3,391 research outputs found
A Primal-Dual Parallel Method with Convergence for Constrained Composite Convex Programs
This paper considers large scale constrained convex (possibly composite and
non-separable) programs, which are usually difficult to solve by interior point
methods or other Newton-type methods due to the non-smoothness or the
prohibitive computation and storage complexity for Hessians and matrix
inversions. Instead, they are often solved by first order gradient based
methods or decomposition based methods. The conventional primal-dual
subgradient method, also known as the Arrow-Hurwicz-Uzawa subgradient method,
is a low complexity algorithm with an convergence time.
Recently, a new Lagrangian dual type algorithm with a faster
convergence time is proposed in Yu and Neely (2017). However, if the objective
or constraint functions are not separable, each iteration of the Lagrangian
dual type method in Yu and Neely (2017) requires to solve a unconstrained
convex program, which can have huge complexity. This paper proposes a new
primal-dual type algorithm with convergence for general
constrained convex programs. Each iteration of the new algorithm can be
implemented in parallel with low complexity even when the original problem is
composite and non-separable.Comment: 23 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1604.0221
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
First-order methods for constrained convex programming based on linearized augmented Lagrangian function
First-order methods have been popularly used for solving large-scale
problems. However, many existing works only consider unconstrained problems or
those with simple constraint. In this paper, we develop two first-order methods
for constrained convex programs, for which the constraint set is represented by
affine equations and smooth nonlinear inequalities. Both methods are based on
the classic augmented Lagrangian function. They update the multipliers in the
same way as the augmented Lagrangian method (ALM) but employ different primal
variable updates. The first method, at each iteration, performs a single
proximal gradient step to the primal variable, and the second method is a block
update version of the first one.
For the first method, we establish its global iterate convergence as well as
global sublinear and local linear convergence, and for the second method, we
show a global sublinear convergence result in expectation. Numerical
experiments are carried out on the basis pursuit denoising and a convex
quadratically constrained quadratic program to show the empirical performance
of the proposed methods. Their numerical behaviors closely match the
established theoretical results
A Simple Parallel Algorithm with an Convergence Rate for General Convex Programs
This paper considers convex programs with a general (possibly
non-differentiable) convex objective function and Lipschitz continuous convex
inequality constraint functions. A simple algorithm is developed and achieves
an convergence rate. Similar to the classical dual subgradient
algorithm and the ADMM algorithm, the new algorithm has a parallel
implementation when the objective and constraint functions are separable.
However, the new algorithm has a faster convergence rate compared with
the best known convergence rate for the dual subgradient
algorithm with primal averaging. Further, it can solve convex programs with
nonlinear constraints, which cannot be handled by the ADMM algorithm. The new
algorithm is applied to a multipath network utility maximization problem and
yields a decentralized flow control algorithm with the fast
convergence rate.Comment: Published in SIAM Journal on Optimization, 2017. (This version also
corrected a minor iteration index typo in the description of the ADMM
algorithm at the top of page 4.
FBstab: A Stabilized Semismooth Quadratic Programming Algorithm with Applications in Model Predictive Control
This paper introduces the proximally stabilized Fischer-Burmeister method
(FBstab); a new algorithm for convex quadratic programming that synergistically
combines the proximal point algorithm with a primal-dual semismooth Newton-type
method. FBstab is numerically robust, easy to warmstart, handles degenerate
primal-dual solutions, detects infeasibility/unboundedness and requires only
that the Hessian matrix be positive semidefinite. We outline the algorithm,
provide convergence and convergence rate proofs, report some numerical results
from model predictive control benchmarks, and also include experimental
results. We show that FBstab is competitive with and often superior to, state
of the art methods, has attractive scaling properties, and is especially
promising for model predictive control applications
Stochastic Primal-Dual Coordinate Method with Large Step Size for Composite Optimization with Composite Cone-constraints
We introduce a stochastic coordinate extension of the first-order primal-dual
method studied by Cohen and Zhu (1984) and Zhao and Zhu (2018) to solve
Composite Optimization with Composite Cone-constraints (COCC). 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
COCC. We obtain almost surely convergence and O(1/t) expected convergence rate
in this work. The high probability complexity bound is also derived in this
paper.Comment: arXiv admin note: substantial text overlap with arXiv:1804.0080
An Inexact Interior-Point Lagrangian Decomposition Algorithm with Inexact Oracles
We develop a new inexact interior-point Lagrangian decomposition method to
solve a wide range class of constrained composite convex optimization problems.
