80 research outputs found
An Extragradient-Based Alternating Direction Method for Convex Minimization
In this paper, we consider the problem of minimizing the sum of two convex
functions subject to linear linking constraints. The classical alternating
direction type methods usually assume that the two convex functions have
relatively easy proximal mappings. However, many problems arising from
statistics, image processing and other fields have the structure that while one
of the two functions has easy proximal mapping, the other function is smoothly
convex but does not have an easy proximal mapping. Therefore, the classical
alternating direction methods cannot be applied. To deal with the difficulty,
we propose in this paper an alternating direction method based on
extragradients. Under the assumption that the smooth function has a Lipschitz
continuous gradient, we prove that the proposed method returns an
-optimal solution within iterations. We apply the
proposed method to solve a new statistical model called fused logistic
regression. Our numerical experiments show that the proposed method performs
very well when solving the test problems. We also test the performance of the
proposed method through solving the lasso problem arising from statistics and
compare the result with several existing efficient solvers for this problem;
the results are very encouraging indeed
Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games
The goal in this paper is to approximate the Price of Stability (PoS) in
stochastic Nash games using stochastic approximation (SA) schemes. PoS is
amongst the most popular metrics in game theory and provides an avenue for
estimating the efficiency of Nash games. In particular, knowing the value of
PoS can help with designing efficient networked systems, including
transportation networks and power market mechanisms. Motivated by the lack of
efficient methods for computing the PoS, first we consider stochastic
optimization problems with a nonsmooth and merely convex objective function and
a merely monotone stochastic variational inequality (SVI) constraint. This
problem appears in the numerator of the PoS ratio. We develop a randomized
block-coordinate stochastic extra-(sub)gradient method where we employ a novel
iterative penalization scheme to account for the mapping of the SVI in each of
the two gradient updates of the algorithm. We obtain an iteration complexity of
the order that appears to be best known result for this class
of constrained stochastic optimization problems, where denotes an
arbitrary bound on suitably defined infeasibility and suboptimality metrics.
Second, we develop an SA-based scheme for approximating the PoS and derive
lower and upper bounds on the approximation error. To validate the theoretical
findings, we provide preliminary simulation results on a networked stochastic
Nash Cournot competition
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