a newton cg augmented lagrangian method for semidefinite programming we consider a newton cg augmented lagrangian method for solving semidefinite programming sdp problems from the perspective of approximate semismooth newton methods in order to analyze the rate of convergence of our proposed method we characterize the lipschitz continuity of the corresponding solution mapping at the origin for the inner problems we show that the positive definiteness of the generalized hessian of the objective function in these inner problems a key property for ensuring the efficiency of using an inexact semismooth newton cg method to solve the inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems numerical experiments on a variety of large scale sdp problems with the matrix dimension n up to 4 110 and the number of equality constraints m up to 2 156 544 show that the proposed method is very efficient we are also able to solve the sdp problem fap36 with n 4 110 and m 1 154 467 in the seventh dimacs implementation challenge much more accurately than in previous attempts

constructing robust good lattice rules for computational finance the valuation of many financial derivatives leads to high dimensional integrals the constructions of robust or universal good lattice rules for financial applications are both important and challenging an important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low dimensional functions i e functions that depend on only a small number of variables usually just two variables for numerical integration of such functions the quality of the low order i e low dimensional projections of the node set is crucial in this paper we propose methods to construct good lattice points with optimal low order projections the quality of a point set is measured by a new measure called elementary order ell discrepancy which measures the quality of all order ell projections and is more informative than usual measures two constructions namely the korobov and the component by component constructions are studied such that the low order projections are optimized numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order 2 projections that are better than those of the sobol points and random points and with higher order projections that are no worse while the sobol points lost the advantage over random points in order 2 projections on the average the new lattice rules have the potential to improve upon the accuracy for favorable functions while doing no harm for unfavorable ones their applications for pricing path dependent options and american options based on the least square monte carlo method are studied and their high efficiency is demonstrated a nice surprise revealed is the robustness property of such lattice rules the good projection property and the suitability for a large range of problems the potential possibility and limitations of good lattice points in achieving good quality of moderate and high order projections is investigated the reason why classical lattice rules may not be efficient for high dimensional finance problems is also discussed

semismoothness of solutions to generalized equations and the moreau yosida regularization we show that a locally lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point we present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible nonpolyhedral convex sets we prove that the semismoothness of solutions to the moreau yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function

pricing derivative securities using integrated quasi monte carlo methods with dimension reduction and discontinuity realignment the dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi monte carlo qmc method the first factor implies that the qmc method can have high performance on a problem with low effective dimension even though its nominal dimension can be very large the second factor suggests that the qmc method can be very effective on a problem that is smooth or even on a problem that is discontinuous as long as its discontinuities are parallel to the axes motivated by these findings methods such as the linear transformation lt method of j imai and k s tan j comput finance 10 2006 pp 129 155 and the orthogonal transformation ot method of x wang and k s tan management sci 59 2013 pp 376 389 have been proposed to increase the efficiency of qmc however both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the lt method enhances qmc by effective dimension reduction while the o

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

a smoothing newton method for a type of inverse semi definite quadratic programming problem we consider an inverse problem arising from the semi definite quadratic programming sdqp problem we represent this problem as a cone constrained minimization problem and its dual denoted isdqd is a semismoothly differentiable sc 1 convex programming problem with fewer variables than the original one the karush kuhn tucker conditions of the dual problem isdqd can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi definite matrices a smoothing newton method is given for getting a karush kuhn tucker point of isdqd the proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration the quadratic convergence of the smoothing newton method is proved under a suitable condition numerical experiments are reported to show that the smoothing newton method is very effective for solving this type of inverse quadratic programming problems

a quasi monte carlo approach to particle simulation of the heat equation the convergence of the monte carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi random in this paper the convergence of monte carlo particle simulation is studied when these quasi random sequences are used for the one dimensional heat equation discretized in both space and time convergence is proved for a quasi random simulation using reordering of the particles according to their position experimental results are presented for the spatially continuous heat equation in one and two dimensions the results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard monte carlo simulations for certain low dimensional problems

a quasi shrinking rectangle algorithm for complex zeros of a function presented here is a quasi shrinking rectangle algorithm that computes a complex zero of a polynomial or transcendental function this method has been implemented using several quasi random number sequences and the results of several test examples are presented the performance of this algorithm is based on a ranked comparison of relative error and time complexity versus several pseudo random number sequences

on the r superlinear convergence of the kkt residuals generated by the augmented lagrangian method for convex composite conic programming due to the possible lack of primal dual type error bounds it was not clear whether the karush kuhn tucker kkt residuals of the sequence generated by the augmented lagrangian method alm for solving convex composite conic programming cccp problems converge superlinearly in this paper we resolve this issue by establishing the r superlinear convergence of the kkt residuals generated by the alm under only a mild quadratic growth condition on the dual of cccp with easy to implement stopping criteria for the augmented lagrangian subproblems this discovery may help to explain the good numerical performance of several recently developed semismooth newton cg based alm solvers for linear and convex quadratic semidefinite programming

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

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

practical aspects of the moreau yosida regularization theoretical preliminaries when computing the infimal convolution of a convex function f with the squared norm the so called moreau yosida regularization of f is obtained among other things this function has a lipschitzian gradient we investigate some more of its properties relevant for optimization the most important part of our study concerns second order differentiability existence of a second order development of f implies that its regularization has a hessian for the converse we disclose the importance of the decomposition of bbb r n along cal u the subspace where f is smooth and cal v the subspace parallel to the subdifferential of f

strong tractability of multivariate integration using quasi monte carlo algorithms we study quasi monte carlo algorithms based on low discrepancy sequences for multivariate integration we consider the problem of how the minimal number of function evaluations needed to reduce the worst case error from its initial error by a factor of e depends on e 1 and the dimension s strong tractability means that it does not depend on s and is bounded by a polynomial in e 1 the least possible value of the power of e 1 is called the e exponent of strong tractability sloan and wozniakowski established a necessary and sufficient condition of strong tractability in weighted sobolev spaces and showed that the e exponent of strong tractability is between 1 and 2 however their proof is not constructive in this paper we prove in a constructive way that multivariate integration in some weighted sobolev spaces is strongly tractable with e exponent equal to 1 which is the best possible value under a stronger assumption than sloan and wozniakowski s assumption we show that quasi monte carlo algorithms using niederreiter s t s sequences and sobol sequences achieve the optimal convergence order o n 1 for any 0 independent of the dimension with a worst case deterministic guarantee where n is the number of function evaluations this implies that strong tractability with the best e exponent can be achieved in appropriate weighted sobolev spaces by using niederreiter s t s sequences and sobol sequences

on approximation problems with zero trace matrices abstract we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms we deal with special cases of approximation of a matrix by zero trace matrices moreover some characterizations of zero trace matrices are given by means of matrix approximation problems

a dynamically bi orthogonal method for time dependent stochastic partial differential equations i derivation and algorithms we propose a dynamically bi orthogonal method dybo to solve time dependent stochastic partial differential equations spdes the objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi orthogonal basis it is well known that the karhunen loeve expansion kle minimizes the total mean squared error and gives the sparsest representation of stochastic solutions however the computation of the kl expansion could be quite expensive since we need to form a covariance matrix and solve a large scale eigenvalue problem the main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the kl expansion unlike other reduced model methods our method constructs the reduced basis on the fly without the need to form the covariance matrix or to compute its eigendecomposition in the first part of our paper we introduce the derivation of the dynamically bi orthogonal formulation for spdes discuss several theoretical issues such as the dynamic bi orthogonality preservation and some preliminary error analysis of the dybo method we also give some numerical implementation details of the dybo methods including the representation of stochastic basis and techniques to deal with eigenvalue crossing in the second part of our paper 11 we will present an adaptive strategy to dynamically remove or add modes perform a detailed complexity analysis and discuss various generalizations of this approach an extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the dybo method

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems it is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable a classical result of ky fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions which implies that the eigenvalues are semismooth functions based on a recent result of the authors it is further proved in this paper that the eigenvalues of a symmetric matrix are strongly semismooth everywhere as an application it is demonstrated how this result can be used to analyze the quadratic convergence of newton s method for solving inverse eigenvalue problems ieps and generalized ieps with multiple eigenvalues

inverse semidefinite quadratic programming problem with l1 norm measure abstract we consider an inverse problem arising from a semidefinite quadratic programming sdqp problem which is a minimization problem involving l 1 vector norm with positive semidefinite cone constraint by using convex optimization theory the first order optimality condition of the problem can be formulated as a semismooth equation under two assumptions we prove that any element of the generalized jacobian of the equation at its solution is nonsingular based on this a smoothing approximation operator is given and a smoothing newton method is proposed for solving the solution of the semismooth equation we need to compute the directional derivative of the smoothing operator at the corresponding point and to solve one linear system per iteration in the newton method and its global convergence is demonstrated finally we give the numerical results to show the effectiveness and stability of the smoothing newton method for this inverse problem

the asymptotic efficiency of randomized nets for quadrature an l 2 type discrepancy arises in the average and worst case error analyses for multidimensional quadrature rules this discrepancy is uniquely defined by k x y which serves as the covariance kernel for the space of random functions in the average case analysis and a reproducing kernel for the space of functions in the worst case analysis this article investigates the asymptotic order of the root mean square discrepancy for randomized 0 m s nets in base b for moderately smooth k x y the discrepancy is o n 1 log n s 1 2 and for k x y with greater smoothness the discrepancy is o n 3 2 log n s 1 2 where n b m is the number of points in the net numerical experiments indicate that the t m s nets of faure niederreiter and sobol do not necessarily attain the higher order of decay for sufficiently smooth kernels however niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions

relating homogeneous cones and positive definite cones via t algebras t algebras are nonassociative algebras defined by vinberg in the early 1960s for the purpose of studying homogeneous cones vinberg defined a cone k cal a for each t algebra cal a and proved that every homogeneous cone is isomorphic to one such k cal a we relate each t algebra cal a with a space of linear operators in such a way that k cal a is isomorphic to the cone of positive definite self adjoint operators together with vinberg s result we conclude that every homogeneous cone is isomorphic to a slice of a cone of positive definite matrices

ectropy of diversity measures for populations in euclidean space measures to evaluate the diversity of a set of points population in euclidean space play an important role in a variety of areas of science and engineering well known measures are often used without a clear insight into their quality and many of them do not appropriately penalize populations with a few distant groups of collocated or closely located points to the best of our knowledge there is a lack of rigorous criteria to compare diversity measures and help select an appropriate one in this work we define a mathematical notion of ectropy for classifying diversity measures in terms of the extent to which they tend to penalize point collocation we investigate the advantages and disadvantages of several known measures and we propose some novel ones that exhibit a good ectropic behavior in particular we introduce a quasi entropy measure based on a geometric covering problem three measures based on discrepancy from uniform distribution and one based on euclidean minimum spanning trees all considered measures are tested and compared on a large set of random and structured populations special attention is also devoted to the complexity of computing the measures most of the novel measures compare favorably with the classical ones in terms of ectropy the measure based on euclidean minimum spanning trees turns out to be the most promising one in terms of the tradeoff between the ectropic behavior and the computational complexity

quasi random initial population for genetic algorithms the selection of the initial population in a population based heuristic optimizationmethod is important since it affects the search for several iterations and often has an influence on the final solution if no a priori information about the optima is available the initial population is often selected randomly using pseudorandom numbers usually however it is more important that the points are as evenly distributed as possible than that they imitate random points in this paper we study the use of quasi random sequences in the initial population of a genetic algorithm sample points in a quasi random sequence are designed to have good distribution properties here a modified genetic algorithm using quasi random sequences in the initial population is tested by solving a large number of continuous benchmark problems from the literature the numerical results of two implementations of genetic algorithms using different quasi random sequences are compared to those of a traditional implementation using pseudorandom numbers the results obtained are promising

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

conditional quasi monte carlo methods and dimension reduction for option pricing and hedging with discontinuous functions abstract quasi monte carlo qmc methods are efficient simulation tools for multidimensional integrations in financial engineering discontinuous functions occur commonly in pricing or hedging financial options high dimensionality and the lack of smoothness may cause an adverse impact on the performance of qmc in this paper we develop a two step procedure to handle the discontinuities and to reduce the effective dimensions successively the first step is to smooth the discontinuous functions by using conditional qmc methods which are deterministic versions of conditional monte carlo methods we show how the discontinuities can be removed completely and how the required conditional expectations can be calculated analytically for some typical pricing and hedging problems the second step is to reduce the effective dimensions of the resulting functions by using a new dimension reduction method based on principal component analysis pca rather than aiming at decomposing the covariance matrix of the brownian motion as in the traditional pca the new dimension reduction method implements pca on the gradients of the smoothed functions at a number of points in order to learn the inherent structure information a suitable orthogonal transformation is obtained to reduce the effective dimensions numerical experiments show that the two step procedure significantly improves qmc methods in pricing or hedging options with discontinuous functions

quasi monte carlo methods in financial engineering an equivalence principle and dimension reduction quasi monte carlo qmc methods are playing an increasingly important role in the pricing of complex financial derivatives for models in which the prices of the underlying assets are driven by brownian motions the performance of qmc methods is known to depend crucially on the construction of brownian motions this paper focuses on the impact of various constructions although the brownian bridge bb construction often yields very good results as papageorgiou pointed out there are financial derivatives for which the bb construction performs badly papageorgiou a 2002 the brownian bridge does not offer a consistent advantage in quasi monte carlo integration j complexity18 1 171 186 in this paper we first extend papageorgiou s analysis to establish an equivalence principle if the bb construction or any other construction is the preferred construction for a particular financial derivative then for any other construction there is another financial derivative for which the latter construction is the preferred one in this sense all methods of construction are equivalent and no method is consistently superior to others it all depends on the particular financial derivative we then show how to find a good construction for a particular class of financial derivatives in practice our strategy is to find a good construction for an easy problem and then apply it to more complicated problems related to the easy one this strategy is applied to the arithmetic asian options including bermudan asian options based on the weighted average of the stock prices we do this by studying a simpler problem namely the geometric asian option for which the best construction is easily available and applying it to the arithmetic asian option numerical experiments confirm the success of this strategy whereas in qmc all the common methods the standard method bb and principal component analysis may lose their power in some situations the new method behaves very well in all cases further large variance reduction can be achieved in combination with a control variate the new method can be interpreted as a practical way of reducing the effective dimension for some class of functions

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

the k steiner ratio in graphs a steiner minimum tree smt is the shortest length tree in a metric space interconnecting a set of points called the regular points possibly using additional vertices a k size steiner minimum tree ksmt is one that can be split into components where all regular points are leaves and all components have at most k leaves the k steiner ratio rho k is the infimum of the ratios smt ksmt over all finite sets of regular points in all possible metric spaces where the distances are given by a complete graph previously only rho 2 and rho 3 were known exactly in graphs and some bounds were known for other values of k in this paper we determine rho k exactly for all k from this we prove a better approximation ratio for the steiner tree problem in graphs

a semismooth newton cg based dual ppa for matrix spectral norm approximation problems we consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints these problems arise often in numerical algebra engineering and other areas such as finding chebyshev polynomials of matrices and fastest mixing markov chain models based on classical analysis of proximal point algorithms ppas and recent developments on semismooth analysis of nonseparable spectral operators we propose a semismooth newton cg based dual ppa for solving the matrix norm approximation problems furthermore when the primal constraint nondegeneracy condition holds for the subproblems our semismooth newton cg method is proven to have at least a superlinear convergence rate we also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers admm the results show that our algorithm which is warm started with an initial point obtained from the admm substantially outperforms the pure admm especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy

constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming it is known that the karush kuhn tucker kkt conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices we show in this paper that the primal and dual constraint nondegeneracies the strong regularity the nonsingularity of the b subdifferential of this nonsmooth system and the nonsingularity of the corresponding clarke s generalized jacobian at a kkt point are all equivalent moreover we prove the equivalence between each of these conditions and the nonsingularity of clarke s generalized jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing newton methods in particular we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies but without the strict complementarity

a note on optimal point distributions in 0 1 s in this note we determine the infimum of the l 2 the star and the extreme discrepancy taken over all 2 element point sets in the s dimensional unit cube moreover we give very good bounds on the infimum of the isotropic discrepancy taken over all s 1 element point sets in 0 1 s

on the equivalence between kernel quadrature rules and random feature expansions we show that kernel based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels for a particular decomposition that always exists for such kernels we provide a theoretical analysis of the number of required samples for a given approximation error leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms in particular we show that the upper bound may be obtained from independent and identically distributed samples from a specific non uniform distribution while the lower bound if valid for any set of points applying our results to kernel based quadrature while our results are fairly general we recover known upper and lower bounds for the special cases of sobolev spaces moreover our results extend to the more general problem of full function approximations beyond simply computing an integral with results in l2 and l norm that match known results for special cases applying our results to random features we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with lipshitz continuous losses

second harmonics effects in random duffing oscillators we consider a stochastic model for duffing oscillators where the nonlinearity and the randomness are scaled in such a way that they interact strongly a typical feature is the appearance of second harmonics effects an asymptotic statistical analysis for these oscillators is performed in the diffusion limit when a suitable absorbing boundary condition is imposed according to the underlying physical problem the related fokker planck equation has been numerically solved to obtain the first two moments of the oscillator s displacement from its rest position dependence on the nonlinearity strength and on the location of the absorbing boundary has also been investigated such results have been compared with those computed solving the corresponding stochastic ito differential equations by a monte carlo method where quasi random sequences of numbers have been efficiently used

properties of the moreau yosida regularization of a piecewise c2 convex function in this paper we discuss second order properties of the moreau yosida regularization f of a piecewise twice continuously differentiable convex function f we introduce a new constraint qualification in order to prove that the gradient of f is piecewise continuously differentiable in addition we discuss conditions depending on the hessians of the pieces that guarantee positive definiteness of the generalized jacobians of the gradient of f

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

a globally and superlinearly convergent algorithm for nonsmooth convex minimization it is well known that a possibly nondifferentiable convex minimization problem can be transformed into a differentiable convex minimization problem by way of the moreau yosida regularization this paper presents a globally convergent algorithm that is designed to solve the latter problem under additional semismoothness and regularity assumptions the proposed algorithm is shown to have a q superlinear rate of convergence

path generation for quasi monte carlo simulation of mortgage backed securities monte carlo simulation is playing an increasingly important role in the pricing and hedging of complex path dependent financial instruments low discrepancy simulation methods offer the potential to provide faster rates of convergence than those of standard monte carlo methods however in high dimensional problems special methods are required to ensure that the faster convergence rates hold indeed ninomiya and tezuka 1996 have shown highdimensional examples in which low discrepancy methods perform worse than monte carlo methods the principal component construction introduced by acworth et al 1998 provides one solution to this problem however the computational effort required to generate each path grows quadratically with the dimension of the problem this article presents two new methods that offer accuracy equivalent in terms of explained variability to the principal components construction with computational requirements that are linearly related to the problem dimension one method is to take into account knowledge about the payoff function which makes it more flexible than the brownian bridge construction numerical results are presented that show the benefits of such adjustments the different methods are compared for the case of pricing mortgage backed securities using the hull white term structure model

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

low discrepancy sequences in high dimensions how well are their projections distributed quasi monte carlo qmc methods have been successfully used to compute high dimensional integrals arising in many applications especially in finance to understand the success and the potential limitation of qmc this paper focuses on quality measures of point sets in high dimensions we introduce the email protected superposition and truncation discrepancies which measure the quality of selected projections of a point set on lower dimensional spaces these measures are more informative than the classical ones we study their relationships with the integration errors and study the tractability issues we present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the sobol nets with that of the sets of latin hypercube sampling and random points numerical results show that in high dimensions the superiority of the sobol nets mainly derives from the one dimensional projections and the projections associated with the earlier dimensions for order 2 and higher order projections all these point sets have similar behavior on the average in weighted cases with fast decaying weights the sobol nets have a better performance than the other two point sets the investigation enables us to better understand the properties of qmc and throws new light on when and why qmc can have a better or no better performance than monte carlo for multivariate integration in high dimensions

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

hybrid deterministic stochastic methods for data fitting many structured data fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum these methods can make great progress initially but often slow as they approach a solution in contrast full gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration we explore hybrid methods that exhibit the benefits of both approaches rate of convergence analysis shows that by controlling the sample size in an incremental gradient algorithm it is possible to maintain the steady convergence rates of full gradient methods we detail a practical quasi newton implementation based on this approach numerical experiments illustrate its potential benefits

probabilistically induced domain decomposition methods for elliptic boundary value problems monte carlo as well as quasi monte carlo methods are used to generate only few interfacial values in two dimensional domains where boundary value elliptic problems are formulated this allows for a domain decomposition of the domain a continuous approximation of the solution is obtained interpolating on such interfaces and then used as boundary data to split the original problem into fully decoupled subproblems the numerical treatment can then be continued implementing any deterministic algorithm on each subdomain both monte carlo or quasi monte carlo simulations and the domain decomposition strategy allow for exploiting parallel architectures scalability and natural fault tolerance are peculiarities of the present algorithm examples concern helmholtz and poisson equations whose probabilistic treatment presents additional complications with respect to the case of homogeneous elliptic problems without any potential term and source

domain decomposition solution of elliptic boundary value problems via monte carlo and quasi monte carlo methods domain decomposition of two dimensional domains on which boundary value elliptic problems are formulated is accomplished by probabilistic monte carlo as well as by quasi monte carlo methods generating only a few interfacial values and interpolating on them continuous approximations for the trace of solution are thus obtained to be used as boundary data for the subproblems the numerical treatment can then proceed by standard deterministic algorithms separately in each of the so obtained subdomains monte carlo and quasi monte carlo simulations may naturally exploit multiprocessor architectures leading to parallel computing as well as the ensuing domain decomposition does the advantages such as scalability obtained by increasing the number of processors are shown both theoretically and experimentally in a number of test examples and the possibility of using clusters of computers grid computing is emphasized

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

approximating the selected internal steiner tree in this paper we consider a variant of the well known steiner tree problem given a complete graph g v e with a cost function c e r and two subsets r and r satisfying r email protected v a selected internal steiner tree is a steiner tree which contains or spans all the vertices in r such that each vertex in r cannot be a leaf the selected internal steiner tree problem is to find a selected internal steiner tree with the minimum cost in this paper we present a email protected algorithm for the problem where r is the best known approximation ratio for the steiner tree problem

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

construction algorithms for digital nets with low weighted star discrepancy we introduce a new construction method for digital nets which yield point sets in the s dimensional unit cube with low star discrepancy the digital nets are constructed using polynomials over finite fields it has long been known that there exist polynomials which yield point sets with low unweighted star discrepancy this result was obtained by niederreiter by the means of averaging over all polynomials hence concrete examples of good polynomials were not known in many cases here we show that good polynomials can be found by computer search the search algorithm introduced in this paper is based on minimizing a quantity closely related to the star discrepancy it has been pointed out that many integration problems can be modeled by weighted function spaces and it has been shown that in this case point sets with low weighted discrepancy are required hence it is particularly useful to be able to adjust a point set to some given weights we are able to extend our results from the unweighted case to show that this can be done using our construction algorithms this way we can find point sets with low weighted star discrepancy making such point sets especially useful for many applications

good permutations for deterministic scrambled halton sequences in terms of l2 discrepancy one of the best known low discrepancy sequences used by many practitioners is the halton sequence unfortunately there seems to exist quite some correlation between the points from the higher dimensions a possible solution to this problem is the so called scrambling in this paper we give an overview of known scrambling methods and we propose a new way of scrambling which gives good results compared to the others in terms of l 2 discrepancy on top of that our new scrambling method is very easy to implement

generating quasi random paths for stochastic processes the need to simulate stochastic processes numerically arises in many fields frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness this paper addresses the question of how to use quasi monte carlo methods to improve this simulation special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps is required two such techniques the generalized brownian bridge and particle reordering are described here these methods are applied to a problem from finance the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate when expressed as an integral this problem is nominally 360 dimensional the analysis of the integrand presented here explains the effectiveness of the quasi random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi random sequences

the error bounds and tractability of quasi monte carlo algorithms in infinite dimension dimensionally unbounded problems are frequently encountered in practice such as in simulations of stochastic processes in particle and light transport problems and in the problems of mathematical finance this paper considers quasi monte carlo integration algorithms for weighted classes of functions of infinitely many variables in which the dependence of functions on successive variables is increasingly limited the dependence is modeled by a sequence of weights the integrands belong to rather general reproducing kernel hilbert spaces that can be decomposed as the direct sum of a series of their subspaces each subspace containing functions of only a finite number of variables the theory of reproducing kernels is used to derive a quadrature error bound which is the product of two terms the generalized discrepancy and the generalized variation tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by ce p for some exponent p and some positive constant c the e exponent of tractability is defined as the smallest power of e 1 in these bounds it is shown by using monte carlo quadrature that the e exponent is no greater than 2 for these weighted classes of integrands under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the halton sequence that the e exponent of tractability is 1 which implies that infinite dimensional integration is no harder than one dimensional integration

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

solving large scale semidefinite programs via an iterative solver on the augmented systems the search directions in an interior point method for large scale semidefinite programming sdp can be computed by applying a krylov iterative method to either the schur complement equation sce or the augmented equation both methods suffer from slow convergence as interior point iterates approach optimality numerical experiments have shown that a diagonally preconditioned conjugate residual method on the sce typically takes a huge number of steps to converge however it is difficult to incorporate cheap and effective preconditioners into the sce this paper proposes to apply the preconditioned symmetric quasi minimal residual psqmr method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior point iterates numerical experiments on sdp problems arising from maximum clique and selected sdplib sdp library problems show that moderately accurate solutions can be obtained with a modest number of psqmr steps using the proposed preconditioned reduced augmented equation an sdp problem with 127600 constraints is solved in about 6 5 hours to an accuracy of 10 6 in relative duality gap

largest j simplices in d cubes some relatives of the hadamard maximum determinant problem abstract this paper studies the computationally difficult problem of finding a largest j dimensional simplex in a given d dimensional cube the case in which j d is of special interest for it is equivalent to the hadamard maximum determinant problem it has been solved for infinitely many values of d but not for d 14 the subcase in which j d 3 mod 4 subsumes the famous problem on the existence of hadamard matrices the known results for the case j d are here summarized and used but the main focus is on fixed small values of j when j 1 the problem is trivial and when j 2 or j 3 it is here solved completely i e for all d beyond that the results are fragmentary but numerous and they lead to several attractive conjectures some other problems involving simplices in cubes are mentioned and the relationship of largest simplices to d optimal weighing designs is discussed

a generalization of the proximal point algorithm the problem considered in this paper is to find a solution to the generalized equation 0 in t x y where t is a maximal monotone operator on the product h 1 times h 2 of two hilbert spaces h 1 and h 2 we give a generalization of the proximal map and the proximal point algorithm in which the proposed iterative procedure is based on just one variable applying to convex programming problems instead of adding a quadratic term for all variables as in the proximal point algorithm a quadratic term for a subset of variables is added this paper proves that under a mild assumption our algorithm has the same convergence properties as the regular proximal point algorithm

an implementable proximal point algorithmic framework for nuclear norm minimization the nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints such a problem often arises from the convex relaxation of a rank minimization problem with noisy data and arises in many fields of engineering and science in this paper we study inexact proximal point algorithms in the primal dual and primal dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints we design efficient implementations of these algorithms and present comprehensive convergence results in particular we investigate the performance of our proposed algorithms in which the inner sub problems are approximately solved by the gradient projection method or the accelerated proximal gradient method our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state of the art algorithms interestingly our proposed algorithms are connected with other algorithms that have been studied in the literature

an augmented lagrangian dual optimization approach to the h weighted model updating problem abstract model updating for the quadratic eigenvalue problem aims to update the model q 2 m c k by given eigendata in this paper we consider the h weighted model updating problem which can not only preserve the symmetry and definiteness of the original model but also express our confidence in the original model through assigning different confidence weights we propose an augmented lagrangian dual method for the h weighted model updating problem under some mild assumptions our method is shown to converge at least linearly numerical results illustrate the effectiveness of our method in addition we compare our method with the semi definite programming sdp method numerical results illustrate that when the scale of the model becomes large our method still works but the sdp method failed to converge

