6,689 research outputs found
Convergence property of the Iri-Imai algorithm for some smooth convex programming problems
In this paper, the Iri-Imai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity, called the harmonic convexity in this paper. A characterization of this convexity condition is given. The same convexity condition was used by Mehrotra and Sun to prove the convergence of a path-following algorithm. The Iri-Imai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior-point algorithms for smooth convex programming are mainly based on the path-following approach
On the Global Linear Convergence of the ADMM with Multi-Block Variables
The alternating direction method of multipliers (ADMM) has been widely used
for solving structured convex optimization problems. In particular, the ADMM
can solve convex programs that minimize the sum of convex functions with
-block variables linked by some linear constraints. While the convergence of
the ADMM for was well established in the literature, it remained an open
problem for a long time whether or not the ADMM for is still
convergent. Recently, it was shown in [3] that without further conditions the
ADMM for may actually fail to converge. In this paper, we show that
under some easily verifiable and reasonable conditions the global linear
convergence of the ADMM when can still be assured, which is important
since the ADMM is a popular method for solving large scale multi-block
optimization models and is known to perform very well in practice even when
. Our study aims to offer an explanation for this phenomenon
Message-Passing Algorithms for Quadratic Programming Formulations of MAP Estimation
Computing maximum a posteriori (MAP) estimation in graphical models is an
important inference problem with many applications. We present message-passing
algorithms for quadratic programming (QP) formulations of MAP estimation for
pairwise Markov random fields. In particular, we use the concave-convex
procedure (CCCP) to obtain a locally optimal algorithm for the non-convex QP
formulation. A similar technique is used to derive a globally convergent
algorithm for the convex QP relaxation of MAP. We also show that a recently
developed expectation-maximization (EM) algorithm for the QP formulation of MAP
can be derived from the CCCP perspective. Experiments on synthetic and
real-world problems confirm that our new approach is competitive with
max-product and its variations. Compared with CPLEX, we achieve more than an
order-of-magnitude speedup in solving optimally the convex QP relaxation
Bregman Proximal Gradient Algorithm with Extrapolation for a class of Nonconvex Nonsmooth Minimization Problems
In this paper, we consider an accelerated method for solving nonconvex and
nonsmooth minimization problems. We propose a Bregman Proximal Gradient
algorithm with extrapolation(BPGe). This algorithm extends and accelerates the
Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive
global Lipschitz gradient continuity assumption needed in Proximal Gradient
algorithms (PG). The BPGe algorithm has higher generality than the recently
introduced Proximal Gradient algorithm with extrapolation(PGe), and besides,
due to the extrapolation step, BPGe converges faster than BPG algorithm.
Analyzing the convergence, we prove that any limit point of the sequence
generated by BPGe is a stationary point of the problem by choosing parameters
properly. Besides, assuming Kurdyka-{\'L}ojasiewicz property, we prove the
whole sequences generated by BPGe converges to a stationary point. Finally, to
illustrate the potential of the new method BPGe, we apply it to two important
practical problems that arise in many fundamental applications (and that not
satisfy global Lipschitz gradient continuity assumption): Poisson linear
inverse problems and quadratic inverse problems. In the tests the accelerated
BPGe algorithm shows faster convergence results, giving an interesting new
algorithm.Comment: Preprint submitted for publication, February 14, 201
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