevancy transformation ret operator it is shown that the form of this ret operator is dependent upon the identity value of the uninorm used in the aggregation step we provide some general formulations for this class of operators we show how the well known forms of fuzzy inference are special cases of our uninorm based approach