9,660 research outputs found
On inexact Newton directions in interior point methods for linear optimization
In each iteration of the interior point method (IPM) at least one linear system
has to be solved. The main computational effort of IPMs consists in the computation
of these linear systems. Solving the corresponding linear systems
with a direct method becomes very expensive for large scale problems.
In this thesis, we have been concerned with using an iterative method for
solving the reduced KKT systems arising in IPMs for linear programming.
The augmented system form of this linear system has a number of advantages,
notably a higher degree of sparsity than the normal equations form.
We design a block triangular preconditioner for this system which is constructed
by using a nonsingular basis matrix identified from an estimate of
the optimal partition in the linear program. We use the preconditioned conjugate
gradients (PCG) method to solve the augmented system. Although
the augmented system is indefinite, short recurrence iterative methods such
as PCG can be applied to indefinite system in certain situations. This approach
has been implemented within the HOPDM interior point solver.
The KKT system is solved approximately. Therefore, it becomes necessary
to study the convergence of IPM for this inexact case. We present the
convergence analysis of the inexact infeasible path-following algorithm, prove
the global convergence of this method and provide complexity analysis
Convergence Analysis of an Inexact Feasible Interior Point Method for Convex Quadratic Programming
In this paper we will discuss two variants of an inexact feasible interior
point algorithm for convex quadratic programming. We will consider two
different neighbourhoods: a (small) one induced by the use of the Euclidean
norm which yields a short-step algorithm and a symmetric one induced by the use
of the infinity norm which yields a (practical) long-step algorithm. Both
algorithms allow for the Newton equation system to be solved inexactly. For
both algorithms we will provide conditions for the level of error acceptable in
the Newton equation and establish the worst-case complexity results
Convergence analysis of an Inexact Infeasible Interior Point method for Semidefinite Programming
In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima,Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is prove
A distributed primal-dual interior-point method for loosely coupled problems using ADMM
In this paper we propose an efficient distributed algorithm for solving
loosely coupled convex optimization problems. The algorithm is based on a
primal-dual interior-point method in which we use the alternating direction
method of multipliers (ADMM) to compute the primal-dual directions at each
iteration of the method. This enables us to join the exceptional convergence
properties of primal-dual interior-point methods with the remarkable
parallelizability of ADMM. The resulting algorithm has superior computational
properties with respect to ADMM directly applied to our problem. The amount of
computations that needs to be conducted by each computing agent is far less. In
particular, the updates for all variables can be expressed in closed form,
irrespective of the type of optimization problem. The most expensive
computational burden of the algorithm occur in the updates of the primal
variables and can be precomputed in each iteration of the interior-point
method. We verify and compare our method to ADMM in numerical experiments.Comment: extended version, 50 pages, 9 figure
Harmonic and Refined Harmonic Shift-Invert Residual Arnoldi and Jacobi--Davidson Methods for Interior Eigenvalue Problems
This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) and
Jacobi--Davidson (HJD) methods as well as their refined variants RHSIRA and
RHJD for the interior eigenvalue problem. Each method needs to solve an inner
linear system to expand the subspace successively. When the linear systems are
solved only approximately, we are led to the inexact methods. We prove that the
inexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts well
when the inner linear systems are solved with only low or modest accuracy. We
show that (i) the exact HSIRA and HJD expand subspaces better than the exact
SIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than the
exact HSIRA and HJD. Based on the theory, we design stopping criteria for inner
solves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJD
algorithms. Numerical results demonstrate that these algorithms are much more
efficient than the restarted standard SIRA and JD algorithms and furthermore
the refined harmonic algorithms outperform the harmonic ones very
substantially.Comment: 15 pages, 4 figure
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given
Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility
Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted
significant interest as in several instances they provably yield the global
optimum to the original non-convex problem. If, however, the relaxation is
inexact, the obtained solution is not AC-feasible. The quality of the obtained
solution is essential for several practical applications of AC-OPF, but
detailed analyses are lacking in existing literature. This paper aims to cover
this gap. We provide an in-depth investigation of the solution characteristics
when convex relaxations are inexact, we assess the most promising AC
feasibility recovery methods for large-scale systems, and we propose two new
metrics that lead to a better understanding of the quality of the identified
solutions. We perform a comprehensive assessment on 96 different test cases,
ranging from 14 to 3120 buses, and we show the following: (i) Despite an
optimality gap of less than 1%, several test cases still exhibit substantial
distances to both AC feasibility and local optimality and the newly proposed
metrics characterize these deviations. (ii) Penalization methods fail to
recover an AC-feasible solution in 15 out of 45 cases, and using the proposed
metrics, we show that most failed test instances exhibit substantial distances
to both AC-feasibility and local optimality. For failed test instances with
small distances, we show how our proposed metrics inform a fine-tuning of
penalty weights to obtain AC-feasible solutions. (iii) The computational
benefits of warm-starting non-convex solvers have significant variation, but a
computational speedup exists in over 75% of the cases
Fast algorithms for large scale generalized distance weighted discrimination
High dimension low sample size statistical analysis is important in a wide
range of applications. In such situations, the highly appealing discrimination
method, support vector machine, can be improved to alleviate data piling at the
margin. This leads naturally to the development of distance weighted
discrimination (DWD), which can be modeled as a second-order cone programming
problem and solved by interior-point methods when the scale (in sample size and
feature dimension) of the data is moderate. Here, we design a scalable and
robust algorithm for solving large scale generalized DWD problems. Numerical
experiments on real data sets from the UCI repository demonstrate that our
algorithm is highly efficient in solving large scale problems, and sometimes
even more efficient than the highly optimized LIBLINEAR and LIBSVM for solving
the corresponding SVM problems
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