Benders Decomposition for a Class of Mathematical Programs with Constraints on Dual Variables

Interdependent systems with mutual feedback are best represented as a multi-level mathematical programming, in which the leader decisions determine the follower operations, which subsequently affect the leader operations. For instance, a recent paper formulated the optimization of electricity and natural gas systems with physical and economic couplings as a tri-level program that incorporates some constraints cutting off first-level solutions (e.g., unit commitment decisions) based on the dual solutions of the third-level problem (e.g., natural gas prices) to ensure economic viability. This tri-level program can be reformulated as a single-level Mixed-Integer Second-Order Cone Program (MISOCP), which is equivalent to a "standard" MISOCP for a joint electricity-gas system with additional constraints linking the first-level variables and the dual variables of the inner-continuous problem. This paper studies how to apply Benders decomposition to this class of mathematical programs. Since a traditional Benders decomposition results in computationally difficult subproblems, the paper proposes a dedicated Benders decomposition where the subproblem is further decomposed into two more tractable subproblems. The paper also shows that traditional acceleration techniques, such as the normalization of Benders feasibility cuts, can be adapted to this setting. Experimental results on a gas-aware unit commitment for coupled electricity and gas networks demonstrate the computational benefits of the approach compared to a state-of-the-art mathematical programming solver and an advanced Benders method with acceleration schemes.