20 research outputs found
Primal-dual accelerated gradient methods with small-dimensional relaxation oracle
In this paper, a new variant of accelerated gradient descent is proposed. The
pro-posed method does not require any information about the objective function,
usesexact line search for the practical accelerations of convergence, converges
accordingto the well-known lower bounds for both convex and non-convex
objective functions,possesses primal-dual properties and can be applied in the
non-euclidian set-up. Asfar as we know this is the rst such method possessing
all of the above properties atthe same time. We also present a universal
version of the method which is applicableto non-smooth problems. We demonstrate
how in practice one can efficiently use thecombination of line-search and
primal-duality by considering a convex optimizationproblem with a simple
structure (for example, linearly constrained)
Scalable Convex Methods for Phase Retrieval
This paper describes scalable convex optimization methods for phase retrieval. The main characteristics of these methods are the cheap per-iteration complexity and the low-memory footprint. With a variant of the original PhaseLift formulation, we first illustrate how to leverage the scalable Frank-Wolfe (FW) method (also known as the conditional gradient algorithm), which requires a tuning parameter. We demonstrate that we can estimate the tuning parameter of the FW algorithm directly from the measurements, with rigorous theoretical guarantees. We then illustrate numerically that recent advances in universal primal-dual convex optimization methods offer significant scalability improvements over the FW method, by recovering full HD resolution color images from their quadratic measurements
A universal accelerated primal-dual method for convex optimization problems
This work presents a universal accelerated first-order primal-dual method for
affinely constrained convex optimization problems. It can handle both Lipschitz
and H\"{o}lder gradients but does not need to know the smoothness level of the
objective function. In line search part, it uses dynamically decreasing
parameters and produces approximate Lipschitz constant with moderate magnitude.
In addition, based on a suitable discrete Lyapunov function and tight decay
estimates of some differential/difference inequalities, a universal optimal
mixed-type convergence rate is established. Some numerical tests are provided
to confirm the efficiency of the proposed method