Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization

Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point

Phase-Resolved Timing Analysis of GRS 1915+105 in Its {\rho} State

We made a phase-resolved timing analysis of GRS 1915+105 in its {\rho} state and obtained detailed {\rho} cycle evolutions of the frequency, the amplitude and the coherence of low-frequency quasi-periodic oscillation (LFQPO). We combined our timing results with the spectral study by Neilsen et al. to perform an elaborate comparison analysis. Our analyses show that the LFQPO frequency does not scale with the inner disk radius, but it is related to the spectral index, indicating a possible correlation between the LFQPOs and the corona. The LFQPO amplitude spectrum and other results are naturally explained by tying the LFQPO to the corona. The similarities of the spectra of variability parameters between the LFQPO from {\rho} state and those from more steady states indicate that the LFQPOs of GRS 1915+105 in very different states seem to share the same origin.Comment: 21 pages, 4 figures, accepted for publication in Ap