This research studies two computational techniques that improve the practical performance of existing implementations of interior point methods for linear
programming. Both are based on the concept of symmetric neighbourhood as the
driving tool for the analysis of the good performance of some practical algorithms.
The symmetric neighbourhood adds explicit upper bounds on the complementarity pairs, besides the lower bound already present in the common N−1 neighbourhood. This allows the algorithm to keep under control the spread among
complementarity pairs and reduce it with the barrier parameter μ. We show that
a long-step feasible algorithm based on this neighbourhood is globally convergent
and converges in O(nL) iterations.
The use of the symmetric neighbourhood and the recent theoretical under-
standing of the behaviour of Mehrotra’s corrector direction motivate the introduction of a weighting mechanism that can be applied to any corrector direction,
whether originating from Mehrotra’s predictor–corrector algorithm or as part of
the multiple centrality correctors technique. Such modification in the way a correction is applied aims to ensure that any computed search direction can positively
contribute to a successful iteration by increasing the overall stepsize, thus avoid-
ing the case that a corrector is rejected. The usefulness of the weighting strategy is
documented through complete numerical experiments on various sets of publicly
available test problems. The implementation within the hopdm interior point
code shows remarkable time savings for large-scale linear programming problems.
The second technique develops an efficient way of constructing a starting point
for structured large-scale stochastic linear programs. We generate a computation-
ally viable warm-start point by solving to low accuracy a stochastic problem of
much smaller dimension. The reduced problem is the deterministic equivalent
program corresponding to an event tree composed of a restricted number of scenarios. The solution to the reduced problem is then expanded to the size of the
problem instance, and used to initialise the interior point algorithm. We present
theoretical conditions that the warm-start iterate has to satisfy in order to be
successful. We implemented this technique in both the hopdm and the oops
frameworks, and its performance is verified through a series of tests on problem
instances coming from various stochastic programming sources
