Local Lipschitz Research Articles

On robustness in nonconvex optimization with application to defense planning

We estimate the increase in minimum value for a decision that is robust to parameter perturbations as compared to the value of a nominal nonconvex problem. The estimates rely on expressions for subgradients and local Lipschitz moduli of min-value functions and require only the solution of the nominal problem. Across 54 mixed-integer optimization models, the median error in estimating the increase in minimum value is 12%. The results inform analysts about the possibility of obtaining cost-effective, parameter-robust decisions.

Geometry of planar curves intersecting many lines at a few points

The local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is σ \sigma -finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.

From Perspective Maps to Epigraphical Projections

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus, the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map—including local Lipschitz continuity, directional differentiability, and semismoothness—are derived. An SC1 optimization framework for computing epigraphical and level-set projections is, thus, established. Numerical experiments on one-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC). M.P. Friedlander was supported by NSERC discovery grant [Grant RGPIN-2017-04461]. T. Hoheisel was supported by NSERC discovery grant [Grant RGPIN-2017-04035]. A. Goodwin’s work was partially supported by an NSERC summer research stipend.

Almost sure exponential stability of nonlinear stochastic delay hybrid systems driven by G-Brownian motion

G-Brownian motion has potential applications in uncertainty problems and risk measures, which has attracted the attention of many scholars. This study investigates the almost sure exponential stability of nonlinear stochastic delay hybrid systems driven by G-Brownian motion. Due to the non-linearity of G-expectation and distribution uncertainty of G-Brownian motion, it is difficult to study this issue. Firstly, the existence of the global unique solution is derived under the linear growth condition and local Lipschitz condition. Secondly, the almost sure exponential stability of the system is analyzed by applying the G-Lyapunov function and G-Itô formula. Finally, an example is introduced to illustrate the stability. The conclusions of this paper can be applied to the stability and risk management of uncertain financial markets.

Global Stabilization for Stochastic Continuous Cascade Nonlinear Systems Subject to SISS Inverse Dynamics and Time-Delay: A Dynamic Gain Approach

This article is devoted to the global continuous control for stochastic low-order cascade nonlinear systems with time-varying delay and stochastic inverse dynamics. Compared with existing results, the nature of only continuous, but nonsmooth, is unfolded since the power of the stochastic cascade system is of low order; and all the traditional growth conditions on unknown drift and diffusion nonlinearities and local Lipschitz condition are quitted, which largely extends the scope of application. Combining with stochastic input-to-state stability, two new lemmas are developed with rigorous proofs to deal with uncertain nonlinear terms and unmeasurable stochastic inverse dynamics. A continuous control scheme consisting of a delay-independent partial state feedback controller and a serial of dynamic update laws is proposed to guarantee the globally asymptotical stability of the closed-loop system.

Vlasov equations on digraph measures

Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs. In this paper, we are interested in IPS on a sequence of directed graphs, or digraphs for short. It is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL. This paper studies VEs on a generalized digraph, regarded as limit of a sequence of digraphs, which we refer to as a digraph measure (DGM) to emphasize that we work with its limit via measures. We provide (i) unique existence of solutions of the VE on continuous DGMs, and (ii) discretization of the solution of the VE by empirical distributions supported on solutions of an IPS via ODEs coupled on a sequence of digraphs converging to the given DGM. Our result extends existing results on one-dimensional Kuramoto-type networks coupled on dense graphs. Here we allow the underlying digraphs to be not necessarily dense which include many interesting graphical structures such that stars, trees and rings, which have been frequently used in many sparse network models in finance, telecommunications, physics, genetics, neuroscience, and social sciences. A key contribution of this paper is a nontrivial generalization of Neunzert's in-cell-particle approach for all-to-all coupled indistinguishable IPS with global Lipschitz continuity in Euclidean spaces to distinguishable IPS on heterogeneous digraphs with local Lipschitz continuity, via a measure-theoretic viewpoint. The approach together with the metrics is different from the known techniques in Lp-functions using graphons and their generalization by means of harmonic analysis of locally compact Abelian groups. Finally, to demonstrate the wide applicability, we apply our results to various models in higher-dimensional Euclidean spaces in epidemiology, ecology, and social sciences.

