constraint qualifications in quasidifferentiable optimization the classical linearization procedure for differentiable nonlinear programming problems can be naturally generalized to the quasidifferentiable case as in the classical case one has to impose so called constraint qualifications on the constraint functions in order to ensure that optimality of a feasible point implies optimality of the nullvector for the corresponding quasilinearized problem we present various constraint qualifications in a unified setting propose a new one and investigate the relations between these conditions

pricing and hedging with discontinuous functions quasi monte carlo methods and dimension reduction quasi monte carlo qmc methods are important numerical tools in the pricing and hedging of complex financial instruments the effectiveness of qmc methods crucially depends on the discontinuity and the dimension of the problem this paper shows how the two fundamental limitations can be overcome in some cases we first study how path generation methods pgms affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of qmc methods the insight is that the discontinuities can be qmc friendly i e aligned with the coordinate axes or not depending on the pgm the pgms that offer the best performance in qmc methods are those that make the discontinuities qmc friendly the structure of discontinuities can affect the accuracy of qmc methods more significantly than the effective dimension this insight motivates us to propose a novel way of handling the discontinuities the basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in qmc methods for pricing options and for estimating greeks it also reduces the effective dimension of the problem this paper was accepted by assaf zeevi stochastic models and simulation

fixed point theorems and morse s lemma for lipschitzian functions abstract we prove a fixed point theorem for set valued mappings defined on a nonempty compact subset x of r n which can be represented by inequality constraints i e x x r n f x 0 f locally lipschitzian and satisfying a nondegeneracy assumption outside of x this class of sets extends significantly the class of convex compact sets with a nonempty interior topological properties of such sets x are proved continuous deformation retract of a ball acyclicity as a consequence of a generalization of morse s lemma for lipschitzian real valued function defined on r n a result also of interest for itself

quasi monte carlo methods for linear two stage stochastic programming problems quasi monte carlo qmc algorithms are studied for generating scenarios to solve two stage linear stochastic programming problems their integrands are piecewise linear quadratic but do not belong to the function spaces considered for qmc error analysis we show that under some weak geometric condition on the two stage model all terms of their anova decomposition except the one of highest order are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to l 2 l2 this implies that randomly shifted lattice rules may achieve the optimal rate of convergence o n 1 delta o n 1 with delta in 0 frac 1 2 0 12 and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two the geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution is normal we discuss effective dimensions and techniques for dimension reduction numerical experiments for a production planning model with normal inputs show that indeed convergence rates close to the optimal rate are achieved when using randomly shifted lattice rules or scrambled sobol point sets accompanied with principal component analysis for dimension reduction

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

estimating mean dimensionality of analysis of variance decompositions analysis of variance anova is now often applied to functions defined on the unit cube where it serves as a tool for the exploratory analysis of functions the mean dimension of a function defined as a natural weighted combination of its anova mean squares provides one measure of how hard or easy it is to integrate the function by quasi monte carlo sampling this article presents some new identities relating the mean dimension and some analogously defined higher moments to the variance importance measures of i m sobol as a result we are able to measure the mean dimension of certain functions arising in computational finance we produce an unbiased and nonnegative estimate of the variance contribution of the highest order interaction that avoids the cancellation problems of previous estimates in an application to extreme value theory we find that among other things the minimum of d independent u 0 1 random variables has a mean dimension of 2 d 1 d 3

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

quasi monte carlo methods for two stage stochastic mixed integer programs we consider randomized qmc methods for approximating the expected recourse in two stage stochastic optimization problems containing mixed integer decisions in the second stage it is known that the second stage optimal value function is piecewise linear quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets this structure is exploited to provide conditions implying that first and higher order terms of the integrand s anova decomposition math comp 79 2010 953 966 have mixed weak first order partial derivatives this leads to a good smooth approximation of the integrand and hence to good convergence rates of randomized qmc methods if the effective superposition dimension is low

morse theory for some lower c 2 functions in finite dimension using inf regularization methods we prove that morse inequalities hold for some lower c2 functions for this purpose we first recall some properties of the class of lower c2 functions and of their moreau yosida approximations then we establish under some qualification conditions on the critical points that it is possible to define a morse index for a lower c2 functionf this index is preserved by the moreau yosida approximation process we prove in particular that the moreau yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space in a last step morse inequalities are written for moreau yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower c2 functions

