maximum principles extension problem and inversion for nonlocal one sided equations abstract we study one sided nonlocal equations of the form x 0 u x u x 0 x x 0 1 d x f x 0 on the real line notice that to compute this nonlocal operator of order 0 1 at a point x 0 we need to know the values of u x to the right of x 0 that is for x x 0 we show that the operator above corresponds to a fractional power of a one sided first order derivative maximum principles and a characterization with an extension problem in the spirit of caffarelli silvestre and stinga torrea are proved it is also shown that these fractional equations can be solved in the general setting of weighted one sided spaces in this regard we present suitable inversion results along the way we are able to unify and clarify several notions of fractional derivatives found in the literature

higher order generalized invexity and duality in nondifferentiable mathematical programming in this paper we consider a number of higher order duals to a nondifferentiable programming problem and establish duality under the higher order generalized invexity conditions introduced in an earlier work by mishra and rueda

a note on higher order symmetric duality in this paper a pair of wolfe type higher order symmetric dual multiobjective programs is formulated strong and converse duality theorems are established under invexity assumptions duality relations for mond weir type dual models have also been obtained under pseudoinvexity assumptions

weighted inequalities for one sided maximal functions let m be the maximal operator defined by g mb f x sup f t lg t dt f g t dt where g is a positive locally integrable function on r we characterize the pairs of nonnegative functions u v for which mb applies lp v in lp u or in weaklp u our results generalize sawyer s case g 1 but our proofs are different and we do not use hardy s inequalities which makes the proofs of the inequalities self contained

multiobjective symmetric duality involving cones abstract in this paper a pair of multiobjective symmetric dual programs over arbitrary cones are formulated for cone convex functions weak strong converse and self duality theorems are proved for these programs

two weight weak and extra weak type inequalities for the one sided maximal operator let be the one sided maximal operator and let be a convex non decreasing function on 0 0 0 we present necessary and sufficient conditions on a couple of weight functions such that the integral inqualities of weak type and of extra weak type hold our proofs do not refer to the theory of orlicz spaces

higher order symmetric duality in vector optimization problem involving generalized cone invex functions in this paper a pair of higher order symmetric dual model in vector optimization problem is formulated the higher order cone pseudoinvex and higher order strongly cone pseudoinvex functions are defined the weak strong and converse duality theorems are established using these defined functions

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

minimax and symmetric duality for a class of multiobjective variational mixed integer programming problems abstract in this paper we formulate a pair of multiobjective variational mixed integer programs for arbitrary cones under the separability and partial invexity assumptions on the functions involved we establish the weak strong converse and self duality theorems related to efficient solution

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

higher order duality in multiobjective fractional programming with support functions abstract in this paper a new class of higher order f type i functions for a multiobjective programming problem is introduced which subsumes several known studied classes higher order mond weir and schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in r n weak and strong duality results are studied in both the cases assuming the involved functions to be higher order f type i a number of previously studied problems appear as special cases

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

cauchy problems for keller segel type time space fractional diffusion equation abstract this paper investigates cauchy problems for nonlinear fractional time space generalized keller segel equation d t 0 c 2 b 0 where caputo derivative d t 0 c models memory effects in time fractional laplacian 2 represents levy diffusion and b s n r n x y x y n 2 y d y is the riesz potential with a singular kernel which takes into account the long rang interaction we first establish l r l q estimates and weighted estimates of the fundamental solutions p x t y x t or equivalently the solution operators s t t t then we prove the existence and uniqueness of the mild solutions when initial data are in l p spaces or the weighted spaces similar to keller segel equations if the initial data are small in critical space l p c r n p c n 2 we construct the global existence furthermore we prove the l 1 integrability and integral preservation when the initial data are in l 1 r n l p r n or l 1 r n l p c r n finally some important properties of the mild solutions including the nonnegativity preservation mass conservation and blowup behaviors are established

on the solution of a two ball trust region subproblem in this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution the method decomposes the subproblem and allows the application of efficient well studied methods for the solution of trust region subproblems arising in unconstrained optimization in the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem

strong duality for the cdt subproblem a necessary and sufficient condition in this paper we consider the problem of minimizing a nonconvex quadratic function subject to two quadratic inequality constraints as an application such a quadratic program plays an important role in the trust region method for nonlinear optimization such a problem is known as the celis dennis and tapia cdt subproblem in the literature the lagrangian dual of the cdt subproblem is a semidefinite program sdp hence convex and solvable however a positive duality gap may exist between the cdt subproblem and its lagrangian dual because the cdt subproblem itself is nonconvex in this paper we present a necessary and sufficient condition to characterize when the cdt subproblem and its lagrangian dual admits no duality gap i e the strong duality holds this necessary and sufficient condition is easy verifiable and involves only one any optimal solution of the sdp relaxation for the cdt subproblem moreover the condition reveals that it is actually rare to render a positive duality gap for the cdt subproblems in general moreover if the strong duality holds then an optimal solution for the cdt problem can be retrieved from an optimal solution of the sdp relaxation by means of a matrix rank one decomposition procedure the same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the lagrangian multipliers at a kkt point furthermore we show that the condition is numerically easy to work with approximatively

multiobjective nonlinear second order symmetric duality with k f pseudoconvexity abstract the purpose of this paper is to introduce a new class of generalized cone pseudoconvex functions and strongly cone pseudoconvex functions called second order k f pseudoconvex functions and strongly second order k f pseudoconvex functions a pair of second order symmetric dual multiobjective nonlinear programs is formulated over arbitrary generalized cone pseudoconvex functions for these second order symmetric dual programs the weak strong and converse duality theorems are established using the above generalization of cone pseudoconvex functions a self duality theorem is also given by assuming the functions involved to be skew symmetric

symmetric duality for minimax mixed integer programming problems with pseudo invexity in this paper we consider the pair of symmetric dual multiobjective variational mixed integer programs proposed by chen and yang x chen j yang symmetric duality for minimax multiobjective variational mixed integer programming problems with partial invexity european journal of operational research 181 2007 76 85 and extend some of their results under the assumptions of partial pseudo invexity and separability on the functions involved these results include several results available in literature as special cases

multiobjective second order symmetric duality with f convexity we suggest a pair of second order symmetric dual programs in multiobjective nonlinear programming for these second order symmetric dual programs we prove the weak strong and converse duality theorems under f convexity conditions

computing a celis dennis tapia trust region step for equality constrained optimization we study an approach for minimizing a convex quadratic function subject to two quadratic constraints this problem stems from computing a trust region step for an sqp algorithm proposed by celis dennis and tapia 1985 for equality constrained optimization our approach is to reformulate the problem into a univariate nonlinear equation 0 where the function is continuous at least piecewise differentiable and monotone well established methods then can be readily applied we also consider an extension of our approach to a class of non convex quadratic functions and show that our approach is applicable to reduced hessian sqp algorithms numerical results are presented indicating that our algorithm is reliable robust and has the potential to be used as a building block to construct trust region algorithms for small sized problems in constrained optimization

on second order symmetric duality in nondifferentiable programming this article is concerned with a pair of second order symmetric dual non differentiable programs and second order f pseudo convexity we establish the weak and the strong duality theorems for the new pair of dual models under the f pseudo convexity assumption several known results including mond and schechter as well as others are obtained as special cases

one sided extrapolation at infinity and singular integrals in this paper we prove on one hand extrapolation from infinity for the one sided classes ap a p q a p q and on the other hand the bmo boundedness for one sided singular integrals we also provide several applications of our results

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

second order symmetric duality for nonlinear multiobjective mixed integer programming we formulate two pairs of second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones by using the concepts of efficiency and second order invexity we establish weak strong converse and self duality theorems for the dual models several known results are obtained as special cases

one sided fractional derivatives fractional laplacians and weighted sobolev spaces abstract we characterize one sided weighted sobolev spaces w 1 p r where is a one sided sawyer weight in terms of a e and weighted l p limits as 1 of marchaud fractional derivatives of order similar results for weighted sobolev spaces w 2 p r n where is an a p muckenhoupt weight are proved in terms of limits as s 1 of fractional laplacians s these are bourgain brezis mironescu type characterizations for weighted sobolev spaces we also complement their work by studying a e and weighted l p limits as s 0

symmetric duality in multiobjective programming involving generalized cone invex functions abstract in this paper cone pseudoinvex and strongly cone pseudoinvex functions are defined a pair of mond weir type symmetric dual multiobjective programs is formulated over arbitrary cones weak duality strong duality and converse duality theorems are established using the above defined functions a self duality theorem is also given by assuming the functions involved to be skew symmetric

second order symmetric duality in non differentiable multiobjective programming with f convexity abstract this paper is concerned with a pair of mond weir type second order symmetric dual non differentiable multiobjective programming problems we establish the weak and strong duality theorems for the new pair of dual models under second order f convexity assumptions several results including many recent works are obtained as special cases

