alternating projections on manifolds we prove that if two smooth manifolds intersect transversally then the method of alternating projections converges locally at a linear rate we bound the speed of convergence in terms of the angle between the manifolds which in turn we relate to the modulus of metric regularity for the intersection problem a natural measure of conditioning we discuss a variety of problem classes where the projections are computationally tractable and we illustrate the method numerically on a problem of finding a low rank solution of a matrix equation

on identification of active constraints ii the nonconvex case in this paper the results of burke and more 5 on the identification of active constraints are extended to the nonconvex constrained nonlinear programming problem the approach is motivated by the geometric structure of a certain polyhedral convex linearization of the constraint region at each iteration as in burke and more 5 questions of constraint identification are couched in terms of the faces of these polyhedra the main result employs a nondegeneracy condition due to dunn 7 and the linear independence condition to obtain a characterization of those algorithms that identify the optimal active constraints in a finite number of iterations the role of the linear independence condition is carefully examined and it is argued that it is required within the context of a first order theory of constraint identification in conclusion the characterization theorem is applied to the wilson han powell sequential quadratic programming algorithm j optim theory appl 22 1977 pp 297 309 proceedin

baire measurable paradoxical decompositions via matchings abstract we show that every locally finite bipartite borel graph satisfying a strengthening of hall s condition has a borel perfect matching on some comeager invariant borel set we apply this to show that if a group acting by borel automorphisms on a polish space has a paradoxical decomposition then it admits a paradoxical decomposition using pieces having the baire property this strengthens a theorem of dougherty and foreman who showed that there is a paradoxical decomposition of the unit ball in r 3 using baire measurable pieces we also obtain a baire category solution to the dynamical von neumann day problem if a is a nonamenable action of a group on a polish space x by borel automorphisms then there is a free baire measurable action of f 2 on x which is lipschitz with respect to a

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

measurable versions of vizing s theorem abstract we establish two versions of vizing s theorem for borel multi graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively and the approximate version states that for any borel probability measure on the edge set and any 0 we can properly colour all but fraction of edges with colours in a borel way the measurable version which is our main result states that if additionally the measure is invariant then there is a measurable proper edge colouring of the whole edge set with at most colours

a robust and accurate finite difference method for a generalized black scholes equation in this paper we present a numerical method for a generalized black scholes equation which is used for option pricing the method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique our scheme is stable for arbitrary volatility and arbitrary interest rate and is second order convergent with respect to the spatial variable furthermore the present paper efficiently treats the singularities of the non smooth payoff function numerical results support the theoretical results

semiconcave solutions of partial differential inclusions abstract we obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions these results will be used to study the behavior of singularities of the value function for a reflected control problem

an upwind numerical approach for an american and european option pricing model the numerical solution of several mathematical models arising in financial economics for the valuation of both european and american call options on different types of assets is considered all the models are based on the black scholes partial differential equation in the case of european options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented when treating with american options an additional inequality constraint leads to a discretized linear complementarity problem in each case the numerical approximation of option values is computed by means of optimization algorithms in particular uzawa s method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics

numerical approximation of a time fractional black scholes equation abstract in this paper a time fractional black scholes equation is examined we transform the initial value problem into an equivalent integral differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization a rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first order convergence with respect to the time variable is proved for overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the black scholes differential operator we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization it is proved that the scheme is stable and second order convergent with respect to the spatial variable numerical experiments support these theoretical results

numerically pricing american options under the generalized mixed fractional brownian motion model abstract in this paper we introduce a robust numerical method based on the upwind scheme for the pricing of american puts under the generalized mixed fractional brownian motion gmfbm model by using portfolio analysis and applying the wick ito formula a partial differential equation pde governing the prices of vanilla options under the gmfbm is successfully derived for the first time based on this we formulate the pricing of american puts under the current model as a linear complementarity problem lcp unlike the classical black scholes b s model or the generalized b s model discussed in cen and le 2011 the newly obtained lcp under the gmfbm model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing pde as time approaches zero to overcome this difficulty a numerical approach based on the upwind scheme is adopted it is shown that the coefficient matrix of the current method is an m matrix which ensures its stability in the maximum norm sense remarkably we have managed to provide a sharp theoretic error estimate for the current method which is further verified numerically the results of various numerical experiments also suggest that this new approach is quite accurate and can be easily extended to price other types of financial derivatives with an american style exercise feature under the gmfbm model

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

new metric properties for prox regular sets in this paper we present diverse new metric properties that prox regular sets shared with convex ones at the heart of our work lie the legendre fenchel transform and complements of balls first we show that a connected prox regular set is completely determined by the legendre fenchel transform of a suitable perturbation of its indicator function then we prove that such a function is also the right tool to extend to the context of prox regular sets the famous connection between the distance function and the support function of a convex set on the other hand given a prox regular set we examine the intersection of complements of open balls containing the set we establish that the distance of a point to a prox regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point this is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes to the best of our knowledge these results are new in the literature and show that the class of prox regular sets have good properties known in convex analysis mathematics subject classification 2010 49j52 49j53

invariant sets and lyapunov pairs for differential inclusions with maximal monotone operators we give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on hilbert spaces which is subject to a lipschitz continuous perturbation depending on the state these sets are not necessarily weakly closed as in 3 4 while the invariance criteria are still written by using only the data of the system so no need to the explicit knowledge of neither the solution of this differential inclusion nor the semi group generated by the maximal monotone operator these invariant viability results are next applied to derive explicit criteria for a lyapunov pairs of lower semi continuous not necessarily weakly lsc functions associated to these differential inclusions the lack of differentiability of the candidate lyapunov functions and the consideration of general invariant sets possibly not convex or smooth are carried out by using techniques from nonsmooth analysis

option pricing using a computational method based on reproducing kernel abstract one of the most important subject in financial mathematics is the option pricing the most famous result in this area is black scholes formula for pricing european options this paper is concerned with a method for solving a generalized black scholes equation in a reproducing kernel hilbert space subsequently the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method furthermore the error estimates for obtained approximation in reproducing kernel hilbert space are presented finally a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

estimating tangent and normal cones without calculus we study simple estimates of the tangent and normal cones to a set with a nonsmooth boundary we seek estimates that are robust relative to the base point and that depend only on collections of nearby points known to lie inside and outside of the set

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

a new higher order compact finite difference method for generalised black scholes partial differential equation european call option abstract this paper presents a high order numerical method based on a uniform mesh to obtain a highly accurate result for generalized black scholes equation arising in the financial market we first semi discretize the underlying problem in the temporal direction by using crank nicolson scheme and then discretize the resulting set of equations by using a high order compact finite difference method hocfdm the stability of the scheme is analyzed convergence of the suggested method is proved it is shown that the method is of order o t 2 h 4 some numerical experiments are carried out in order to illustrate the applicability and accuracy of the proposed method and to validate the theoretical results as well it is shown that hocfdm solution matches very well with the exact solution further the rate of convergence predicted theoretically is the same as that obtained numerically the computational time for our method in matlab programming language is provided

identifiable surfaces in constrained optimization the concept of a class c p identifiable surface of a convex set in euclidean space is introduced the paper shows how the smoothness of these surfaces is related to the smoothness of the projection operator and presents finite identification results for certain algorithms for minimization of a function over this set the work uses a partially geometric view of constrained optimization to generalize previous finite identification results

a sixth order numerical method and its convergence for generalized black scholes pde abstract the main aim of this paper is to construct a new computational approach for the numerical solution of generalized black scholes equation in this approach the temporal variable is discretized using crank nicolson scheme and spatial variable is discretized using sextic b spline collocation method convergence analysis of the method is discussed the proposed method is proved to be stable and has second order convergence with respect to time variable and sixth order convergence with respect to space variable to illustrate the applicability and efficiency of the proposed method we consider some benchmark problems describing european call options numerical results verify the orders of convergence predicted by the analysis numerical results reveal that the present method provides better results as compared to some existing numerical methods based on b spline collocation approach

convergence of a fitted finite volume method for the penalized black scholes equation governing european and american option pricing in this paper we present an analysis of a numerical method for a degenerate partial differential equation called the black scholes equation governing american and european option pricing the method is based on a fitted finite volume spatial discretization and an implicit time stepping technique the analysis is performed within the framework of the vertical method of lines where the spatial discretization is formulated as a petrov galerkin finite element method with each basis function of the trial space being determined by a set of two point boundary value problems we establish the stability and an error bound for the solutions of the fully discretized system numerical results are presented to validate the theoretical results

