calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

extended karush kuhn tucker condition for constrained interval optimization problems and its application in support vector machines abstract in this paper we correct some errors in the extended first and second gordan s theorems in a recent paper information sciences 504 2019 276 292

linear programming with a possibilistic objective function abstract this paper proposes some ways for dealing with a linear program when the coefficients of the objective function are subject to possibilistic imprecision i e they are characterized by fuzzy restrictions emphasis is placed upon a passive approach that yields a satisfying solution via an appropriate semi infinite program and an active one that allows to reach a solution with a high possibility level of optimality extensions to the possibilistic constraints case and to the case of multiple objective programming problems with possibilistic coefficients are also hinted we end up with some concluding remarks and indicate axes for further developments

a self dual polar factorization for vector fields we show that any nondegenerate vector field u in where is a bounded domain in can be written as where s is a measure preserving point transformation on such that a e an involution and is a globally lipschitz antisymmetric convex concave hamiltonian moreover u is a monotone map if and only if s can be taken to be the identity which suggests that our result is a self dual version of brenier s polar decomposition for the vector field as where is convex and s is a measure preserving transformation we also describe how our polar decomposition can be reformulated as a self dual mass transport problem 2012 wiley periodicals inc

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

multiple objective linear programming models with interval coefficients an illustrated overview in most real world situations the coefficients of decision support models are not exactly known in this context it is convenient to consider an extension of traditional mathematical programming models incorporating their intrinsic uncertainty without assuming the exactness of the model coefficients interval programming is one of the tools to tackle uncertainty in mathematical programming models moreover most real world problems inherently impose the need to consider multiple conflicting and incommensurate objective functions this paper provides an illustrated overview of the state of the art of interval programming in the context of multiple objective linear programming models

minimax regret solution to linear programming problems with an interval objective function in this paper a linear programming problem with an interval objective function is treated first the previous approaches to this problem are reviewed and the drawbacks are pointed out to improve the drawbacks a new approach to this problem is proposed by introducing the minimax regret criterion as used in decision theory the properties of minimax regret solution are investigated in order to obtain the minimax regret solution a method of solution by a relaxation procedure is proposed it is shown that the solution is obtained by repetitional use of the simplex method a numerical example is given to illustrate the proposed solution method

a characterization of m accretivity and an application to nonlinear schrodinger type equations on caracterise la m accretivite et la monotonicite cyclique et on demontre le theoreme d existence des equations de type schrodinger non lineaires par la methode des semigroupes

violation analysis on two step method for interval linear programming in many real world problems system parameters or model coefficients may be bounded between lower and upper bounds due to a variety of uncertainties over the past decades intensive research efforts have focused on interval linear programming ilp to tackle such uncertainties as one of the most popular methods for solving ilp problems two step method tsm proposed by huang et al in 1995 allows uncertain information to be directly communicated into the optimization process and resulting solutions such that decision alternatives could be generated through the interpretation of interval solutions however part of optimum solution points obtained through tsm may go beyond the decision space in some cases this phenomenon referred to as solution violation may mislead decision makers to unreasonable policies plans or strategies which play important roles in the social and economic development therefore this study first investigates and identifies the essential cause for the solution violation of tsm following that an improved solution method namely itsm is proposed to avoid resulting violation by introducing extra constraints in the solving process a numeric example is then presented to demonstrate the effectiveness of itsm in handling solution violation

wirtinger type integral inequalities for interval valued functions abstract this study presents some wirtinger type integral inequalities for interval valued functions by means of the generalized hukuhara differentiability and the pompeiu hausdorff metric these integral inequalities generalize their respective versions for real valued functions numerical examples that illustrate the applicability of the theory developed herein are also provided

the karush kuhn tucker optimality conditions in multiobjective programming problems with interval valued objective functions the kkt conditions in multiobjective programming problems with interval valued objective functions are derived in this paper many concepts of pareto optimal solutions are proposed by considering two orderings on the class of all closed intervals in order to consider the differentiation of an interval valued function we invoke the hausdorff metric to define the distance between two closed intervals and the hukuhara difference to define the difference of two closed intervals under these settings we are able to consider the continuity and differentiability of an interval valued function the kkt optimality conditions can then be naturally elicited

the schur algorithm and reproducing kernel hilbert spaces in the ball abstract using reproducing kernel hilbert spaces methods we develop a schur type algorithm for a subclass of the functions analytic and contractive in the ball we also consider the nevanlinna pick interpolation problem in that class