variance bounds and existence results for randomly shifted lattice rules we study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions we consider integrands that are square integrable but whose fourier series are not necessarily absolutely convergent for such integrands a bound on the variance is expressed through a certain type of weighted discrepancy we prove existence and construction results for randomly shifted lattice rules such that the variance bounds are almost o n a where n is the number of function evaluations and a 1 depends on our assumptions on the convergence speed of the fourier coefficients these results hold for general weights arbitrary n and any dimension with additional conditions on the weights we obtain a convergence that holds uniformly in the dimension and this provides sufficient conditions for strong tractability of the integration problem we also show that lattice rules that satisfy these bounds are not difficult to construct explicitly and we provide numerical illustrations of the behaviour of construction algorithms

the smoothing effect of integration in r d and the anova decomposition this paper studies the anova decomposition of a d variate function f dened on the whole of r d where f is the maximum of a smooth function and zero or f could be the absolute value of a smooth function our study is motivated by option pricing problems we show that under suitable conditions all terms of the anova decomposition except the one of highest order can have unlimited smoothness in particular this is the case for arithmetic asian options with both the standard and brownian bridge constructions of the brownian motion 2010 mathematics subject classication primary 41a63 41a99 secondary 65d30

homeomorphism conditions for coherently oriented piecewise affine mappings this article is mainly concerned with the homeomorphism problem for piecewise affine mappings pa maps i e mappings which coincide with an affine mapping on each polyhedron of some finite polyhedral subdivision of rn in the first part we prove that a pa map can be defined without referring to a subdivision of rn as a continuous mapping which coincides at every point x rn with at least one function from a finite collection of affine functions the second part studies the recession function of a pa map it is shown that the recession function is piecewise linear and that a coherently oriented pa map is a homeomorphism if and only if its recession function is a homeomorphism in the last part we prove that a coherently oriented pa map is a homeomorphism if it admits a corresponding polyhedral subdivision of rn such that for some number k 2 n every face of codimension k is contained in at most 2k polyhedra provided the subdivision contains at least one face of codimension k

minimal conjunctive normal expression of continuous piecewise affine functions continuous piecewise affine pwa functions arise in many aspects of control for this kind of function we propose the minimal conjunctive normal expression cne the cne can be expressed as the minimum of a collection of terms each of which is the maximum of a set of affine functions the minimal cne is defined to contain the smallest number of parameters analogous to boolean algebra we propose implicants and prime implicants for continuous pwa functions after obtaining all prime implicants the problem of finding minimal cnes can then be cast as a binary programming problem a sharp bound on the number of boolean variables in the binary programming problem is given in two worked examples minimal cnes are derived for given continuous pwa functions

quasi monte carlo integration over ℝ in this paper we show that a wide class of integrals over r d with a probability weight function can be evaluated using a quasi monte carlo algorithm based on a proper decomposition of the domain r d and arranging low discrepancy points over a series of hierarchical hypercubes for certain classes of power exponential decaying weights the algorithm is of optimal order

tensor analysis of anova decomposition abstract the analogy between anova for general n factor crossed layouts and the anova type decomposition of square integrable statistics is demonstrated using the notions and notations of tensor analysis and multilinear algebra a theory of tensors is developed in such a way that a it can be immediately applied in computer programs and b it can be easily generalized to l 2 spaces

local topological equivalence of nonsmooth functions in hilbert spaces in this paper we investigate the relation between nonsmooth functions with domain in a hilbert space and their local approximations we consider lipschitz functions and define an approximation model with directional derivatives the qualitative behaviour of the approximation is studied by means of the concept of topological equivalence using this concept we establish the existence of a local coordinate transformation between the original function and the local approximation

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

irredundant lattice representations of continuous piecewise affine functions in this paper we revisit the full lattice representation of continuous piecewise affine pwa functions and give a formal proof of its representation ability based on this we derive the irredundant lattice pwa representations through removal of redundant terms and literals necessary and sufficient conditions for irredundancy are proposed besides we explain how to remove terms and literals in order to ensure irredundancy an algorithm is given to obtain an irredundant lattice pwa representation in the worked examples the irredundant lattice pwa representations are used to express the optimal solution of explicit model predictive control problems and the results turn out to be much more compact than those given by a state of the art algorithm

optimal randomized multilevel algorithms for infinite dimensional integration on function spaces with anova type decomposition in this paper we consider the infinite dimensional integration problem on weighted reproducing kernel hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type the weights model the relative importance of different groups of variables we present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error furthermore we provide in this setting the first nontrivial lower error bounds for general randomized algorithms which in particular may be adaptive or nonlinear these lower bounds show that our multilevel algorithms are optimal our analysis refines and extends the analysis provided in f j hickernell t muller gronbach b niu and k ritter j complexity 26 2010 pp 229 254 and our error bounds improve substantially on the error bounds presented there as an illustrative example we discuss the unanchored sobolev space and employ randomized quasi monte carlo multilevel a

