460 research outputs found
Interior-point solver for convex separable block-angular problems
Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner
may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the precondi-
tioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using l1 and l2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to l1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.Preprin
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
Improving an interior-point algorithm for multicommodity flows by quadratic regularizations
One of the best approaches for some classes of multicommodity flow problems is a specialized interior-point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned
conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work the authors improved this algorithm (i.e., reduced the spectral radius) for general block-angular problems by adding a quadratic
regularization to the logarithmic barrier. This barrier was shown to be self-concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work we focus on linear multicommodity problems, a particular case of primal block-angular ones. General results
are tailored for multicommodity flows, allowing a local sensitivity analysis
on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior-point algorithm is more efficient than the nonregularized one.
From this work it can be concluded that, if interior-point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones.Preprin
Quadratic regularizations in an interior-point method for primal block-angular problems
One of the most efficient interior-point methods for some classes of primal block-angular problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. Its efficiency depends on the spectral radius—in [0,1)— of a certain matrix in the definition of the preconditioner. Spectral radius close to 1 degrade the performance of the approach. The purpose of this work is twofold. First, to show that a separable quadratic regularization term in the objective reduces the spectral radius, significantly improving the overall performance in some classes of instances. Second, to consider a regularization term which decreases with the barrier function, thus with no need for an extra parameter. Computational experience with some primal block-angular problems confirms the efficiency of the regularized approach. In particular, for some difficult problems, the solution time is reduced by a factor of two to ten by the regularization term, outperforming state-of-the-art commercial solvers.Peer ReviewedPostprint (author’s final draft
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Improving an interior-point approach for large block-angular problems by hybrid preconditioners
The computational time required by interior-point methods
is often domi-
nated by the solution of linear systems of equations. An efficient spec
ialized
interior-point algorithm for primal block-angular proble
ms has been used to
solve these systems by combining Cholesky factorizations for the
block con-
straints and a conjugate gradient based on a power series precon
ditioner for
the linking constraints. In some problems this power series prec
onditioner re-
sulted to be inefficient on the last interior-point iterations, wh
en the systems
became ill-conditioned. In this work this approach is combi
ned with a split-
ting preconditioner based on LU factorization, which is main
ly appropriate
for the last interior-point iterations. Computational result
s are provided for
three classes of problems: multicommodity flows (oriented and no
noriented),
minimum-distance controlled tabular adjustment for statistic
al data protec-
tion, and the minimum congestion problem. The results show that
, in most
cases, the hybrid preconditioner improves the performance an
d robustness of
the interior-point solver. In particular, for some block-ang
ular problems the
solution time is reduced by a factor of 10.Peer ReviewedPreprin
Solving -CTA in 3D tables by an interior-point method for primal block-angular problems
The purpose of the field of statistical disclosure control is to avoid that no
confidential information can be derived from statistical data released by, mainly, national
statistical agencies. Controlled tabular adjustment (CTA) is an emerging technique
for the protection of statistical tabular data. Given a table to be protected, CTA
looks for the closest safe table. In this work we focus on CTA for three-dimensional
tables using the L1 norm for the distance between the original and protected tables.
Three L1-CTA models are presented, giving rise to six different primal block-angular
structures of the constraint matrices. The resulting linear programming problems are
solved by a specialized interior-point algorithm for this constraints structure, which
solves the normal equations by a combination of Cholesky factorization and preconditioned
conjugate gradients (PCG). In the past this algorithm shown to be one of
the most efficient approaches for some classes of block-angular problems. The effect
of quadratic regularizations is also analyzed, showing that for three of the six
primal block-angular structures the performance of PCG is guaranteed to improve.
Computational results are reported for a set of large instances, which provide linear
optimization problems of up to 50 millions of variables and 25 millions of constraints.
The specialized interior-point algorithm is compared with the state-of-the-art barrier
solver of the CPLEX 12.1 package, showing to be a more efficient choice for very
large L1-CTA instances.Preprin
- …