The numerical algorithm for stochastic age-dependent population system with Lévy noise in a polluted environment

Environmental pollution has significant effects on population systems worldwide. In this study, we established a numerical algorithm for modelling stochastic age-dependent population systems driven by Lévy noise in a polluted environment. A modified explicit Euler–Maruyama (EM) discretization scheme was employed for ensuring the positiveness of the numerical solution used for modelling the stochastic age-dependent population hybrid system with Lévy noise in a polluted environment. The convergence of the numerical solution was confirmed by It's formula and a projection operator under the condition that the drift coefficient satisfies the local Lipschitz and global one-sided Lipschitz conditions. The theoretical results were finally verified with a numerical example.

On some Lipschitz-type functions

In recent years Lipschitz-type functions have been studied in an extensive manner. Particularly, we have locally Lipschitz functions (both the ball radius and the local Lipschitz constant depend on the point), uniformly locally Lipschitz functions (the ball radius is independent of the point) and Lipschitz in the small functions (both the ball radius and the local Lipschitz constant are independent of the point), etc. In this article, another interesting class of Lipschitz-type functions is studied in which the ball radius may depend on the point but the local Lipschitz constant can be chosen independently. Necessary and sufficient conditions on the domain space are established under which this class will contain or coincide with other well-known classes of Lipschitz-type functions.

Razumikhin Theorems on Polynomial Stability of Neutral Stochastic Pantograph Differential Equations with Markovian Switching

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work.

Tamed Euler–Maruyama approximation of McKean–Vlasov stochastic differential equations with super-linear drift and Hölder diffusion coefficients

This paper establishes the strong convergence of the tamed Euler–Maruyama (EM) approximation for McKean–Vlasov stochastic differential equations (SDEs) with super-linear drift and Hölder diffusion coefficients. By the Yamada-Watanable technique, this paper deals with the Hölder diffusion coefficient. To obtain the desired approximation, this paper also proves the existence and uniqueness of strong solution for this class of McKean-Vlasov SDEs. When one-sided local Lipschitz condition is replaced by the global condition, the convergence rate can be obtained. Finally, two examples are presented to illustrate our results.

Least-squares finite element method for ordinary differential equations

We consider the least-squares finite element method (lsfem) for systems of nonlinear ordinary differential equations, and establish an optimal error estimate for this method when piecewise linear elements are used. The main assumptions are that the vector field is sufficiently smooth and that the local Lipschitz constant as well as the operator norm of the Jacobian matrix associated with the nonlinearity are sufficiently small, when restricted to a suitable neighborhood of the true solution for the considered initial value problem. This theoretic optimality is further illustrated numerically, along with evidence of possible extension to higher-order basis elements. Examples are also presented to show the advantages of lsfem compared with finite difference methods in various scenarios. Suitable modifications for adaptive time-stepping are discussed as well.

Local Lipschitz continuity for [formula omitted]PDEs with explicit [formula omitted]dependence

This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑i=1n∂∂xiaix,u,Du=bx,u,Duunder p,q−growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable.

Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth

Abstract We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard ( p , q ) {(p,q)} -growth with the source term in the Lorentz space L ⁢ ( N , 1 ) {L(N,1)} under the restriction q < p + 1 + p ⁢ min ⁡ { 1 N , 2 ⁢ ( p - 1 ) N ⁢ p - 2 ⁢ p + 2 } . q<p+1+p\min\Bigl{\{}\frac{1}{N},\frac{2(p-1)}{Np-2p+2}\Bigr{\}}. This extends the recent work by Beck and Mingione to bounded minimizers under weaker hypothesis and is sharp for some special ranges of p, q and N.

Proximal Gradient Algorithms Under Local Lipschitz Gradient Continuity

Composite optimization offers a powerful modeling tool for a variety of applications and is often numerically solved by means of proximal gradient methods. In this paper, we consider fully nonconvex composite problems under only local Lipschitz gradient continuity for the smooth part of the objective function. We investigate an adaptive scheme for PANOC-type methods (Stella et al. in Proceedings of the IEEE 56th CDC, 2017), namely accelerated linesearch algorithms requiring only the simple oracle of proximal gradient. While including the classical proximal gradient method, our theoretical results cover a broader class of algorithms and provide convergence guarantees for accelerated methods with possibly inexact computation of the proximal mapping. These findings have also significant practical impact, as they widen scope and performance of existing, and possibly future, general purpose optimization software that invoke PANOC as inner solver.

