Date of Completion
Peter B. Luh
Field of Study
Doctor of Philosophy
For many important mixed-integer programming (MIP) problems, the goal is to obtain near-optimal solutions with quantifiable quality in a computationally efficient manner (within, e.g., 5, 10 or 20 minutes). A traditional method to solve such problems has been Lagrangian relaxation, but the method suffers from zigzagging of multipliers and slow convergence. When solving mixed-integer linear programming (MILP) problems, the recently adopted branch-and-cut may also suffer from slow convergence because when the convex hull of the problems has complicated facial structures, facet-defining cuts are typically difficult to obtain, and the method relies mostly on time-consuming branching operations. In this thesis, the novel Surrogate Lagrangian Relaxation method is developed and its convergence is proved to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations a stepsizing formula, that guarantees convergence without requiring the optimal dual value, has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. At the same time, stepsizes are kept sufficiently large so that the algorithm does not terminate prematurely. At convergence, the lower-bound property of the surrogate dual is guaranteed. To solve MIP problems, based on Surrogate Lagrangian Relaxation, stable and accelerated convergence is ensured by introducing quadratic penalty terms motivated by Augmented Lagrangian relaxation. Convergence of Lagrange multipliers to their optimal values is significantly improved. When solving MILP problems, through the novel V-shape linearization, the relaxed problem is linearized to ensure that solutions consistent with those of the nonlinear relaxed problem can be obtained by branch-and-cut, thereby ensuring that convergence characteristics are very similar to those of Augmented Lagrangian relaxation. When solving block-structured MILP problems, the V-shaped relaxed problem allows the decomposition into much smaller MILP subproblems with exponentially reduced complexity as compared to the original problem thereby drastically reducing computational requirements. Moreover, analytical subproblem solutions can be obtained thereby making the reduction of computational requirements even more drastic. When solving large-scale unit commitment problems with combined cycle units as well as other problems for which facet defining cuts are difficult to obtain, it is demonstrated the new method is robust and much more efficient as compared to frequently-used branch-and-cut and surrogate Lagrangian relaxation, and represents a major step forward to solve difficult MIP and MILP problems.
Bragin, Mikhail, "An Efficient Solution Methodology for Mixed-Integer Programming Problems Arising in Power Systems" (2016). Doctoral Dissertations. 1318.