Automated tight Lyapunov analysis for first-order methods

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We provide a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality that amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle-Pock method when the linear operator is the identity mapping

International scientific collaboration for a network mission to Mars

For the past 15 years here has been a strong consensus within the international scientific community in favor of sending a network of geophysical observatories to Mars. These landers would probe the interior of the planet through seismic monitoring, magnetic sounding, and measurements of its rotational dynamics. They would study the global circulation and regional dynamics of the atmosphere and through imaging would reveal the geology of a host of new places on the surface. The enthusiasm for this endeavor has been manifested in a number of mission studies, includiung MESUR Network, MarsNet and InterMarsNet. At the beginning of the 21st century the goal was close to being realized with the NetLander mission, which was sponsored by a broad consortium of nations in Europa and America. Unfortunately, budget difficulties and programmatic conflicts at CNES and NASA resulted in recent cancellation. However the scientific priorities for understanding the origin, evolution, environment and habitat of Mars remain, and a geophysical network is clearly required to make the measurements necessary to address these questions. A large group of scientists remain committed to goal of deploying an international Geoscience Observatory on Mars, whose data would be available to the world science community, and we welcome the participation of all who share this aim. To this end, we are exploring all international options, including ESA's Aurora program, the Scout programm of NASA's Mars Exploration Program, and NASA's new exploration initiatives