Sub-linear convergence of a stochastic proximal iteration method in Hilbert space

We consider a stochastic version of the proximal point algorithm for convex optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in this form. Indeed, most related results are confined to the finite-dimensional setting, where error bounds could depend on the dimension of the space. On the other hand, the few existing results in the infinite-dimensional setting only prove very weak types of convergence, owing to weak assumptions on the problem. In particular, there are no results that show strong convergence with a rate. In this article, we bridge these two worlds by assuming more regularity of the optimization problem, which allows us to prove convergence with an (optimal) sub-linear rate also in an infinite-dimensional setting. In particular, we assume that the objective function is the expected value of a family of convex differentiable functions. While we require that the full objective function is strongly convex, we do not assume that its constituent parts are so. Further, we require that the gradient satisfies a weak local Lipschitz continuity property, where the Lipschitz constant may grow polynomially given certain guarantees on the variance and higher moments near the minimum. We illustrate these results by discretizing a concrete infinite-dimensional classification problem with varying degrees of accuracy.

Convolutional neural network based simulation and analysis for backward stochastic partial differential equations

We develop a generic convolutional neural network (CNN) based numerical scheme to simulate the 2-tuple adapted strong solution to a unified system of backward stochastic partial differential equations (B-SPDEs) driven by Brownian motions, which can be used to model many real-world system dynamics such as optimal control and differential game problems. The dynamics of the scheme is modeled by a CNN through conditional expectation projection. It consists of two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with mean-square error estimation and almost sure (a.s.) convergence supported by both theoretical proof and numerical examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution to the system under both conventional and Malliavin derivatives with general local Lipschitz and linear growth conditions.

Quantitative Sensitivity Bounds for Nonlinear Programming and Time-Varying Optimization

Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit expressions for the local and global Lipschitz constants of the solution map of nonconvex or convex optimization problems, respectively. Our results are geared towards the study of time-varying optimization problems, which are commonplace in various applications of online optimization, including power systems, robotics, signal processing, and more. In this context, our results can be used to bound the rate of change of the optimizer. To illustrate the use of our sensitivity bounds we generalize existing arguments to quantify the tracking performance of continuous-time, monotone running algorithms. Furthermore, we introduce a new continuous-time running algorithm for time-varying constrained optimization, which we model as a so-called perturbed sweeping process. For this discontinuous scheme we establish an explicit bound on the asymptotic solution tracking for a class of convex problems.

Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders

In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

Numerical analysis of split-step [formula omitted] methods with truncated Wiener process for a stochastic SIS epidemic model

In this paper, the split-step θ methods for the stochastic SIS model are considered, and then the numerical positivity, convergence and dynamical behaviors are analyzed. Obviously, the numerical positivity makes sense biologically and is achieved by the truncated Wiener processes. Because the global Lipschitz condition is not satisfied to diffusion terms of the stochastic SIS model, an alternative approach is implemented for the convergence by the local Lipschitz condition and the numerical boundedness in the almost sure sense. Moreover, numerical extinction and persistence are presented by introducing a localized stochastic stability function, which is different from traditional methods. By using the exponential estimation of the localized stability function and the large number law for martingales, numerical extinction and persistence are obtained. Finally, numerical examples are given to illustrate our results.

Characterizing Sobolev spaces of vector-valued functions

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Ω⊂RN and a Banach space V, we characterize the functions in the Sobolev-Reshetnyak space R1,p(Ω,V), where 1≤p≤∞, in terms of the existence of partial metric derivatives or partial w⁎-derivatives with suitable integrability properties. In the case p=∞ the Sobolev-Reshetnyak space R1,∞(Ω,V) is characterized in terms of a uniform local Lipschitz property. We also consider the special case of the space V=ℓ∞.