finite difference scheme with a moving mesh for pricing asian options in this paper we propose a stable numerical method for pricing asian call options which is based on a central difference scheme with a moving mesh in the spatial discretization and the rannacher time stepping scheme in the time discretization at each time mesh point we make a piecewise uniform mesh to discretise the space interval which ensures that the matrix associated with the discrete operator is an m matrix hence the spatial discretization scheme is maximum norm stable for arbitrary volatility and arbitrary interest rate we show that the scheme is second order convergent with respect to both time and spatial variables numerical results support the theoretical results

numerical solution of generalized black scholes model abstract this paper presents a numerical scheme that approximates the option prices for different option styles governed by the generalized black scholes equation in its degenerate form the proposed method uses the hodie scheme in the spacial direction and the two step backward differentiation formula in the temporal direction it is proved that the method has second order convergence in space as well as in time numerical experiments validate the theoretical results

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

an efficient primal dual interior point method for minimizing a sum of euclidean norms the problem of minimizing a sum of euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature this nonsmooth optimization problem arises in many different kinds of modern scientific applications we derive a primal dual interior point algorithm for the problem by applying newton s method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition the main work at each step consists of solving a system of linear equations the schur complement equations this schur complement matrix is not symmetric unlike in linear programming we incorporate a mehrotra type predictor corrector scheme and present some experimental results comparing several variations of the algorithm including as one option explicit symmetrization of the schur complement with a skew corrector term we also present results obtained from a code implemented to solve large sparse problems using a symmetrized schur complement this has been applied to problems arising in plastic collapse analysis with hundreds of thousands of variables and millions of nonzeros in the constraint matrix the algorithm typically finds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable

mixed semicontinuous perturbations of nonconvex sweeping processes in this paper we prove a theorem on the existence of a global solution of a differential inclusion governed by a class of nonconvex sweeping processes with unbounded pertubations the perturbations are not required to be convex valued

linear regularity for a collection of subsmooth sets in banach spaces using variational analysis we study the linear regularity for a collection of finitely many closed sets in particular we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting moreover the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption

countable abelian group actions and hyperfinite equivalence relations an equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations e n where all e n equivalence classs are finite in this article we establish the following theorem if a countable abelian group acts on a standard borel space in a borel manner then the orbit equivalence relation is hyperfinite the proof uses constructions and analysis of borel marker sets and regions in the space 2 mathbb z omega this technique is also applied to a problem of finding borel chromatic numbers for invariant borel subspaces of 2 mathbb z n

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

active set newton methods and partial smoothness diverse optimization algorithms correctly identify in finite time intrinsic constraints that must be active at optimality analogous behavior extends beyond optimization to systems involving part

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

a comparison study of adi and operator splitting methods on option pricing models in this paper we perform a comparison study of alternating direction implicit adi and operator splitting os methods on multi dimensional black scholes option pricing models the adi method is used extensively in mathematical finance for numerically solving multi factor option pricing problems however numerical results from the adi scheme show oscillatory solution behaviors with nonsmooth payoffs or discontinuous derivatives at the exercise price with large time steps in the adi scheme there are source terms which include y derivatives when we solve x derivative involving equations then due to the nonsmooth payoffs source terms contain abrupt changes which are not in the range of implicit discrete operators and this leads to difficulty in solving the problem on the other hand the os method does not contain the other variable s derivatives in the source terms we provide computational results showing the performance of the methods for two asset option pricing problems the results show that the os method is very efficient and gives better accuracy and robustness than the adi method with large time steps

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

measurable versions of the lovasz local lemma and measurable graph colorings abstract in this paper we investigate the extent to which the lovasz local lemma an important tool in probabilistic combinatorics can be adapted for the measurable setting in most applications the lovasz local lemma is used to produce a function f x y with certain properties where x is some underlying combinatorial structure and y is a typically finite set can this function f be chosen to be borel or measurable for some probability borel measure on x assuming that x is a standard borel space in the positive direction we prove that if the set of constraints put on f is in a certain sense locally finite then there is always a borel choice for f that is e close to satisfying these constraints for any e 0 moreover if the combinatorial structure on x is induced by the 0 1 shift action of a countable group then even without any local finiteness assumptions there is a borel choice for f which satisfies the constraints on an invariant conull set i e with e 0 a direct corollary of our results is an upper bound on the measurable chromatic number of the graph g n generated by the shift action of the free group f n that is asymptotically tight up to a factor of at most 2 which answers a question of lyons and nazarov on the other hand our result for structures induced by measure preserving group actions is at least for amenable groups sharp in the following sense a probability measure preserving action of a countably infinite amenable group satisfies the measurable version of the lovasz local lemma if and only if it admits a factor map to the 0 1 shift action to prove this we combine the tools of the ornstein weiss theory of entropy for actions of amenable groups with concepts from computability theory specifically kolmogorov complexity

optimality identifiability and sensitivity around a solution of an optimization problem an identifiable subset of the feasible region is one containing all nearby solutions after small perturbations to the problem a quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are minimal this new notion lays a broad and intuitive variational analytic foundation for optimality conditions sensitivity and active set methods

generalised prox regularity in reflexive smooth banach spaces with smooth dual norm abstract in this paper we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth banach spaces with smooth dual norm started by the first author in 9 we define a new class of nonconvex sets in banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm

quintic b spline collocation approach for solving generalized black scholes equation governing option pricing we develop a numerical algorithm for solving generalized black scholes partial differential equation which arises in european option pricing the method comprises the horizontal method of lines for time integration and method 1 2 1 1 corresponds to the back ward euler method and 1 2 corresponds to the crank nicolson method to discretize in temporal direction and the quintic b spline collocation method in uniform spatial direction the convergence analysis and stability of proposed method are discussed in detail it is justifying that the approximate solution converges to the exact solution of orders o k h 3 for the back ward euler method and o k 2 h 3 for the crank nicolson method where k and h are mesh sizes in the time and space directions respectively the proposed method is also shown to be unconditionally stable this scheme applied on some test examples the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence results shown by this method are found to be in good agreement with the known exact solutions the produced results are also seen to be more accurate than some available results given in the literature

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

estimation of local volatilities in a generalized black scholes model this paper studies a parameter estimation problem for a generalized black scholes equation which is used for option pricing in estimating the volatility function from a set of market observations we use an implicit finite difference scheme the function space parameter estimation convergence fspec is proved and numerical simulations were performed

on finite convergence and constraint identification of subgradient projection methods this paper deals with a continuous time subgradient projection algorithm shown to generate trajectories that accumulate to the solution set under a strong convexity assumption we show that convergence is exponential in norm a sharpness condition yields convergence in finite time and the necessary lapse is estimated invoking a constraint qualification and a non degeneracy assumption we demonstrate that optimally active constraints are identified in finite time

a determinacy approach to borel combinatorics we introduce a new method involving infinite games and borel determinacy which we use to answer several well known questions in borel combinatorics

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

second order sweeping process with a lipschitz perturbation abstract in this paper we prove a theorem concerning the existence of solutions for a second order sweeping process with a lipschitz single valued perturbation in hilbert space

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

an extension of attouch s theorem and its application to second order epi differentiation of convexly composite functions in 1977 hedy attouch established that a sequence of closed proper convex functions epi converges to a convex function if and only if the graphs of the subdierentials converge in the mosco sense to the subdifferential of the limiting function and roughly speaking there is a condition that fixes the constant of integration we show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping in addition a constraint qualification is imposed

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

manifold identification in dual averaging for regularized stochastic online learning iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets when the objective consists of a loss function plus a nonsmooth regularization term the solution often lies on a low dimensional manifold of parameter space along which the regularizer is smooth when an l1 regularizer is used to induce sparsity in the solution for example this manifold is defined by the set of nonzero components of the parameter vector this paper shows that a regularized dual averaging algorithm can identify this manifold with high probability before reaching the solution this observation motivates an algorithmic strategy in which once an iterate is suspected of lying on an optimal or near optimal manifold we switch to a local phase that searches in this manifold thus converging rapidly to a near optimal point computational results are presented to verify the identification property and to illustrate the effectiveness of this approach

a radial basis function hermite finite difference approach to tackle cash or nothing and asset or nothing options abstract a novel local meshfree method is proposed for simulating cash or nothing and asset or nothing options at which the initial condition is discontinuous the novelty of the scheme is in the use of hermite radial basis function rbf interpolation on set of local nodes we also construct the rbf generated hermite finite difference hfd formulas and subsequently obtain their associated differentiation matrices dms the developed accuracy equipped with a shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm