Lipschitz Continuous Derivative Research Articles

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p∈[1,∞) a transformed Milstein-type scheme reaches an Lp-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

Large and moderate deviation principles for path-distribution-dependent stochastic differential equations

In this paper, by weak convergence method, large and moderate deviation principles are established for path-distribution-dependent stochastic differential equations. To prove large deviation principle, both of the drift and diffusion coefficients are required to be Lipschitz continuous in the space variable as well as the distribution term, uniformly with respect to the time parameter $ t $. In further, to establish the moderate deviation principle, the drift coefficient is assumed to be Frechet differentiable with Lipschitz continuous derivative in the space variable.

Self-adjointness of Toeplitz operators on the Segal-Bargmann space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.

Local convergence of tensor methods

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.

A strong order 3/4 method for SDEs with discontinuous drift coefficient

Abstract In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently, it has been shown in Müller-Gronbach & Yaroslavtseva (2020, On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. Henri Poincaré Probab. Stat., 56, 1162–1178) that for scalar SDEs with a piecewise Lipschitz drift coefficient, and a Lipschitz diffusion coefficient that is nonzero at the discontinuity points of the drift coefficient the classical Euler–Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty )$. Up to now this was the best $L_p$-error rate available in the literature for equations of that type. In the present paper we construct a method based on finitely many evaluations of the driving Brownian motion that even achieves an $L_p$-error rate of at least $3/4$ for all $p\in [1,\infty )$ under additional piecewise smoothness assumptions on the coefficients. This is the first higher order method known in the literature for SDEs of that type. We add that the $L_p$-error rate $3/4$ cannot be improved in general. To obtain the upper error bound for our new method, we prove in particular that a quasi-Milstein scheme achieves an $L_p$-error rate of at least $3/4$ in the case of coefficients that are both Lipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives, which is of interest in itself. The latter error rates are obtained via a detailed analysis of the average size of increments of the time-continuous quasi-Milstein scheme over time intervals in which the scheme crosses a point of nondifferentiability of the coefficients.

Local Convergence and Complex Dynamics of a Uni-parametric Family of Iterative Schemes

Using the Lipschitz continuous derivative, we study the local convergence analysis of a third and fourth-order convergent uni-parametric class of iterative schemes. This technique avoids the usual practice of Taylor expansion in convergence analysis and extends the applicability of the family by using the assumption based on the first-order derivative only. Our study provides the radii of balls of convergence and computable error bounds along with the uniqueness of the solution. Also, complex dynamical properties of the family are discussed to select good schemes in the view of numerical stability. Numerical illustrations show that our analysis is useful in solving such problems where previous studies fail to solve.

A Hybrid Proximal Extragradient Self-Concordant Primal Barrier Method for Monotone Variational Inequalities

This paper presents a hybrid proximal extragradient (HPE) self-concordant primal barrier method for solving a monotone variational inequality over a closed convex set endowed with a self-concordant barrier and with an underlying map that has Lipschitz continuous derivative. In contrast to the iteration of a previous method developed by the first and third authors that has to compute an approximate solution of a linearized variational inequality, the one of the present method solves a simpler Newton system of linear equations. The method performs two types of iterations, namely, those that follow ever changing interior paths and those that correspond to large-step HPE iterations. Due to its first-order nature, the present method is shown to have a better iteration-complexity than its zeroth order counterparts such as Korpelevich's algorithm and Tseng's modified forward-backward splitting method, although its work per iteration is larger than the one for the latter methods.

A Further Research on the Convergence of Wu-Schaback’s Multi-quadric Quasi-Interpolation

The paper discusses the error estimate of Wu-Schaback's quasi-interpolant for a wider class of approximated functions (the functions with lower smoothness order). Three cases are considered: a function with a Lipschitz continuous first-order derivative, a continuous function and a Lipschitz continuous function, respectively.

