arXiv subjects

Andreas Grothey

Publications and source records attributed to Andreas Grothey.

4 recordsLinked to original sources

arXiv2021

Primal heuristics for Dantzig-Wolfe decomposition for unit commitment

The unit commitment problem is a short-term planning problem in the energy industry. Dantzig-Wolfe decomposition is a popular approach to solve the problem. This paper focuses on primal heuristics used with Dantzig-Wolfe decomposition. We propose two primal heuristics: one based on decomposition and one based on machine learning. The first one uses the fractional solution to the restricted master problem to fix a subset of the integer variables. In each iteration of the column generation procedure, the primal heuristic obtains the fractional solution, checks whether each binary variable satisfies the integrality constraint and fix those which do. The remaining variables are then optimised quickly by a solver to find a feasible, near-optimal solution to the original instance. The other primal heuristic based on machine learning is of interest when the problems are to be solved repeatedly with different demand data but with the same problem structure. The primal heuristic uses a pre-trained neural network to fix a subset of the integer variables. In the training phase, a neural network is trained to predict for any demand data and for each binary variable how likely it is that the variable takes each of two possible values. After the training, given an instance to be solved, the prediction of the model is used with a rounding threshold to fix some binary variables. Our numerical experiments compare our methods with solving the undecomposed problem and also with other primal heuristics from the literature. The experiments reveal that the primal heuristic based on machine learning is superior when the suboptimality tolerance is relatively large, such as 0.5% or 0.25%, while the decomposition is the best when the tolerance is small, for example 0.1%.

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arXiv2021

Incremental cutting-plane method and its application

We consider regularized cutting-plane methods to minimize a convex function that is the sum of a large number of component functions. One important example is the dual problem obtained from Lagrangian relaxation on a decomposable problem. In this paper, we focus on an incremental variant of the regularized cutting-plane methods, which only evaluates a subset of the component functions in each iteration. We first consider a limited-memory setup where the method deletes cuts after a finite number of iterations. The convergence properties of the limited-memory methods are studied under various conditions on regularization. We then provide numerical experiments where the incremental method is applied to the dual problems derived from large-scale unit commitment problems. In many settings, the incremental method is able to find a solution of high precision in a shorter time than the non-incremental method.

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arXiv2021

Use of Machine Learning Models to Warmstart Column Generation for Unit Commitment

The unit commitment problem is an important optimization problem in the energy industry used to compute the most economical operating schedules of power plants. Typically, this problem has to be solved repeatedly with different data but with the same problem structure. Machine learning techniques have been applied in this context to find primal feasible solutions. On the other hand, Dantzig-Wolfe decomposition with a column generation procedure has been shown to be successful in solving the unit commitment problem to tight tolerance. We propose the use of machine learning models not to find primal feasible solutions directly but to generate initial dual values for the column generation procedure. Our numerical experiments compare machine learning based methods for warmstarting the column generation procedure with three baselines: column pre-population, the linear programming relaxation and coldstart. The experiments reveal that the machine learning approaches are able to find both tight lower bounds and accurate primal feasible solutions in a shorter time compared to the baselines. Furthermore, these approaches scale well to handle large instances.

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arXiv2020

On the Effectiveness of Sequential Linear Programming for the Pooling Problem

The aim of this paper is to compare the performance of a local solution technique -- namely Sequential Linear Programming (SLP) employing random starting points -- with state-of-the-art global solvers such as Baron and more sophisticated local solvers such as Sequential Quadratic Programming and Interior Point for the pooling problem. These problems can have many local optima, and we present a small example that illustrates how this can occur. We demonstrate that SLP -- usually deemed obsolete since the arrival of fast reliable QP solvers, Interior Point Methods and sophisticated global solvers -- is still the method of choice for an important class of pooling problem when the criterion is the quality of the solution found within a given acceptable time budget. In addition we introduce a new formulation, the qq-formulation, for the case of fixed demands, that exclusively uses proportional variables. We compare the performance of SLP and the global solver Baron on the qq-formulation and other common formulations. While Baron with the qq-formulation generates weaker bounds than with the other formulations tested, for both SLP and Baron the qq-formulation finds the best solutions within a given time budget. The qq-formulation can be strengthened by pq-like cuts in which case the same bounds as for the pq-formulation are obtained. However the associated time penalty due to the additional constraints results in poorer solution quality within the time budget.

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