Daoli Zhu

Daoli Zhu, Lei Zhao, Shuzhong Zhang

This paper presents a unified analysis for the proximal subgradient method (Prox-SubGrad) type approach to minimize an overall objective of $f(x)+r(x)$, subject to convex constraints, where both $f$ and $r$ are weakly convex, nonsmooth, and non-Lipschitz. Leveraging on the properties of the Moreau envelope of weakly convex functions, we are able to relate error-bound conditions, the growth conditions of the subgradients of the objective, and the behavior of the proximal subgradient iterates on some remarkably broad classes of objective functions. Various existing as well as new bounding conditions are studied, leading to novel iteration complexity results. The terrain of our exploration expands to stochastic proximal subgradient algorithms.

Lei Zhao, Daoli Zhu, Shuzhong Zhang

In this paper we consider a non-monotone (mixed) variational inequality model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle-point system for the VI model in question, we introduce an augmented Lagrangian primal-dual method, to be called ALAVI in the current paper, for solving a general constrained VI model. Under an assumption, to be called the primal-dual variational coherence condition in the paper, we prove the convergence of ALAVI. Next, we show that many existing generalized monotonicity properties are sufficient -- though by no means necessary -- to imply the above mentioned coherence condition, thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an $o(1/\sqrt{k})$ global rate of convergence where $k$ is the iteration count. By introducing a new gap function, this rate further improves to be $O(1/k)$ if the mapping is monotone. Finally, we show that under a metric subregularity condition, even if the VI model may be non-monotone the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and non-monotone VI problems show practical efficacy of the newly proposed method.

Lei Zhao, Ding Chen, Daoli Zhu et al.

Coordinate-type subgradient methods for addressing nonsmooth optimization problems are relatively underexplored due to the set-valued nature of the subdifferential. In this work, our study focuses on nonsmooth composite optimization problems, encompassing a wide class of convex and weakly convex (nonconvex nonsmooth) problems. By utilizing the chain rule of the composite structure properly, we introduce the Randomized Coordinate Subgradient method (RCS) for tackling this problem class. To the best of our knowledge, this is the first coordinate subgradient method for solving general nonsmooth composite optimization problems. In theory, we consider the linearly bounded subgradients assumption for the objective function, which is more general than the traditional Lipschitz continuity assumption, to account for practical scenarios. We then conduct convergence analysis for RCS in both convex and weakly convex cases based on this generalized Lipschitz-type assumption. Specifically, we establish the $\widetilde{\mathcal{O}}$$(1/\sqrt{k})$ convergence rate in expectation and the $\tilde o(1/\sqrt{k})$ almost sure asymptotic convergence rate in terms of the suboptimality gap when $f$ is convex. For the case when $f$ is weakly convex and its subdifferential satisfies the global metric subregularity property, we derive the $\mathcal{O}(\varepsilon^{-4})$ iteration complexity in expectation. We also establish an asymptotic convergence result. To justify the global metric subregularity property utilized in the analysis, we establish this error bound condition for the concrete (real-valued) robust phase retrieval problem. We also provide a convergence lemma and the relationship between the global metric subregularity properties of a weakly convex function and its Moreau envelope. Finally, we conduct several experiments to demonstrate the possible superiority of RCS over the subgradient method.

Xiao Li, Lei Zhao, Daoli Zhu et al.

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this work, we first extend the typical iteration complexity results for the subgradient method to cover non-Lipschitz convex and weakly convex minimization. Specifically, for the convex case, we can drive the suboptimality gap to below $\varepsilon$ in $\mathcal{O}( \varepsilon^{-2} )$ iterations; for the weakly convex case, we can drive the gradient norm of the Moreau envelope of the objective function to below $\varepsilon$ in $\mathcal{O}( \varepsilon^{-4} )$ iterations. Then, we provide convergence results for the subgradient method in the non-Lipschitz setting when proper diminishing rules on the step size are used. In particular, when $f$ is convex, we establish an $\mathcal{O}(\log(k)/\sqrt{k})$ rate of convergence in terms of the suboptimality gap, where $k$ represents the iteration count. With an additional quadratic growth property, the rate is improved to $\mathcal{O}(1/k)$ in terms of the squared distance to the optimal solution set. When $f$ is weakly convex, asymptotic convergence is established. Our results neither require any modification to the subgradient method nor impose any growth condition on the subgradients, while our analysis is surprisingly simple. To further illustrate the wide applicability of our framework, we extend the aforementioned iteration complexity results to cover the truncated subgradient, the stochastic subgradient, and the proximal subgradient methods for non-Lipschitz convex / weakly convex objective functions.