polyhedral sweeping processes with unbounded nonconvex valued perturbation abstract a polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable hilbert space polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes we consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time the widely used assumption of lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of h lipschitzness the existence of solutions is proved for this sweeping process

self dual regularization of monotone operators via the resolvent average we give two self dual regularizations of maximal monotone operators on hilbert spaces these regularizations and their set valued inverses are strongly monotone single valued and lipschitz with full domain moreover these regularizations graphically converge to the original monotone operator if a maximal monotone operator has nonempty zeros these self dual regularizations can be used to find its least norm solution when the maximal monotone operator is the subdifferential of a proper lower semicontinuous convex function with nonempty minimizers this translates to finding the least norm minimizer

a unified exact analysis for the poynting effects of cylindrical tubes made of hill s class of hookean compressible elastic materials at finite strain abstract a direct natural extension of hooke s law to finite strain was achieved by r hill in 1978 employing the notion of work conjugate measures of stress and strain with seth hill doyle ericksen class of finite strain measures this extension actually defines a broad class of compressible hyperelastic materials at finite strain each of which retains the simple linear structure of hooke s law as stress strain relationship several known simple elasticity models at finite strain are included as its particular examples with a novel idea of utilizing a suitable parametric variable here we present a unified study of the free end torsion problem poynting effects of thin walled cylindrical tubes made of the foregoing hill s class of hookean type hyperelastic materials we show that it is possible to derive a unified exact solution to the nonlinear coupling equations relating the torque the shear stress and the controlling deformation quantities including in particular the axial length change discussions and comparisons concerning various hookean type elasticity models are made based on the exact solution obtained

weak topologies for the closed subsets of a metrizable space the purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space it can be shown that all of the standard hyperspace topologies including the hausdorff metric topology the vietoris topology the attouch wets topology the fell topology the locally finite topology and the topology of mosco convergence arise as weak topologies generated by families of geometric functionals defined on closed sets a key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of hausdorff excess functionals

set convergences an attempt of classification we endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics such structure is completely deter mined by a class which is a family of closed sets and a type which is a semi metric two types are sufficient to define and classify almost all convergences known till now these two types offer the possibility of defining other set con vergences

a constructive proof of the existence of a utility in revealed preference theory within the context of the standard model of rationality within economic modelling we show the existence of a utility function that rationalises a demand correspondence hence completely characterizes the associated preference structure by taking a dense demand sample this resolves the problem of revealed preferences under some very mild assumptions on the demand correspondence which are closely related to a number of established axioms in preference theory the proof establishes the existence of a limit of a sequences of indirect utilities that rationalise finite data sets where the sample size increases to infinity this limiting utility provides a rationalisation of the demand relation up to a rescaling this limiting indirect utility is unique and continuous on a set of full measure

stability results for ekeland s e variational principle and cone extremal solutions given x a banach space and f x a a proper lower semicontinuous function which is bounded from below the ekeland s e variational principle asserts the existence of a point xi in x which we call e extremal with respect to f which satisfies fu fxi e u xi for all u x u xi by using set convergence notions kuratowski painleve mosco bounded hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e ext f the set of such e extremal points the key for the geometrical understanding of such properties is to consider the equivalent phelps extremization principle which given a closed set d in x and a partial ordering with respect to a pointed cone associates the set of elements of d maximal with respect to this order direct or potential applications are given in various fields multicriteria optimization numerical algorithmic calculus of variations

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

quantitative stability of variational systems ii a framework for nonlinear conditioning stability results of lipschitz and holder type are obtained for the solutions and optimal values of optimization problems when perturbations are measured in terms of the rho epi distance

isometries for the legendre fenchel transform it is shown that on the space of lower semicontinuous convex functions defined on r the conjugation map the legendre fenchel transform is an isometry with respect to some metrics consistent with the epi topology we also obtain isometries for the infinite dimensional case hilbert space and reflexive banach space but this time they correspond to topologies finer than the mosco epi topology

path integral for system with spin a path integral representation for the system with spin is considered explicit path integral representations are given in classical phase space for spin realized as the complex kahler manifold and as the grassmann algebra for the spin case

convergence of subdifferentials and normal cones in locally uniformly convex banach space abstract in this paper we study the behaviour of normal cones and subdifferentials with respect to two types of convergence of sets and functions mosco and attouch wets convergences our analysis is devoted to proximal frechet and mordukhovich limiting normal cones and subdifferentials the results obtained can be seen as extensions of the attouch theorem to the context of non convex functions on locally uniformly convex banach space they also generalize to sequences of subsmooth sets or functions various results in the literature

another isometry for the legendre fenchel transform abstract a new distance is introduced on the space of extended real valued lower semicontinuous convex functions defined on reflexive banach spaces it is shown that with this notion of distance the legendre fenchel is an isometry

persistence and stability of solutions of hamilton jacobi equations abstract given a convergent sequence of hamiltonians h n and a convergent sequence of initial data g n for the first order evolutionary hamilton jacobi equation we look for conditions ensuring that the sequences u n and v n of lax solutions and hopf solutions respectively converge the convergences we deal with are variational convergences we take advantage of several recent results giving criteria for the continuity of usual operations

epiconvergence of convexly composite functions in banach spaces this paper gives conditions ensuring the epiconvergence or gamma convergence of sequences of convexly composite functions via the study of painleve kuratowski convergence of inverse images of sets the convergence of multipliers of optimization problems associated with convexly composite functions is also established

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

the epi distance topology continuity and stability results with applications to convex optimization problems let i x denote the proper lower semicontinuous convex functions on a banach space x equipped with the completely metrizable topology of uniform convergence of distance functions on bounded sets we show subject to some standard constraint qualifications that the operations of addition and restriction are continuous these results are applied to convex well posed optimization problems as well as to convergence of approximated solutions in infinite dimensional convex programming to linear functionals exposing convex sets and to metric projections also for any given function in i x we obtain results regarding the convergence of its inf convolution with smoothing kernels

epi convergent discretizations of stochastic programs via integration quadratures the simplest and the best known method for numerical approximation of high dimensional integrals is the monte carlo method mc i e random sampling mc has also become the most popular method for constructing numerically solvable approximations of stochastic programs however certain modern integration quadratures are often superior to crude mc in high dimensional integration so it seems natural to try to use them also in discretization of stochastic programs this paper derives conditions that guarantee the epi convergence of the resulting objectives to the original one our epi convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure as examples we prove epi convergence of quadrature based discretizations of three different models of portfolio management and we study their behavior numerically besides mc our discretizations are the only existing ones with guaranteed epi convergence for these problem classes in our tests modern quadratures seem to result in faster convergence of optimal values than mc

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems