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Xianfu Wang

Publications and source records attributed to Xianfu Wang.

At least 19 recordsLinked to original sources
arXiv2026

Modulus of conically averaged mappings and its applications to angles between two subspaces

Conically averaged mappings, a generalization of averaged mappings, are important in a wide range of Optimization Algorithms. In this paper, we propose the modulus of conical averagedness to classify conical averaged mappings. Introducing the monotone and comonotone values of generalized monotone mappings, we investigate their connections to the modulus of conical averagedness. In the linear setting, we completely characterize conically averaged matrices, and derive explicit and pleasing formulae for computing their modulus of averagedness. As applications, we compute the Dixmier and Friedrichs angles between two subspaces. Nonlinear results are established as extensions of the linear case. Conical averagedness of proximal and reflection mappings of hypoconvex functions are also studied.

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arXiv2026

On Characterizations of (Almost) Strictly Convex Functions

In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is subdifferentiable on its domain, then it is strictly convex if and only if its subdifferential is strictly monotone, equivalently, almost strictly monotone. Rockafellar-Wets' characterizations of almost strictly convex functions via almost differentiability of Fenchel conjugates and strict monotonicity of subdifferentials are extended from a finite-dimensional space to a Hilbert space. We also establish similar results for paramonotone operators.

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arXiv2026

Level proximal subdifferential, variational convexity, and pointwise quadratic approximation

Level proximal subdifferential was introduced by Rockafellar recently for studying proximal mappings of possibly nonconvex functions. In this paper a systematic study of level proximal subdifferential is given. We characterize variational convexity of a function by local firm nonexpansiveness of proximal mappings or local relative monotonicity of level proximal subdifferential, and use them to study local convergence of proximal gradient method and others for variationally convex functions. Variational sufficiency guarantees that proximal gradient method converges to local minimizers rather than just critical points. We also investigate the existence, single-valuedness and integration of level proximal subdifferential, and quantify pointwise quadratic approximation (or Lipschitz smoothness) of a function. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization.

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arXiv2026

A Near-Optimal Total Complexity for the Inexact Accelerated Proximal Gradient Method via Quadratic Growth

We consider the optimization problem $\min_{x\in \mathbb R^n}{F(x):=f(x)+ω(Ax)}$, where $f$ is an $L$-Lipschitz smooth function, and $ω$ is a proper, lower semicontinuous, and convex function. We prove in this paper that when $ω$ is a conic polyhedral function, the inexact accelerated proximal gradient method (IAPG), employed in a double-loop structure, achieves a total complexity of $\mathcal O(\ln(1/\varepsilon)/\sqrt{\varepsilon})$ measured by the total number of calls to the proximal operator of the convex conjugate $ω^\star$ and the gradient of $f$ to achieve $\varepsilon$-optimality in function value. To the best of our knowledge, this improves upon the best-known complexity for IAPG. The key theoretical ingredient is a quadratic growth condition on the dual of the inexact proximal problem, which arises from the conic polyhedral structure of $ω$ and implies linear convergence of the inner proximal gradient loop. To validate these findings, we conduct numerical experiments on a robust TV-$\ell_2$ signal recovery problem, demonstrating fast convergence.

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arXiv2025

On Bauschke-Bendit-Moursi modulus of averagedness and classifications of averaged nonexpansive operators

Averaged operators are important in Convex Analysis and Optimization Algorithms. In this paper, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators using the Bauschke-Bendit-Moursi modulus of averagedness. We show that if an operator is averaged with a constant less than 1/2, then it is a bi-Lipschitz homeomorphism. Amazingly the proximal operator of a convex function has its modulus of averagedness less than 1/2 if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximal operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results.

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arXiv2025

On the Bredies-Chenchene-Lorenz-Naldi algorithm

Monotone inclusion problems occur in many areas of optimization and variational analysis. Splitting methods, which utilize resolvents or proximal mappings of the underlying operators, are often applied to solve these problems. In 2022, Bredies, Chenchene, Lorenz, and Naldi introduced a new elegant algorithmic framework that encompasses various well known algorithms including Douglas-Rachford and Chambolle-Pock. They obtained powerful weak and strong convergence results, where the latter type relies on additional strong monotonicity assumptions. In this paper, we complement the analysis by Bredies et al. by relating the projections of the fixed point sets of the underlying operators that generate the (reduced and original) preconditioned proximal point sequences. We also obtain strong convergence results in the case of linear relations. Various examples are provided to illustrate the applicability of our results.

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arXiv2025

Relaxed Weak Accelerated Proximal Gradient Method: a Unified Framework for Nesterov's Accelerations

This paper is devoted to the study of accelerated proximal gradient methods where the sequence that controls the momentum term doesn't follow Nesterov's rule. We propose a relaxed weak accelerated proximal gradient (R-WAPG) method, a generic algorithm that unifies the convergence results for strongly convex and convex problems where the extrapolation constant is characterized by a sequence that is much weaker than Nesterov's rule. Our R-WAPG provides a unified framework for several notable Euclidean variants of FISTA and verifies their convergences. In addition, we provide the convergence rate of strongly convex objective with a constant momentum term. Without using the idea of restarting, we also reformulate R-WAPG as ``Free R-WAPG" so that it doesn't require any parameter. Explorative numerical experiments were conducted to show its competitive advantages.