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

sparse pseudospectral approximation method multivariate global polynomial approximations such as polynomial chaos or stochastic collocation methods are now in widespread use for sensitivity analysis and uncertainty quantification the pseudospectral variety of these methods uses a numerical integration rule to approximate the fourier type coefficients of a truncated expansion in orthogonal polynomials for problems in more than two or three dimensions a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation however when using a sparse rule to approximately integrate these coefficients one often finds unacceptable errors in the coefficients associated with higher degree polynomials by reexamining smolyak s algorithm and exploiting the connections between interpolation and projection in tensor product spaces we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule the compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation

quasi monte carlo integration the standard monte carlo approach to evaluating multidimensional integrals using pseudo random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement as an alternative to the random methods it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo random sequences with more uniformly distributed sequences known as quasi random in this paper quasi random halton sobol and faure and pseudo random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand including variance variation smoothness and dimension the results show that variation which plays an important role in the theoretical upper bound given by the koksma hlawka inequality does not affect convergence while variance the determining factor in random monte carlo is shown to provide a rough upper bound but does not accurately predict performance in general quasi monte carlo methods are superior to random monte carlo but the advantage may be slight particularly in high dimensions or for integrands that are not smooth for discontinuous integrands we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi monte carlo is only slightly larger than 1 2 in high dimensions more 21 refs 6 figs 5 tabs less

handling discontinuities in financial engineering good path simulation and smoothing discontinuities are common in the pricing and hedging of complex financial derivatives quasi monte carlo qmc methods for high dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality interestingly path simulation method psm may affect both factors implying its significance in qmc methods what defines a good psm for problems with discontinuities the ability to align the discontinuities with the coordinate axes is a desirable property for a psm we show that for an arbitrary psm there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis parallel discontinuities for which good qmc performance can be expected in this sense any psm can be good in qmc methods for a specific class of problems we analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness up to a permutation of the standard co

sensitivity analysis of composite piecewise smooth equations this paper is a contribution to the sensitivity analysis of piecewise smooth equations a piecewise smooth function is a lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible b derivative at this point we emphasise the role of functions of the typef g h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf g tilde h is a lipschitzian homeomorphism near a given point for every tilde h sufficiently close toh with respect to thec1 topology

randomized dimension reduction for monte carlo simulations we present a new unbiased algorithm that estimates the expected value of f u via monte carlo simulation where u is a vector of d independent random variables and f is a function of d variables

on the qualitative approximation of lipschitzian functions abstract problem of topological equivalence between a function and certain local approximations is studied the study is carried out in a neighbourhood of a critical point with the concept of critical point of clarke s theory the function belongs to a particular class of non b differentiable functions the local approximations are positively homogeneous maps using the concept of topological equivalence we establish the existence of a local coordinate transformation between the original function and the positively homogeneous function as a consequence we obtain sufficient conditions for the existence of local extremes for the initial map

on decompositions of multivariate functions we present formulas that allow us to decompose a function f of d variables into a sum of 2 d terms fu indexed by subsets u of f1 dg where each term fu depends only on the variables with indices in u the decomposition depends on the choice of d commuting projections fpjg d 1 where pj f does not depend on the variable xj we present an explicit formula for fu which is new even for the anova and anchored decompositions both are special cases of the general decomposition we show that the decomposition is minimal in the following sense if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z then for every choice of fpjg d 1 the terms fu 0 for all subsets u containing z furthermore in a reproducing kernel hilbert space setting we give su cient conditions for the terms fu to be mutually orthogonal

on the universal mod p supersingular quotients for gl2 f over f p for a general f qp abstract let f q p be a finite extension we explore the universal supersingular mod p representations of gl 2 f by computing a basis for their spaces of invariants under the pro p iwahori subgroup this generalizes works of breuil and schein from q p and the totally ramified cases to an arbitrary extension f q p using these results we then construct for an unramified f q p quotients of the universal supersingular modules which have as quotients all the supersingular representations of gl 2 f with a gl 2 o f socle that is expected to appear in the mod p local langlands correspondence a construction in the case of an extension of q p with inertia degree 2 and suitable ramification index is also presented