higher order generalized convexity and duality in nondifferentiable multiobjective mathematical programming in this paper a class of generalized convexity is introduced and a unified higher order dual model for nondifferentiable multiobjective programs is described where every component of the objective function contains a term involving the support function of a compact convex set weak duality theorems are established under generalized convexity conditions the well known case of the support function in the form of square root of a positive semidefinite quadratic form and other special cases can be readily derived from our results

pseudo invexity and duality in nonlinear programming abstract the purpose of this paper is to study various duality results in nonlinear programming for pseudo invex functions such results were known in the literature for invex functions

minimax and symmetric duality for nonlinear multiobjective mixed integer programming abstract we formulate two pairs of symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones by using the concept of efficiency we establish the weak strong converse and self duality theorems for our symmetric models several known results are obtained as special cases

a trust region method for norm constrained problems a trust region method for the solution of nonlinear optimization problems with norm constraints is analyzed such optimization problems arise in parameter identification and nonlinear eigenvalue problems the characteristics of these applications motivate the formulation and analysis of the trust region method the algorithms studied here allow for inexact gradient information and the use of subspace methods for the approximate solution of subproblems characterizations and the descent properties of trust region steps are given criteria for the existence of successful iterations under inexact gradient information and under the use of subspace methods are established and global convergence of the method is proven

weighted norm inequalities for the riemann liouville and weyl fractional integral operators caracterisation des fonctions poids pour lesquelles l operateur integral fractionnaire de riemann liouville d ordre 0 est borne

on weights satisfying parabolic muckenhoupt conditions abstract this note collects results related to parabolic muckenhoupt a p weights for a doubly nonlinear parabolic partial differential equation a general approach is proposed which extends the theory beyond the quadratic growth case in particular the natural parabolic geometry for the equation and the unavoidable time lag are incorporated in the definitions and results the limiting muckenhoupt conditions of a type are also discussed and several open questions are posed

converse duality in nonlinear programming with cone constraints abstract the purpose of this paper is to establish various converse duality results for nonlinear programming with cone constraints and its four dual models introduced by chandra and abha s chandra abha a note on pseudo invex and duality in nonlinear programming european journal of operational research 122 2000 161 165

symmetric duality for multiobjective fractional variational problems involving cones in this paper a pair of multiobjective fractional variational symmetric dual problems over cones is formulated weak strong and converse duality theorems are established under generalized f convexity assumptions moreover self duality theorem is also discussed

non differentiable symmetric duality for multiobjective programming with cone constraints abstract two pairs of non differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones which are wolfe type and mond weir type are considered on the basis of weak efficiency with respect to a convex cone we obtain symmetric duality results for the two pairs of problems under cone invexity and cone pseudoinvexity assumptions on the involved functions our results extend the results in khurana s khurana symmetric duality in multiobjective programming involving generalized cone invex functions european journal of operational research 165 2005 592 597 to the non differentiable multiobjective symmetric dual problem

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

weighted inequalities for the one sided hardy littlewood maximal functions let m f x suph 0 l h jx h t 1 idt denote the one sided maximal function of hardy and littlewood for w x 0 on r and 1 0 and a v ll p then m is bounded from lp v to lp w if and only if m x a w 0 the corresponding weak type inequality is also characterized further properties of a weights such as a a e and a p a a1 p are established

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

non differentiable higher order symmetric duality in mathematical programming with generalized invexity a pair of non differentiable higher order symmetric dual model in mathematical programming is formulated the weak and strong duality theorems are established under higher order invexity assumption symmetric minimax mixed integer primal and dual problems are also investigated

mond weir type second order symmetric duality in non differentiable minimax mixed integer programming problems abstract a pair of mond weir type non differentiable second order symmetric minimax mixed integer primal and dual problems in mathematical programming is formulated symmetric and self duality theorems are then established under second order f pseudo convexity assumptions several known results including that of gulati and ahmad eur j oper res 101 1997 122 hou and yang j math anal appl 255 2001 491 and mond and schechter bull aust math soc 53 1996 177 as well as others are obtained as special cases

higher order symmetric duality in nondifferentiable multiobjective programming problems abstract in this paper a pair of nondifferentiable multiobjective programming problems is first formulated where each of the objective functions contains a support function of a compact convex set in rn for a differentiable function h r n r n r we introduce the definitions of the higher order f convexity f pseudo convexity f quasi convexity of function f r n r with respect to h when f and h are taken certain appropriate transformations all known other generalized invexity such as invexity type i invexity and higher order type i invexity can be put into the category of the higher order f invex functions under these the higher order f convexity assumptions we prove the higher order weak higher order strong and higher order converse duality theorems related to a properly efficient solution

