arXiv subjects

Tianxiang Liu

Publications and source records attributed to Tianxiang Liu.

12 recordsLinked to original sources

arXiv2026

An error bound-based convergence analysis framework for a class of randomized algorithms

Classical analyses show that randomized coordinate descent (RCD) and gradient descent (GD) share the same convergence rates in terms of objective gap under convexity and specific global EB-type assumptions. However, this rate preservation phenomenon does not extend to iterate rates, almost-sure rates, or general local error bound conditions. In this paper, we study this phenomenon systematically, not only for RCD but also for a much broader class of algorithms that share the same structures. Specifically, we introduce an abstract class of algorithms, called monotone randomized algorithms (MRAs), and new unified error bound (UEB) conditions that are independent of problem class and accommodate general EB moduli. These UEB conditions subsume many EB- and Kurdyka--Łojasiewicz-type assumptions used in the analysis of algorithms for optimization, convex feasibility, and common fixed point problems. We then develop a new analysis framework and a set of technical tools that yield nonasymptotic in-expectation rates and asymptotic almost-sure rates for both the optimality measure and the iterates under the global UEB condition. Under the local UEB condition, we further establish asymptotic almost-sure rates for both quantities. For comparison, we introduce the monotone deterministic algorithms (MDAs), as the corresponding deterministic counterpart of MRAs. We illustrate the tightness of our MRA analysis by showing that the lower bounds of MDAs match the upper bounds of MRAs qualitatively. We also demonstrate the strength and versatility of our framework through three applications, namely generalized randomized subspace descent for unconstrained minimization, randomized block proximal gradient methods for composite optimization, and the randomized Krasnosel'ski\uı--Mann method for common fixed point problems.

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arXiv2026

A nonmonotone extrapolated proximal gradient-subgradient algorithm beyond global Lipschitz gradient continuity

With the advancement of modern applications, an increasing number of composite optimization problems arise whose smooth component does not possess a globally Lipschitz continuous gradient. This setting prevents the direct use of the proximal gradient (PG) method and its variants, and has motivated a growing body of research on new PG-type methods and their convergence theory, in particular, global convergence analysis without imposing any explicit or implicit boundedness assumptions on the iterates. Until recently, the first complete analysis of this kind has been established for the PG method and its specific nonmonotone variants, which has since stimulated further exploration along this research direction. In this paper, we consider a general composite optimization model beyond the global Lipschitz gradient continuity setting. We propose a novel problem-parameter-free algorithm that incorporates a carefully designed nonmonotone line search to handle the non-global Lipschitz gradient continuity, together with an extrapolation step to achieve potential acceleration. Despite the added technical challenges introduced by combining extrapolation with nonmonotone line search, we establish a refined convergence analysis for the proposed algorithm under the Kurdyka-Ł ojasiewicz property, without requiring any boundedness assumptions on the iterates. This work thus further advances the theoretical understanding of PG-type methods in the non-global Lipschitz gradient continuity setting. Finally, we conduct numerical experiments to illustrate the effectiveness of our algorithm and highlight the advantages of integrating extrapolation with a nonmonotone line search.

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arXiv2026

Concrete convergence rates for common fixed point problems under Karamata regularity

We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number $k$ instead, of say, a function of the iterate $x^k$. While it is well-known that under Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-Hölderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular.

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arXiv2025

On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian

It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.

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arXiv2024

A new dual spectral projected gradient method for log-determinant semidefinite programming with hidden clustering structures

In this paper, we propose a new efficient method for a sparse Gaussian graphical model with hidden clustering structures by extending a dual spectral projected gradient (DSPG) method proposed by Nakagaki et al.~(2020). We establish the global convergence of the proposed method to an optimal solution, and we show that the projection onto the feasible region can be solved with a low computational complexity by the use of the pool-adjacent-violators algorithm. Numerical experiments on synthesis data and real data demonstrate the efficiency of the proposed method. The proposed method takes 0.91 seconds to achieve a similar solution to the direct application of the DSPG method which takes 4361 seconds.

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arXiv2023

Doubly majorized algorithm for sparsity-inducing optimization problems with regularizer-compatible constraints

We consider a class of sparsity-inducing optimization problems whose constraint set is regularizer-compatible, in the sense that, the constraint set becomes easy-to-project-onto after a coordinate transformation induced by the sparsity-inducing regularizer. Our model is general enough to cover, as special cases, the ordered LASSO model and its variants with some commonly used nonconvex sparsity-inducing regularizers. The presence of both the sparsity-inducing regularizer and the constraint set poses challenges on the design of efficient algorithms. In this paper, by exploiting absolute-value symmetry and other properties in the sparsity-inducing regularizer, we propose a new algorithm, called the Doubly Majorized Algorithm (DMA), for this class of problems. The DMA makes use of projections onto the constraint set after the coordinate transformation in each iteration, and hence can be performed efficiently. Without invoking any commonly used constraint qualification conditions such as those based on horizon subdifferentials, we show that any accumulation point of the sequence generated by DMA is a so-called $ψ_{\rm opt}$-stationary point, a new notion of stationarity we define as inspired by the notion of $L$-stationarity. We also show that any global minimizer of our model has to be a $ψ_{\rm opt}$-stationary point, again without imposing any constraint qualification conditions. Finally, we illustrate numerically the performance of DMA on solving variants of ordered LASSO with nonconvex regularizers.

