Advances in Interior Point Methods for Large-Scale Linear Programming

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

Optimization Algorithms for Power Grid Planning and Operational Problems.

The modern electrical grid is an engineering marvel. The power grid is an incredibly complex system that largely functions very reliably. However, aging infrastructure and changing power consumption and generation trends will necessitate that new investments be made and new operational regimes be explored to maintain this level of reliability. One of the primary difficulties in power grid planning is the presence of uncertainty. In this thesis, we address short-term (i.e., day-ahead) and long-term power system planning problems where there is uncertainty in the forecasted demand for power, future renewable generation levels, and/or possible component failures. We initially consider a network capacity design problem where there is uncertainty in the nodal supplies and demands. This robust single-commodity network design problem underlies several applications including power transmission networks. Minimum cost capacity expansion decisions are made to ensure that there exists a feasible network flow solution for alpha% of the demand scenarios in the given set, where alpha is a parameter specified by the user. We next consider a day-ahead planning problem that is specifically applicable to the power grid. We present an extension of the traditional unit commitment problem where we additionally consider (1) a more stringent security requirement and (2) a more flexible set of recovery actions. We require that feasible operation is possible for any simultaneous failure of k generators and/or transmission lines (i.e., N-k security), and transmission switching may be used to recover from a failure event. Finally, we consider a transmission expansion planning problem where there is uncertainty in future loads, renewal generation outputs and line failures, and transmission switching is also allowed as a recovery action. We propose a robust optimization model where feasible operation is required for all loads and renewable generation levels within given ranges, and for all single transmission line failures. For all three of these problems, novel algorithms are presented that enable these problems to be solved even when straight-forward formulations are too large to be tractable. Computational results are presented for each algorithm to provide insight into the advantages and limitations of these algorithms in practicePHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/107076/1/kaschu_1.pd

Robust Capacity Assignment in Telecommunications

Robust optimization, Capacity assignment in telecommunications, Cutting plane methods,