Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds

We consider the problem of decentralized nonconvex optimization over a compact submanifold, where each local agent's objective function defined by the local dataset is smooth. Leveraging the powerful tool of proximal smoothness, we establish local linear convergence of the projected gradient descent method with unit step size for solving the consensus problem over the compact manifold. This serves as the basis for analyzing decentralized algorithms on manifolds. Then, we propose two decentralized methods, namely the decentralized projected Riemannian gradient descent (DPRGD) and the decentralized projected Riemannian gradient tracking (DPRGT) methods. We establish their convergence rates of $\mathcal{O}(1/\sqrt{K})$ and $\mathcal{O}(1/K)$, respectively, to reach a stationary point. To the best of our knowledge, DPRGT is the first decentralized algorithm to achieve exact convergence for solving decentralized optimization over a compact manifold. The key ingredients in the proof are the Lipschitz-type inequalities of the projection operator on the compact manifold and smooth functions on the manifold, which could be of independent interest. Finally, we demonstrate the effectiveness of our proposed methods compared to state-of-the-art ones through numerical experiments on eigenvalue problems and low-rank matrix completion.Comment: 32 page

Decentralized Douglas-Rachford splitting methods for smooth optimization over compact submanifolds

We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized Douglas-Rachford splitting algorithm, DDRS. When the proximal operator of the local loss function does not have a closed-form solution, an inexact version of DDRS, iDDRS, is also presented. Both algorithms rely on an ingenious integration of the nonconvex Douglas-Rachford splitting algorithm with gradient tracking and manifold optimization. We show that our DDRS and iDDRS achieve the best-known convergence rate of $\mathcal{O}(1/K)$. The main challenge in the proof is how to handle the nonconvexity of the manifold constraint. To address this issue, we utilize the concept of proximal smoothness for compact submanifolds. This ensures that the projection onto the submanifold exhibits convexity-like properties, which allows us to control the consensus error across agents. Numerical experiments on the principal component analysis are conducted to demonstrate the effectiveness of our decentralized DRS compared with the state-of-the-art ones

Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

In this paper, we introduce the notion of generalized $\epsilon$-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space. To find a generalized $\epsilon$-stationarity point, we develop a family of Riemannian gradient-type methods based on the Moreau envelope technique with a decreasing sequence of smoothing parameters, namely Riemannian smoothing gradient and Riemannian smoothing stochastic gradient methods. We prove that the Riemannian smoothing gradient method has the iteration complexity of $\mathcal{O}(\epsilon^{-3})$ for driving a generalized $\epsilon$-stationary point. To our knowledge, this is the best-known iteration complexity result for the nonconvex and nonsmooth composite problem on manifolds. For the Riemannian smoothing stochastic gradient method, one can achieve the iteration complexity of $\mathcal{O}(\epsilon^{-5})$ for driving a generalized $\epsilon$-stationary point. Numerical experiments are conducted to validate the superiority of our algorithms

Fault-Tolerant Control for Systems with Unmatched Actuator Faults and Disturbances

A fault-tolerant control (FTC) scheme for a class of nonlinear systems with unmatched actuator redundancy and unmatched disturbances is proposed in this note. A methodology to construct unified smooth sliding mode control laws and update laws is proposed such that the equivalent injections of the first-order time derivatives of the unmatched actuator faults and unmatched disturbances can appear in the unmatched channels. The unmatched actuator faults and unmatched disturbances are completely canceled by these equivalent injections. Based on this methodology and using the backstepping design procedure, a set of smooth FTC sliding surfaces, FTC laws and update laws are then designed. With the help of the FTC law selecting mechanism, the output tracking errors of the closed-loop FTC system converge to zero asymptotically, and time-varying faults and disturbances are reconstructed. A simulation example is presented to illustrate the effectiveness of the proposed FTC method

Measuring Access to Medicines: A Survey of Prices, Availability and Affordability in Shaanxi Province of China

Objective: To measure the prices and availability of selected medicines in Shaanxi Province after the implementation of new healthcare reform in 2009. Methods: Data on the prices and availability of 47 medicines were collected from 50 public and 36 private sector medicine outlets in six regions of Shaanxi Province, Western China using a standardized methodology developed by the World Health Organization and Health Action International from September to October 2010. Medicine prices were compared with international reference prices to obtain a median price ratio. Affordability was measured as the number of days’ wages required for the lowest-paid unskilled government worker to purchase standard treatments for common conditions. Findings: The mean availabilities of originator brands and lowest-priced generics were 8.9% and 26.5% in the public sector, and 18.1% and 43.6% in the private sector, respectively. The public sector procured generics and originator brands at median price ratios of 0.75 and 8.49, respectively, while patients paid 0.97 and 10.16. Final patient prices for lowest-priced generics and originator brands in the private sector were about 1.53 and 8.36 times their international retail prices, respectively. Public sector vendors applied high markups of 30.4% to generics, and 19.6% to originator brands. In the private sector, originator brands cost 390.7% more, on average, than their generic equivalents. Generic medicines were priced 17.3% higher in the private sector than the public sector. The lowest-paid government worker would need 0.1 day’s wages to purchase captopril for lowest-priced generics from private sector, while 6.6 days’ wages for losartan. For originator brands, the costs rise to 1.2 days’ wages for salbutamol inhaler and 15.6 days’ wages for omeprazole. Conclusions: The prices, availability and affordability of medicines in China should be improved to ensure equitable access to basic medical treatments, especially for the poor. This requires multi-faceted interventions, as well as the review and refocusing of policies, regulations and educational interventions