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arXiv2024

Projecting onto rectangular hyperbolic paraboloids in Hilbert space

In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we provide a rigorous analysis of the associated projection. In some cases, finding this projection amounts to finding a certain root of a quintic or cubic polynomial. We also observe when the projection is not a singleton and point out connections to graphical and set convergence.

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arXiv2023

Every proximal mapping is a resolvent of level proximal subdifferential

We propose a level proximal subdifferential for a proper lower semicontinuous function. Level proximal subdifferential is a uniform refinement of the well-known proximal subdifferential, and has the pleasant feature that its resolvent always coincides with the proximal mapping of a function. It turns out that the resolvent representation of proximal mapping in terms of Mordukhovich limiting subdifferential is only valid for hypoconvex functions. We also provide properties of level proximal subdifferential and numerous examples to illustrate our results.

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arXiv2023

Real roots of real cubics and optimization

The solution of the cubic equation has a century-long history; however, the usual presentation is geared towards applications in algebra and is somewhat inconvenient to use in optimization where frequently the main interest lies in real roots. In this note, we present the roots of the cubic in a form that makes them convenient to use and we also focus on information on the location of the real roots. Armed with this, we provide several applications in optimization where we compute Fenchel conjugates, proximal mappings and projections.

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arXiv2022

The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work in a product space with as many factors as there are operators. In groundbreaking recent work by Ryu and by Malitsky and Tam, it was shown that the number of factors can be reduced by one. A similar reduction was achieved recently by Campoy through a clever reformulation originally proposed by Kruger. All three splitting methods guarantee weak convergence to some solution of the underlying sum problem; strong convergence holds in the presence of uniform monotonicity. In this paper, we provide a case study when the operators involved are normal cone operators of subspaces and the solution set is thus the intersection of the subspaces. Even though these operators lack strict convexity, we show that striking conclusions are available in this case: strong (instead of weak) convergence and the solution obtained is (not arbitrary but) the projection onto the intersection. Numerical experiments to illustrate our results are also provided.

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arXiv2022

A mirror inertial forward-reflected-backward splitting: Global convergence and linesearch extension beyond convexity and Lipschitz smoothness

This work investigates a Bregman and inertial extension of the forward-reflected-backward algorithm [Y. Malitsky and M. Tam, SIAM J. Optim., 30 (2020), pp. 1451--1472] applied to structured nonconvex minimization problems under relative smoothness. To this end, the proposed algorithm hinges on two key features: taking inertial steps in the dual space, and allowing for possibly negative inertial values. Our analysis begins with studying an associated envelope function that takes inertial terms into account through a novel product space formulation. Such construction substantially differs from similar objects in the literature and could offer new insights for extensions of splitting algorithms. Global convergence and rates are obtained by appealing to the generalized concave Kurdyka-Lojasiewicz (KL) property, which allows us to describe a sharp upper bound on the total length of iterates. Finally, a linesearch extension is given to enhance the proposed method.

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arXiv2022

A Bregman inertial forward-reflected-backward method for nonconvex minimization

We propose a Bregman inertial forward-reflected-backward (BiFRB) method for nonconvex composite problems. Our analysis relies on a novel approach that imposes general conditions on implicit merit function parameters, which yields a stepsize condition that is independent of inertial parameters. In turn, a question of Malitsky and Tam regarding whether FRB can be equipped with a Nesterov-type acceleration is resolved. Assuming the generalized concave Kurdyka-Łojasiewicz property of a quadratic regularization of the objective, we obtain sequential convergence of BiFRB, as well as convergence rates on both the function value and actual sequence. We also present formulae for the Bregman subproblem, supplementing not only BiFRB but also the work of Boţ-Csetnek-László and Boţ-Csetnek. Numerical simulations are conducted to evaluate the performance of our proposed algorithm.

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arXiv2022

On Carlier's inequality

The Fenchel-Young inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter. In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results.

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arXiv2022

The Bregman proximal average

We provide a proximal average with repect to a $1$-coercive Legendre function. In the sense of Bregman distance, the Bregman envelope of the proximal average is a convex combination of Bregman envelopes of individual functions. The Bregman proximal mapping of the average is a convex combination of convexified proximal mappings of individual functions. Techniques from variational analysis provide the keys for the Bregman proximal average.

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arXiv2022

Roots of the identity operator and proximal mappings: (classical and phantom) cycles and gap vectors

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. In this paper, we extend Simons's lemma to closed convex functions, show its connections to Attouch-Thera duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings.

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arXiv2022

The projection onto the cross

We consider the set of pairs of orthogonal vectors in Hilbert space, which is also called the cross because it is the union of the horizontal and vertical axes in the Euclidean plane when the underlying space is the real line. Crosses, which are nonconvex sets, play a significant role in various branches of nonsmooth analysis such as feasibility problems and optimization problems. In this work, we study crosses and show that in infinite-dimensional settings, they are never weakly (sequentially) closed. Nonetheless, crosses do turn out to be proximinal (i.e., they always admit projections) and we provide explicit formulas for the projection onto the cross in all cases.

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