20 research outputs found
Standard Bundle Methods: Untrusted Models and Duality
We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function
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
Large-scale optimization with the primal-dual column generation method
The primal-dual column generation method (PDCGM) is a general-purpose column
generation technique that relies on the primal-dual interior point method to
solve the restricted master problems. The use of this interior point method
variant allows to obtain suboptimal and well-centered dual solutions which
naturally stabilizes the column generation. As recently presented in the
literature, reductions in the number of calls to the oracle and in the CPU
times are typically observed when compared to the standard column generation,
which relies on extreme optimal dual solutions. However, these results are
based on relatively small problems obtained from linear relaxations of
combinatorial applications. In this paper, we investigate the behaviour of the
PDCGM in a broader context, namely when solving large-scale convex optimization
problems. We have selected applications that arise in important real-life
contexts such as data analysis (multiple kernel learning problem),
decision-making under uncertainty (two-stage stochastic programming problems)
and telecommunication and transportation networks (multicommodity network flow
problem). In the numerical experiments, we use publicly available benchmark
instances to compare the performance of the PDCGM against recent results for
different methods presented in the literature, which were the best available
results to date. The analysis of these results suggests that the PDCGM offers
an attractive alternative over specialized methods since it remains competitive
in terms of number of iterations and CPU times even for large-scale
optimization problems.Comment: 28 pages, 1 figure, minor revision, scaled CPU time
Advances in interior point methods and column generation
In this thesis we study how to efficiently combine the column generation technique (CG)
and interior point methods (IPMs) for solving the relaxation of a selection of integer
programming problems. In order to obtain an efficient method a change in the column
generation technique and a new reoptimization strategy for a primal-dual interior point
method are proposed.
It is well-known that the standard column generation technique suffers from unstable
behaviour due to the use of optimal dual solutions that are extreme points of
the restricted master problem (RMP). This unstable behaviour slows down column
generation so variations of the standard technique which rely on interior points of the
dual feasible set of the RMP have been proposed in the literature. Among these techniques,
there is the primal-dual column generation method (PDCGM) which relies on
sub-optimal and well-centred dual solutions. This technique dynamically adjusts the
column generation tolerance as the method approaches optimality. Also, it relies on
the notion of the symmetric neighbourhood of the central path so sub-optimal and
well-centred solutions are obtained. We provide a thorough theoretical analysis that
guarantees the convergence of the primal-dual approach even though sub-optimal solutions
are used in the course of the algorithm. Additionally, we present a comprehensive
computational study of the solution of linear relaxed formulations obtained after applying
the Dantzig-Wolfe decomposition principle to the cutting stock problem (CSP), the
vehicle routing problem with time windows (VRPTW), and the capacitated lot sizing
problem with setup times (CLSPST). We compare the performance of the PDCGM
with the standard column generation method (SCGM) and the analytic centre cutting
planning method (ACCPM). Overall, the PDCGM achieves the best performance when
compared to the SCGM and the ACCPM when solving challenging instances from a
column generation perspective. One important characteristic of this column generation
strategy is that no speci c tuning is necessary and the algorithm poses the same level
of difficulty as standard column generation method. The natural stabilization available
in the PDCGM due to the use of sub-optimal well-centred interior point solutions is a
very attractive feature of this method. Moreover, the larger the instance, the better is
the relative performance of the PDCGM in terms of column generation iterations and
CPU time.
The second part of this thesis is concerned with the development of a new warmstarting
strategy for the PDCGM. It is well known that taking advantage of the previously
solved RMP could lead to important savings in solving the modified RMP. However,
this is still an open question for applications arising in an integer optimization context
and the PDCGM. Despite the current warmstarting strategy in the PDCGM working
well in practice, it does not guarantee full feasibility restorations nor considers the
quality of the warmstarted iterate after new columns are added. The main motivation
of the design of the new warmstarting strategy presented in this thesis is to close this
theoretical gap. Under suitable assumptions, the warmstarting procedure proposed in this thesis restores primal and dual feasibilities after the addition of new columns in
one step. The direction is determined so that the modi cation of small components at
a particular solution is not large. Additionally, the strategy enables control over the
new duality gap by considering an expanded symmetric neighbourhood of the central
path. As observed from our computational experiments solving CSP and VRPTW, one
can conclude that the warmstarting strategies for the PDCGM are useful when dense
columns are added to the RMP (CSP), since they consistently reduce the CPU time
and also the number of iterations required to solve the RMPs on average. On the other
hand, when sparse columns are added (VRPTW), the coldstart used by the interior
point solver HOPDM becomes very efficient so warmstarting does not make the task
of solving the RMPs any easier
Modeling and solving a vehicle-sharing problem
Motivated by the change in mobility patterns, we present a new modeling
approach for the vehicle-sharing problem. We aim at assigning vehicles to
user-trips so as to maximize savings compared to other modes of transport. We
base our formulations on the minimum-cost and the multi-commodity flow problem.
These formulations make the problem applicable in daily operations. In the
analysis we discuss an optimal composition of a shared fleet, restricted sets
of modes of transport, and variations of the objective function