spectral operators of matrices the class of matrix optimization problems mops has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important ap plications involving structured low rank matrices this trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing the lowner operator which gen erates a matrix valued function by applying a single variable function to each of the singular values of a matrix has played an important role for a long time in solving matrix optimization problems however the classical theory developed for lowner operators has become inadequate in these recent applications the main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the mop this goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions coined as spectral operators of matrices several fundamental properties of spectral operators including the well definedness continuity di rectional differentiability frechet differentiability locally lipschitzian continuity order b ouligand differentiability 0 1 order g semismooth 0 1 and the characterization of clarke s generalized jacobian are systematically studied

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

a dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure the inverse quadratic eigenvalue problem iqep arises in the field of structural dynamics it aims to find three symmetric matrices known as the mass the damping and the stiffness matrices such that they are closest to the given analytical matrices and satisfy the measured data the difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite based on an equivalent dual optimization version of the iqep we present a quadratically convergent newton type method our numerical experiments confirm the high efficiency of the proposed method

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

quasi random sequences and their discrepancies quasi random also called low discrepancy sequences are a deterministic alternative to random sequences for use in monte carlo methods such as integration and particle simulations of transport processes the error in uniformity for such a sequence of n points in the s dimensional unit cube is measured by its discrepancy which is of size log n s n 1 for large n as opposed to discrepancy of size log log n 1 2 n 1 2 for a random sequence i e for almost any randomly chosen sequence several types of discrepancies one of which is new are defined and analyzed a critical discussion of the theoretical bounds on these discrepancies is presented computations of discrepancies are presented for a wide choice of dimension s number of points n and different quasi random sequences in particular for moderate or large s there is an intermediate regime in which the discrepancy of a quasi random sequence is almost exactly the same as that of a randomly chosen sequence a simplified p

constructing sobol sequences with better two dimensional projections direction numbers for generating sobol sequences that satisfy the so called property a in up to 1111 dimensions have previously been given in joe and kuo acm trans math software 29 2003 pp 49 57 however these sobol sequences may have poor two dimensional projections here we provide a new set of direction numbers alleviating this problem these are obtained by treating sobol sequences in d dimensions as t d sequences and then optimizing the t values of the two dimensional projections our target dimension is 21201

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

minimizing the l 2 and l star discrepancies of a single point in the unit hypercube consider a point set that contains a single point x in the s dimensional hypercube we show where to place x in order to minimize the l2 and l star discrepancies measures often used in quasi monte carlo integration

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

acceleration of quasi monte carlo approximations with applications in mathematical finance a procedure to accelerate the convergence of quasi monte carlo approximations is described and investigated the procedure applies a weighted least squares smoothing method to improve the approximations that were obtained using low discrepancy sequences the acceleration procedure is tested computationally on several numerical integration problems including two problems of interest in mathematical finance the results indicate that when halton low discrepancy points are used in the quasi monte carlo method the accelerated sequence often converges a factor of five or more faster than the original quasi monte carlo approximations

convergence of newton s method for convex best interpolation in this paper we consider the problem of finding a convex function which interpolates given points and has a minimal l2 norm of the second derivative this problem reduces to a system of equations involving semismooth functions we study a newton type method utilizing clarke s generalized jacobian and prove that its local convergence is superlinear for a special choice of a matrix in the generalized jacobian we obtain the newton method proposed by irvine et al and settle the question of its convergence by using a line search strategy we present a global extension of the newton method considered the efficiency of the proposed global strategy is confirmed with numerical experiments

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

the full steiner tree problem motivated by the reconstruction of phylogenetic tree in biology we study the full steiner tree problem in this paper given a complete graph g v e with a length function on e and a proper subset r v the problem is to find a full steiner tree of minimum length in g which is a kind of steiner tree with all the vertices of r as its leaves in this paper we show that this problem is np complete and max snp hard even when the lengths of the edges are restricted to either 1 or 2 for the instances with lengths either 1 or 2 we give a 8 5 approximation algorithm to find an approximate solution for the problem