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arXiv2022

Convergence analysis under consistent error bounds

We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and Hölderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.

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arXiv2019

A hybrid penalty method for a class of optimization problems with multiple rank constraints

In this paper, we consider the problem of minimizing a smooth objective over multiple rank constraints on Hankel-structured matrices. This kind of problems arises in system identification, system theory and signal processing, where the rank constraints are typically "hard constraints". To solve these problems, we propose a hybrid penalty method that combines a penalty method with a post-processing scheme. Specifically, we solve the penalty subproblems until the penalty parameter reaches a given threshold, and then switch to a local alternating "pseudo-projection'' method to further reduce constraint violation. Pseudo-projection is a generalization of the concept of projection. We show that a pseudo-projection onto a {\em single} low-rank Hankel-structured matrix constraint can be computed efficiently by existing softwares such as SLRA (Markovsky and Usevich, 2014), under mild assumptions. We also demonstrate how the penalty subproblems in the hybrid penalty method can be solved by pseudo-projection-based optimization methods, and then present some convergence results for our hybrid penalty method. Finally, the efficiency of our method is illustrated by numerical examples.

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arXiv2018

A successive difference-of-convex approximation method for a class of nonconvex nonsmooth optimization problems

We consider a class of nonconvex nonsmooth optimization problems whose objective is the sum of a smooth function and a finite number of nonnegative proper closed possibly nonsmooth functions (whose proximal mappings are easy to compute), some of which are further composed with linear maps. This kind of problems arises naturally in various applications when different regularizers are introduced for inducing simultaneous structures in the solutions. Solving these problems, however, can be challenging because of the coupled nonsmooth functions: the corresponding proximal mapping can be hard to compute so that standard first-order methods such as the proximal gradient algorithm cannot be applied efficiently. In this paper, we propose a successive difference-of-convex approximation method for solving this kind of problems. In this algorithm, we approximate the nonsmooth functions by their Moreau envelopes in each iteration. Making use of the simple observation that Moreau envelopes of nonnegative proper closed functions are continuous {\em difference-of-convex} functions, we can then approximately minimize the approximation function by first-order methods with suitable majorization techniques. These first-order methods can be implemented efficiently thanks to the fact that the proximal mapping of {\em each} nonsmooth function is easy to compute. Under suitable assumptions, we prove that the sequence generated by our method is bounded and any accumulation point is a stationary point of the objective. We also discuss how our method can be applied to concrete applications such as nonconvex fused regularized optimization problems and simultaneously structured matrix optimization problems, and illustrate the performance numerically for these two specific applications.

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arXiv2018

A refined convergence analysis of pDCA$_e$ with applications to simultaneous sparse recovery and outlier detection

We consider the problem of minimizing a difference-of-convex (DC) function, which can be written as the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous possibly nonsmooth concave function. We refine the convergence analysis in [38] for the proximal DC algorithm with extrapolation (pDCA$_e$) and show that the whole sequence generated by the algorithm is convergent when the objective is level-bounded, {\em without} imposing differentiability assumptions in the concave part. Our analysis is based on a new potential function and we assume such a function is a Kurdyka-Łojasiewicz (KL) function. We also establish a relationship between our KL assumption and the one used in [38]. Finally, we demonstrate how the pDCA$_e$ can be applied to a class of simultaneous sparse recovery and outlier detection problems arising from robust compressed sensing in signal processing and least trimmed squares regression in statistics. Specifically, we show that the objectives of these problems can be written as level-bounded DC functions whose concave parts are {\em typically nonsmooth}. Moreover, for a large class of loss functions and regularizers, the KL exponent of the corresponding potential function are shown to be 1/2, which implies that the pDCA$_e$ is locally linearly convergent when applied to these problems. Our numerical experiments show that the pDCA$_e$ usually outperforms the proximal DC algorithm with nonmonotone linesearch [24, Appendix A] in both CPU time and solution quality for this particular application.

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arXiv2017

Peaceman-Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman-Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.

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arXiv2016

Further properties of the forward-backward envelope with applications to difference-of-convex programming

In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-Łojasiewicz function with an exponent of $\frac12$; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale $\ell_{1-2}$ regularized least squares problems [37] illustrate that an implementation of this approach with a limited-memory BFGS scheme usually outperforms standard first-order methods such as the nonmonotone proximal gradient method in [35].

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