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