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

new efficiency conditions for multiobjective interval valued programming problems abstract in this paper we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval valued functions a new concept of fritz john and karush kuhn tucker type points is introduced for this mathematical programming problem based on the gh derivative concept the innovation and importance of these concepts are presented from a practical and computational point of view the problem is approached directly without transforming it into a real valued programming problem thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses we also provide necessary conditions for efficiency which have been inexistent in the relevant literature to date the identification of necessary conditions is important for the development of future computational optimization techniques in an interval valued environment we introduce new generalized convexity notions for gh differentiable interval valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts finally the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper the theoretical developments are illustrated and justified through several numerical examples

metric selfduality and monotone vector fields on manifolds abstract we develop a metrically selfdual variational calculus for c monotone vector fields between general manifolds x and y where c is a coupling on x y remarkably many of the key properties of classical monotone operators known to hold in a linear context extend to this non linear setting this includes an integral representation of c monotone vector fields in terms of c convex selfdual lagrangians their characterization as a partial c gradients of antisymmetric hamiltonians as well as the property that these vector fields are generically single valued we also use a symmetric monge kantorovich transport to associate to any measurable map its closest possible c monotone rearrangement we also explore how this metrically selfdual representation can lead to a global variational approach to the problem of inverting c monotone maps an approach that has proved efficient for resolving non linear equations and evolutions driven by monotone vector fields in a hilbertian setting

fuzzy goal programming a parametric approach narasimhan incorporated fuzzy set theory within goal formulation in 1980 since then much research has been performed in this field and various models for solving fuzzy goal programming have been proposed one of the well known models was proposed by tiwari et al in 1987 19 where an additive model was proposed this paper is an extension to the tiwari et al model that deals with the sum of weighted negative deviations between the desirable achievement degree and the common target here properties of the model are proposed a numerical example is also given to illustrate the approach

multiobjective transportation problem with interval cost source and destination parameters in this paper we focus on the solution procedure of the multiobjective transportation problem motp where the cost coefficients of the objective functions and the source and destination parameters have been expressed as interval values by the decision maker this problem has been transformed into a classical motp where to minimize the interval objective function the order relations that represent the decision maker s preference between interval profits have been defined by the right limit left limit centre and half width of an interval the constraints with interval source and destination parameters have been converted into deterministic ones finally the equivalent transformed problem has been solved by fuzzy programming technique numerical examples have been provided to illustrate the solution procedure for three possible cases of the original problem

goal programming problems with interval coefficients and target intervals abstract in conventional goal programming the coefficients of objective functions and constraints and target values are determined as crisp values however it is not frequent that the coefficients and the target values are known precisely in such cases the coefficients and target values should be represented by intervals reflecting the imprecision this paper treats goal programming problems in which coefficients and target values are given the intervals it is shown that four formulations of the problems can be considered the properties of the four formulated problems are investigated an example is given to demonstrate the differences between the four formulations

the brezis ekeland principle for doubly nonlinear equations the celebrated brezis ekeland principle c r acad sci paris ser a b 282 1976 pp ai a1197 a1198 aii and a971 a974 characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems this note extends this characterization to doubly nonlinear evolution equations driven by convex potentials the characterization is exploited in order to establish approximation results for gradient flows doubly nonlinear equations and rate independent evolutions

fuzzy goal programming with different importance and priorities abstract this paper formulates fuzzy goal programming fgp incorporating different importance and preemptive priorities by using an additive model to maximize the sum of achievement degrees of all fuzzy goals in contrast to previous works the proposed approach allows the decision maker to determine a desirable achievement degree for each fuzzy goal to reflect explicitly the relative importance of these goals this approach can generate a set of achievement degrees consistent with the decision maker s expectations even though the relative importance of the goals may change furthermore in this paper we incorporate the decision maker s preemptive priority structure into a single formulation the resulting solutions satisfy both the preemptive priority structure and have the maximum achievement degrees in sum the proposed approaches effectiveness and computational superiority over the existing approaches are demonstrated and compared with examples from the literature

serial correlation estimation through the imprecise goal programming model abstract the goal programming gp model was used as a time series analysis tool that incorporates a serial correlation where the dependent variable is considered as precise this formulation does not take into consideration the decision maker s preferences however the dependent variable can be imprecise and its value can be expressed through an interval the aim of this paper is to develop a new formulation of the gp model for regression with serial correlation where the dependent variable is imprecise the proposed model will also integrate explicitly the decision maker s preferences a numerical example was used to illustrate our model

