35 research outputs found
Stochastic Primal-Dual Coordinate Method for Nonlinear Convex Cone Programs
Block coordinate descent (BCD) methods and their variants have been widely
used in coping with large-scale nonconstrained optimization problems in many
fields such as imaging processing, machine learning, compress sensing and so
on. For problem with coupling constraints, Nonlinear convex cone programs
(NCCP) are important problems with many practical applications, but these
problems are hard to solve by using existing block coordinate type methods.
This paper introduces a stochastic primal-dual coordinate (SPDC) method for
solving large-scale NCCP. In this method, we randomly choose a block of
variables based on the uniform distribution. The linearization and Bregman-like
function (core function) to that randomly selected block allow us to get simple
parallel primal-dual decomposition for NCCP. The sequence generated by our
algorithm is proved almost surely converge to an optimal solution of primal
problem. Two types of convergence rate with different probability (almost
surely and expected) are also obtained. The probability complexity bound is
also derived in this paper
An Augmented Lagrangian Approach to Conically Constrained Non-monotone Variational Inequality Problems
In this paper we consider a non-monotone (mixed) variational inequality model
with (nonlinear) convex conic constraints. Through developing an equivalent
Lagrangian function-like primal-dual saddle-point system for the VI model in
question, we introduce an augmented Lagrangian primal-dual method, to be called
ALAVI in the current paper, for solving a general constrained VI model. Under
an assumption, to be called the primal-dual variational coherence condition in
the paper, we prove the convergence of ALAVI. Next, we show that many existing
generalized monotonicity properties are sufficient -- though by no means
necessary -- to imply the above mentioned coherence condition, thus are
sufficient to ensure convergence of ALAVI. Under that assumption, we further
show that ALAVI has in fact an global rate of convergence where
is the iteration count. By introducing a new gap function, this rate
further improves to be if the mapping is monotone. Finally, we show
that under a metric subregularity condition, even if the VI model may be
non-monotone the local convergence rate of ALAVI improves to be linear.
Numerical experiments on some randomly generated highly nonlinear and
non-monotone VI problems show practical efficacy of the newly proposed method