A nonmonotone approximate sequence algorithm for unconstrained nonlinear optimization

A new nonmonotone algorithm is proposed and analyzed for unconstrained nonlinear optimization. The nonmonotone techniques applied in this algorithm are based on the estimate sequence proposed by Nesterov (Introductory Lectures on Convex Optimization: A Basic Course, 2004) for convex optimization. Under proper assumptions, global convergence of this algorithm is established for minimizing general nonlinear objective function with Lipschitz continuous derivatives. For convex objective function, this algorithm maintains the optimal convergence rate of convex optimization. In numerical experiments, this algorithm is specified by employing safe-guarded nonlinear conjugate gradient search directions. Numerical results show the nonmonotone algorithm performs significantly better than the corresponding monotone algorithm for solving the unconstrained optimization problems in the CUTEr (Bongartz et al. in ACM Trans. Math. Softw. 21:123---160, 1995) library.

Convergence of a continuation method under Lipschitz continuous derivative in Banach spaces

The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the semilocal convergence of a continuation method combining Chebyshev method and Convex acceleration of Newton’s method for solving nonlinear equations in Banach spaces under the assumption that the first Frechet derivative satisfies the Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in both the examples. Further, we observed that for particular values of α, our analysis reduces to those for Chebyshev method (α=0) and Convex acceleration of Newton’s method (α=1) respectively with improved results.

Central limit theorem for the heat kernel measure on the unitary group

We prove that for a finite collection of real-valued functions f 1 , … , f n on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of ( tr f 1 , … , tr f n ) under the properly scaled heat kernel measure at a given time on the unitary group U ( N ) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N − 1 . In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.

Global convergence of conjugate gradient method

In this article, we consider the global convergence of the Polak–Ribiére–Polyak conjugate gradient method (abbreviated PRP method) for minimizing functions that have Lipschitz continuous partial derivatives. A novel form of non-monotone line search is proposed to guarantee the global convergence of the PRP method. It is also shown that the PRP method has linear convergence rate under some mild conditions when the non-monotone line search reduces to a related monotone line search. The new non-monotone line search needs to estimate the Lipschitz constant of the gradients of objective functions, for which two practical estimations are proposed to help us to find a suitable initial step size for the PRP method. Numerical results show that the new line search approach is efficient in practical computation.

The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights

We obtain Plancherel–Rotach-type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application, we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. We develop a new technique of asymptotic analysis for matrix Riemann–Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.

Convergence of Liu–Storey conjugate gradient method

The conjugate gradient method is a useful and powerful approach for solving large-scale minimization problems. Liu and Storey developed a conjugate gradient method, which has good numerical performance but no global convergence result under traditional line searches such as Armijo, Wolfe and Goldstein line searches. In this paper a convergent version of Liu–Storey conjugate gradient method (LS in short) is proposed for minimizing functions that have Lipschitz continuous partial derivatives. By estimating the Lipschitz constant of the derivative of objective functions, we can find an adequate step size at each iteration so as to guarantee the global convergence and improve the efficiency of LS method in practical computation.

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

We consider the problem of finding a singularity of a differentiable vector field X defined on a complete Riemannian manifold. We prove a unified result for theexistence and local uniqueness of the solution, and for the local convergence of a Riemannian version of Newton's method. Our approach relies on Kantorovich's majorant principle: under suitable conditions, we construct an auxiliary scalar equation φ(r) = 0 which dominates the original equation X(p) = 0 in the sense that the Riemannian-Newton method for the latter inherits several features of the real Newton method applied to the former. The majorant φ is derived from an adequate radial parametrization of a Lipschitz-type continuity property of the covariant derivative of X, a technique inspired by the previous work of Zabrejko and Nguen on Newton's method in Banach spaces. We show how different specializations of the main result recover Riemannian versions of Kantorovich's theorem and Smale's α-theorem, and, at least partially, the Euclidean self-concordant theory of Nesterov and Nemirovskii. In the specific case of analytic vector fields, we improve recent developments inthis area by Dedieu et al. (IMA J. Numer. Anal., Vol. 23, 2003, pp. 395-419). Some Riemannian techniques used here were previously introduced by Ferreira and Svaiter (J. Complexity, Vol. 18, 2002, pp. 304-329) in the context of Kantorovich's theorem for vector fields with Lipschitz continuous covariant derivatives.

