arXiv subjects

Kaiwen Meng

Publications and source records attributed to Kaiwen Meng.

7 recordsLinked to original sources

arXiv2025

Lipschitz continuity of solution multifunctions of extended $\ell_1$ regularization problems

The Lasso and the basis pursuit in compressed sensing and machine learning are convex optimization problems with three parameters: the regularization scalar, the observation vector and the data matrix. Relative to the first two parameters, we obtain the Lipschitz continuity of the solution multifunction on its convex domain. When the data matrix of the Lasso also perturbs, where non-polyhedral structure may display, we obtain full characterizations for the Lipschitz continuity of the solution multifunction on the product of a compact and convex set in the space of first two parameters and a neighborhood of the fixed data matrix. Moreover for the solution multifunction of the Lasso, we show that the Lipschitz continuity implies its single-valuedness. Our analysis is based on polyhedron theory, a sufficient condition that ensures the Lipschitz continuity of a polyhedral multifunction with a convex domain, and an explicit representation of the solution multifunction, where the latter is a consequence of the Lipschitz continuity of the solution multifunction relative to the first two parameters.

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arXiv2023

Variational Analysis of Kurdyka-Łojasiewicz Property, Exponent and Modulus

The Kurdyka-Łojasiewicz (KŁ) property, exponent and modulus have played a very important role in the study of global convergence and rate of convergence for optimal algorithms. In this paper, at a stationary point of a locally lower semicontinuous function, we obtain complete characterizations of the KŁ property and the KŁ modulus via the outer limiting subdifferential of an auxilliary function and a newly-introduced subderivative function respectively. In particular, for a class of prox-regular, twice epi-differentiable and subdifferentially continuous functions, we show that the KŁ property and the KŁ modulus can be described by its Moreau envelopes and a quadratic growth condition. We apply the obtained results to establish the KŁ property with exponent $\frac12$ and to provide calculation of the modulus for a smooth function, the pointwise maximum of finitely many smooth functions and regularized functions respectively. These functions often appear in the modelling of structured optimization problems.

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arXiv2022

Projectional Coderivatives and Calculus Rules

This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to give a complete characterization of the relative Lipschitz-like property under such a setting. Chain rules and sum rules are obtained to facilitate the application of the tool to a wider range of problems.

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arXiv2020

Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.

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arXiv2017

Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems

The $\ell_p$ regularization problem with $0< p< 1$ has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. The proximal gradient algorithm is one of the most popular algorithms for solving the $\ell_p$ regularisation problem. In the present paper, we investigate the linear convergence issue of one inexact descent method and two inexact proximal gradient algorithms (PGA). For this purpose, an optimality condition theorem is explored to provide the equivalences among a local minimum, second-order optimality condition and second-order growth property of the $\ell_p$ regularization problem. By virtue of the second-order optimality condition and second-order growth property, we establish the linear convergence properties of the inexact descent method and inexact PGAs under some simple assumptions. Both linear convergence to a local minimal value and linear convergence to a local minimum are provided. Finally, the linear convergence results of the inexact numerical methods are extended to the infinite-dimensional Hilbert spaces.

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arXiv2016

On Error Bound Moduli for Locally Lipschitz and Regular Functions

In this paper we study local error bound moduli for a locally Lipschitz and regular function via its outer limiting subdifferential set. We show that the distance of 0 from the outer limiting subdifferential of the support function of the subdifferential set, which is essentially the distance of 0 from the end set of the subdifferential set, is an upper estimate of the local error bound modulus. This upper estimate becomes tight for a convex function under some regularity conditions. We show that the distance of 0 from the outer limiting subdifferential set of a lower $\mathcal{C}^1$ function is equal to the local error bound modulus.

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arXiv2016

Group sparse optimization via $\ell_{p,q}$ regularization

In this paper, we investigate a group sparse optimization problem via $\ell_{p,q}$ regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish some oracle property and a global recovery bound of order $O(λ^\frac{2}{2-q})$ for any point in a level set of the $\ell_{p,q}$ regularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order $O(λ^2)$ for a path of local minima. In the algorithmic aspect, we apply the well-known proximal gradient method to solve the $\ell_{p,q}$ regularization problems, either by analytically solving some specific $\ell_{p,q}$ regularization subproblems, or by using the Newton method to solve general $\ell_{p,q}$ regularization subproblems. In particular, we establish the linear convergence rate of the proximal gradient method for solving the $\ell_{1,q}$ regularization problem under some mild conditions. As a consequence, the linear convergence rate of proximal gradient method for solving the usual $\ell_{q}$ regularization problem ($0<q<1$) is obtained. Finally in the aspect of application, we present some numerical results on both the simulated data and the real data in gene transcriptional regulation.

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