Our method relies on four techniques: Lagrangian dual decomposition,
self-concordant barrier smoothing, path-following, and proximal-Newton
technique. It also allows one to approximately compute the solution of the
primal subproblems (called the slave problems), which leads to inexact oracles
(i.e., inexact gradients and Hessians) of the smoothed dual problem (called the
master problem). The smoothed dual problem is nonsmooth, we propose to use an
inexact proximal-Newton method to solve it. By appropriately controlling the
inexact computation at both levels: the slave and master problems, we still
estimate a polynomial-time iteration-complexity of our algorithm as in standard
short-step interior-point methods. We also provide a strategy to recover primal
solutions and establish complexity to achieve an approximate primal solution.
We illustrate our method through two numerical examples on well-known models
with both synthetic and real data and compare it with some existing
state-of-the-art methods.Comment: 34 pages, 2 figures, and 1 tabl
DuQuad: an inexact (augmented) dual first order algorithm for quadratic programming
In this paper we present the solver DuQuad specialized for solving general
convex quadratic problems arising in many engineering applications. When it is
difficult to project on the primal feasible set, we use the (augmented)
Lagrangian relaxation to handle the complicated constraints and then, we apply
dual first order algorithms based on inexact dual gradient information for
solving the corresponding dual problem. The iteration complexity analysis is
based on two types of approximate primal solutions: the primal last iterate and
an average of primal iterates. We provide computational complexity estimates on
the primal suboptimality and feasibility violation of the generated approximate
primal solutions. Then, these algorithms are implemented in the programming
language C in DuQuad, and optimized for low iteration complexity and low memory
footprint. DuQuad has a dynamic Matlab interface which make the process of
testing, comparing, and analyzing the algorithms simple. The algorithms are
implemented using only basic arithmetic and logical operations and are suitable
to run on low cost hardware. It is shown that if an approximate solution is
sufficient for a given application, there exists problems where some of the
implemented algorithms obtain the solution faster than state-of-the-art
commercial solvers.Comment: 25 pages, 6 figure
Asynchronous parallel primal-dual block coordinate update methods for affinely constrained convex programs
Recent several years have witnessed the surge of asynchronous (async-)
parallel computing methods due to the extremely big data involved in many
modern applications and also the advancement of multi-core machines and
computer clusters. In optimization, most works about async-parallel methods are
on unconstrained problems or those with block separable constraints.
In this paper, we propose an async-parallel method based on block coordinate
update (BCU) for solving convex problems with nonseparable linear constraint.
Running on a single node, the method becomes a novel randomized primal-dual BCU
with adaptive stepsize for multi-block affinely constrained problems. For these
problems, Gauss-Seidel cyclic primal-dual BCU needs strong convexity to have
convergence. On the contrary, merely assuming convexity, we show that the
objective value sequence generated by the proposed algorithm converges in
probability to the optimal value and also the constraint residual to zero. In
addition, we establish an ergodic convergence result, where is the
number of iterations. Numerical experiments are performed to demonstrate the
efficiency of the proposed method and significantly better speed-up performance
than its sync-parallel counterpart
Dual Smoothing and Level Set Techniques for Variational Matrix Decomposition
We focus on the robust principal component analysis (RPCA) problem, and
review a range of old and new convex formulations for the problem and its
variants. We then review dual smoothing and level set techniques in convex
optimization, present several novel theoretical results, and apply the
techniques on the RPCA problem. In the final sections, we show a range of
numerical experiments for simulated and real-world problems.Comment: 38 pages, 10 figures. arXiv admin note: text overlap with
arXiv:1406.108
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