optimality conditions for fuzzy constrained programming problems abstract this paper solves optimization problems where both the objective and constraints are given by fuzzy functions in order to get it we first prove that these problems are equivalent to optimization problems where the constraints functions are non fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far this new stationary point concept is based on the gh differentiability and has many computational advantages that we describe it is well known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity and we are in that case due to the fact that the fuzzy numbers intervals space is a nonlinear one in this direction the gh derivative for fuzzy functions is a concept that is more general than hukuhara and level wise derivatives that are usually used in fuzzy optimization so far in the sense that they can be applied to a wider number of fuzzy function classes than above concepts with this new differentiability concept we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far moreover due to the fact that we do not have a linear space for fuzzy numbers the convex concepts and generalized convex fuzzy function notion are very restrictive also this implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful

a self dual variational approach to stochastic partial differential equations abstract unlike many of their deterministic counterparts stochastic partial differential equations are not amenable to the methods of calculus of variations a la euler lagrange in this paper we show how self dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields we construct solutions as minima of suitable non negative and self dual energy functionals on ito spaces of stochastic processes we show how a stochastic version of bolza s duality leads to solutions for equations with additive noise we then use a hamiltonian formulation to construct solutions for non linear equations with non additive noise such as the stochastic navier stokes equations in dimension two

on possibilistic linear programming abstract the problem of processing and combining possibilistic information in the scope of linear programming is approached as apparatus to cope with this problem we use the concepts of more possible value possibly feasible action and possibly efficient action issues regarding the feasibility of the resulting deterministic programs as well as the characterization of their optimal solutions are also discussed

antisymmetric hamiltonians variational resolutions for navier stokes and other nonlinear evolutions the theory of anti self dual asd lagrangians introduced in 6 is developed further to allow for a variational resolution of nonlinear pdes of the form u au u f 0 where is a convex lower semicontinuous function on a reflexive banach space x f x a d a x x is a positive linear operator and where d x x is a nonlinear operator that satisfies suitable continuity and antisymmetry properties asd lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form u t u t au t u t f 0 starting at u 0 u0 in both stationary and dynamic cases the equations associated to the proposed variational principles are not derived from the fact that they are critical points of the action functional but because they are also zeroes of the lagrangian itself for that we establish a general nonlinear variational principle that has many applications in particular to navier stokes type equations to generalized choquard pekar schrodinger equations with nonlocal terms as well as to complex ginsburg landau type initial value problems the case of navier stokes evolutions is more involved and will be dealt with in 9 the general theory of antisymmetric hamiltonians and its applications is developed in detail in an upcoming monograph 7 2007 wiley periodicals inc

a method for solving fuzzy goal programming problems based on minmax approach abstract narasimhan incorporated fuzzy set theory within goal programming formulation in 1980 since then numerous research has been carried out in this field one of the well known models for solving fuzzy goal programming problems was proposed by hannan in 1981 in this paper the conventional minmax approach in goal programming is applied to solve fuzzy goal programming problems it is proved that the proposed model is an extension to hannan model that deals with unbalanced triangular linear membership functions in addition it is shown that the new model is equivalent to a model proposed in 1991 by yang et al moreover a weighted model of the new approach is introduced and is compared with kim and whang s model presented in 1998 a numerical example is given to demonstrate the validity and strengths of the new models

a variational principle for gradient flows we verify after appropriate modifications an old conjecture of brezis ekeland 3 4 concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals our approach is based on a concept of self duality inherent in many parabolic evolution equations and motivated by bolza type problems in the classical calculus of variations the modified principle allows to identify the extremal value which was the missing ingredient in 3 and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation of course but also for quasi linear parabolic equations porous media fast diffusion and more general dissipative evolution equations

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

generalized interval vector spaces and interval optimization this paper presents a method of endowing the generalized interval space with some different structures such as vector space topological space order relations and algebraic calculus we formulate interval optimization problems and relate them to classic multi objective optimization problems we also present a version of the mini max s theorem in the interval space context this paper presents a method for endowing the generalized interval space with some different structures such as vector spaces order relations and an algebraic calculus with these concepts we formulate interval optimization problems and relate them to classic multi objective optimization problems we also present a version of the von neumann s mini max theorem in the interval context

variational principles for operator equations and initial value problems on decrit et on etudie des principes variationnels pour certaines classes d equations operateur et de problemes aux valeurs initiales

optimality conditions for generalized differentiable interval valued functions in this paper we study the optimal solutions set for a generalized differentiable interval valued function necessary and sufficient optimality conditions are established for gh differentiable functions convexity assumptions that are necessary or required to ensure the characterization of the optimal solutions are weaker or less strict than those presented in previous works these convexity assumptions are the weakest to characterize the optimal solutions set known results for classical non interval valued optimization are particular cases of the ones proved here