A Scalable Solution Methodology for Mixed-Integer Linear Programming Problems Arising in Automation

Many operation optimization problems such as scheduling and assignment of interest to the automation community are mixed-integer linear programming (MILP) problems. Because of their combinatorial nature, the effort required to obtain optimal solutions increases drastically as the problem size increases. Such operation optimization problems typically need to be solved several times a day and require short solving times (e.g., 5, 10, or 20 min). The goal is, therefore, to obtain near-optimal solutions with quantifiable quality in a computationally efficient manner. Existing MILP methods, however, suffer from slow convergence and may not efficiently achieve this goal. In this paper, motivated by fast convergence of augmented Lagrangian relaxation (LR), a novel advanced price-based decomposition and coordination “surrogate absolute-value LR” (SAVLR) approach is developed. Within the method, convergence of our recent surrogate LR (SLR), which has overcome all major difficulties of traditional LR, is significantly improved by penalizing constraint violations by adding “absolute-value” penalties. Moreover, such penalties are efficiently linearized in a standard way, thereby enabling the use of MILP solvers. By exploiting the beautiful property of exponential reduction of complexity of subproblems upon decomposition, subproblems are efficiently solved and their solutions are efficiently coordinated by updating Lagrangian multipliers. Convergence is then proved under novel adjustment of penalty coefficients. A series of generalized assignment problems is considered, and for these problems, superior performance of SAVLR over other state-of-the-art and state-of-the-practice methods is demonstrated. Accompanying CPLEX codes, whereby SAVLR is implemented, are also included.
Note to Practitioners
—Examples of important problems that arise in automation community include scheduling and assignment problems. Because of their combinatorial nature, the effort required to obtain optimal solutions increases drastically as the problem size increases. Existing mixed-integer linear programming (MILP) methods, however, may suffer from slow convergence and may not efficiently achieve this goal. The new method revolutionizes the way such problems can be solved with major improvements on the overall performance. It is based on our recent breakthrough “surrogate Lagrangian relaxation” (LR), which has overcome all major difficulties of traditional LR while exploiting the beautiful property of exponential reduction of complexity upon decomposition. To significantly improve convergence while maintaining linearity so as to use MILP solvers, our idea is to penalize violations of relaxed constraints by the infrequently used “absolute-value” penalty functions. Although not differentiable, absolute-value penalties have the advantage of being exactly linearizable through extra variables and constraints. The difficulties caused by those extra constraints, which couple subproblems, are resolved by adaptive adjustment of penalty coefficients. A series of generalized assignment problems is considered and superior performance of the new method against state-of-the-art and state-of-the-practice methods is demonstrated. Accompanying CPLEX codes whereby the new method is implemented are also included.