non differentiable second order symmetric duality in mathematical programming with f convexity abstract a pair of wolfe type non differentiable second order symmetric primal and dual problems in mathematical programming is formulated the weak and strong duality theorems are then established under second order f convexity assumptions symmetric minimax mixed integer primal and dual problems are also investigated

a global convergence theory for the celis dennis tapia trust region algorithm for constrained optimization a global convergence theory for a class of trust region algorithms for solving the equality constrained optimization problem is presented this theory is sufficiently general that it holds for any algorithm that generates steps giving at least a fraction of cauchy decrease in the quadratic model of the constraints and that uses the augmented lagrangian as a merit function this theory is used to establish global convergence of the 1984 celis dennis tapia algorithm with a different scheme for updating the penalty parameter the behavior of the penalty parameter is also discussed

technical note on symmetric duality in multiobjective programming some remarks on recent results abstract certain shortcomings are pointed out in some recent works of das and nanda european journal of operational research 97 1997 167 171 and kim et al european journal of operational research 107 1998 686 691 which show that these studies are highly restricted

mixed symmetric duality in non differentiable multiobjective mathematical programming two mixed symmetric dual models for a class of non differentiable multiobjective nonlinear programming problems with multiple arguments are introduced in this paper these two mixed symmetric dual models unify the four existing multiobjective symmetric dual models in the literature weak and strong duality theorems are established for these models under some mild assumptions of generalized convexity several special cases are also obtained

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

multiobjective symmetric duality with cone constraints we formulate a pair of multiobjective symmetric dual programs for pseudo invex functions and arbitrary cones our model is unifying the wolfe vector symmetric dual and the mond weir vector symmetric dual models we establish the weak strong converse and self duality theorems for our pair of dual models nanda and das results optimization 28 1994 267 eur j oper res 88 1996 572 are obtained as special cases

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

a global convergence theory for dennis el alem and maciel s class of trust region algorithms for constrained optimization without assuming regularity this work presents a convergence theory for dennis el alem and maciel s class of trust region based algorithms for solving the smooth nonlinear programming problem with equality constraints the results are proved under very mild conditions on the quasi normal and tangential components of the trial steps the lagrange multiplier estimates and the hessian estimates are assumed to be bounded no regularity assumption is made in particular linear independence of the gradients of the constraints is not assumed the theory proves global convergence for the class in particular it shows that a subsequence of the iteration sequence satisfies one of four types of mayer bliss stationary conditions in the limit

symmetric and self duality in vector optimization problem abstract in this paper we present the first order symmetric and self duality programs in multiobjective nonlinear programming problems for these first order symmetric and dual programs we introduce the weak strong and converse duality theorems under convexity and concavity conditions where the pair of the symmetric dual multiobjective nonlinear programming problems in this paper different which in d s kim y b yun h kuk second order symmetric and self duality in multiobjective programming appl math lett 10 2 1997 17 22 also we prove the self duality theorem for these first order self dual programs and illustrate its example

optimality criteria in nonlinear programming involving nonconvex functions abstract in this paper a new class of functions called convex quasiconvex and pseudoconvex are defined and a number of sufficient optimality criteria are obtained for nonlinear programming problems involving these functions

higher order generalized invexity and duality in mathematical programming in this paper we introduce the concepts of higher order type i pseudo type i and quasi type i functions and establish various higher order duality results involving these functions

a note on pseudo invexity and duality in nonlinear programming abstract certain shortcomings are pointed out in the recent work of s nanda l n das european journal of operational research 88 1996 572 577 and appropriate modifications are suggested for studying duality under pseudo invexity assumptions

non local fractional derivatives discrete and continuous abstract we prove maximum and comparison principles for the discrete fractional derivatives in the integers regularity results when the space is a mesh of length h and approximation theorems to the continuous fractional derivatives are shown when the functions are good enough holder continuous these approximation procedures give a measure of the order of approximation these results also allow us to prove the coincidence for holder continuous functions of the marchaud and grunwald letnikov derivatives in every point and the speed of convergence to the grunwald letnikov derivative the discrete fractional derivative will be also described as a neumann dirichlet operator defined by a semi discrete extension problem some operators related to the harmonic analysis associated to the discrete derivative will be also considered in particular their behavior in the lebesgue spaces l p z