A Primal-Dual Algorithmic Framework for Constrained Convex Minimization

We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap technique in a structured fashion and unifies it with smoothing and primal-dual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction method-of-multipliers methods as its special cases, and provides optimal convergence rates on the primal objective residual as well as the primal feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure

Global Positioning: the Uniqueness Question and a New Solution Method

We provide a new algebraic solution procedure for the global positioning problem in $n$ dimensions using $m$ satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures. We fill a gap in the literature by giving a proof for the long-held belief that when $m \ge n+2$, the solution is unique for almost all user positions. Even better, when $m \ge 2n+2$, almost all satellite configurations will guarantee a unique solution for all user positions. Some of our results are obtained using tools from algebraic geometry