We consider a class of three-objective mixed-integer linear programs (MILPs) where at least one of the objective functions take only discrete values. These problems commonly occur in MILPs where one or more of the three objective functions contain only integer decision variables. In such problems, the nondominated set consists of the union of nondominated edges and individual nondominated points. The nondominated edges can provide valuable insights on the trade-offs between the two continuous objectives at different levels of the discrete-valued objective. We develop an objective-space search algorithm that keeps partitioning the search space by progressively creating cones in the two dimensional feasible space of the two continuous objectives for relevant values of the discrete-valued objective. The algorithm generates the nondominated points or edges in the non-increasing order of the feasible values of the selected discrete-valued objective. Additionally, the algorithm uncovers all efficient integer variable vectors including different vectors that lead to the same nondominated points or edges. We apply the algorithm to the day-ahead electricity market clearing problem.
- mixed-integer linear program
- objective-space search algorithm