Asymptotic Variational Wave Equations

We investigate the equation (u t +(f(u)) x ) x =f ′ ′(u) (u x )2/2 where f(u) is a given smooth function. Typically f(u)=u 2/2 or u 3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u tt − c(u) (c(u)u x ) x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view. We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.

Convergence of PRP method with new nonmonotone line search

In this paper, we develop a new nonmonotone line search for PRP conjugate gradient method (Polak–Ribiére–Polyak) for minimizing functions having Lipschitz continuous partial derivatives. The nonmonotone line search can guarantee the global convergence of original PRP method under some mild conditions. Numerical experiments show that PRP method with the new nonmonotone line search is available and efficient in practical computation.

Adaptive Dwo Estimator of a Regression Function

We address a problem of non-parametric estimation of an unknown regression function f : [-1/2,1/2] → R at a fixed point x0 ∊ (-1/2,1/2) on the basis of observations (xi, yi), i =l,.., n such that yi = f(xi) + ei, where ei ~ N(0, σ2) is unobservable, Gaussian i.i.d. random noise and xi ∊ [-1/2, 1/2] are given design points. Recently, the Direct Weight Optimization (DWO) method has been proposed to solve a problem of such kind. The properties of the method have been studied for the case when the unknown function f is continuously differentiable with Lipschitz continuous derivative having a priori known Lipschitz constant L. The minimax optimality and adaptivity with respect to the design have been established for the resulting estimator. However, in order to implement the approach, both L and σ are to be known. The subject of the submission is the study of an adaptive version of the DWO estimator which uses a data-driven choice of the method parameter L.

Newton method under weak Lipschitz continuous derivative in Banach spaces

In the present paper, a relax Kantorovich-type condition to guarantee the convergence of Newton method with order 2 in Banach space is obtained, which takes the well-known Kantorovich and Smale conditions as its special cases. The relations between the new condition and the old ones are also studied.

Newton’s algorithm in Euclidean Jordan algebras, with applications to robotics

We consider a convex optimization problem on linearly constrained cones in Euclidean Jordan algebras. The problem is solved using a damped Newton algorithm. Quadratic convergence to the global minimum is shown using an explicit step-size selection. Moreover, we prove that the algorithm is a smooth discretization of a Newton flow with Lipschitz continuous derivative. 1. Introduction. It is a great pleasure to contribute this paper to the special issue, honoring John Moore on the occasion of his 60th birthday. I (U.H.) first met John in 1989 during the end of a short visit of mine at the ANU. Communication between us went along easily right from the start, despite quite dierent backgrounds in mathematics and control. Initial ideas for future research were immediately flying high at fast speed and at the end of the day we had a new Ph.D. student starting to work on balanced realization flows as well as an emerging plan for longer term research collaborations. The excitement and fun in working with John has been steady ever since and added a lot to my own enjoyment of research and life. The motivation for this paper comes from a problem in robotics that John started to work on a few years ago, i.e. that of force optimization in dextrous hand grasp- ing. Here the main goal is to achieve the minimization of contact forces without violating nonlinear friction cone constraints. In earlier work (2), (8), linear program- ming techniques have been used, based on piecewise linear approximations of the cone constraints, but these approaches led to ill conditioning problems. Therefore, an alter- native approach had to be found that avoids approximations of the constraints. In the pioneering joint paper (1) by John Moore with M. Buss and H. Hashimoto, the task was reformulated as a convex optimization problem on linearly constrained cones of positive definite matrices, thus making contact with the well established area of con- vex (semi)definite optimization theory. Linearly convergent gradient flow algorithms were applied to find the global minimum and this led to the first real time solution approach. A challenge remained in finding faster computational schemes. This prob- lem has been resolved in subsequent work (6), where a damped Newton algorithm is