Contributions to dual subgradient optimization and maintenance scheduling

Licentiate thesis, 2013

This thesis analyses two topics within the area of mathematical optimization; dual subgradient methods and maintenance optimization. The first two papers consider dual subgradient methods, and the third paper considers maintenance optimization. In the first paper, we apply a subgradient optimization method to the Lagrangian dual of a general convex and (possibly) nonsmooth optimization problem. The dual subgradient method produce neither primal feasible nor primal optimal solutions automatically. We show that convergence to the set of optimal primal solutions can be obtained by constructing a class of ergodic sequences of Lagrangian subproblem solutions. We generalize previous convergence results for such ergodic sequences and propose a new set of rules for choosing the convexity weights defining the sequences. Numerical results indicate that the application of the new set of rules generates primal solutions of higher quality than those created by the previously developed rules. In the second paper, we analyze the properties of a subgradient method when applied to the Lagrangian dual of an infeasible linear program. We show that the primal-dual linear program is associated with a saddle point problem in which the primal part amounts to finding a solution in the primal space with the minimum amount of infeasibility in the relaxed constraints; the dual part aims to identify a steepest feasible ascent direction. We establish that the composed sequence of primal and dual solutions, generated by a specific primal ergodic sequence and scaled dual iterates, respectively, converges to a saddle point for the primal-dual program. In the third paper, we study the preventive maintenance scheduling problem with interval costs; a problem in which maintenance should be scheduled for the components in a multi-component system. The objective is to minimize the sum of the set-up and interval costs for the system. We show that this problem is NP-hard and present a $0-1$ integer linear programming model for the problem which was originally introduced for the joint replenishment problem. The model is utilized in three case studies from the railway, aircraft, and wind power industries.