To efficiently solve it, a hybrid Benders decomposition (HBD) algorithm combining the strengths of both mathematical programming and constraint programming is developed.
Hybrids of metaheuristics with other optimization techniques, like branch-and-bound, mathematical programming or constraint programming are also increasingly popular.
On the other hand, many papers appeared dealing with bilevel problems such as mathematical programming with equilibrium constraints [22], optimization problems with variational inequality constraints [20], optimization problems with Nash equilibrium constraints [20], optimization problems with equilibrium constraints [23, 24], etc.
To address this problem, we formulate a mathematical program with constraints involving the mixed strategy Nash equilibria of an operator-attacker game, and present a solution approach based on two combinatorial optimization problems, formulated as minimum set cover and maximum set packing problems.
Taking the fixed-point model as a constraint, the multimodal toll design problem is thus formulated as a mathematical programming with equilibrium constraints (MPEC) model.
The design optimization is a formulation of mathematical programming with equilibrium constraints (MPECs).
The methodology uses a decomposition method to solve the IPDC typically formulated as a mathematical programming (optimization with constraints) problem.
The toll design problem is formulated as a mixed-integer mathematical programming with equilibrium constraints (MPEC) model, which is solved by a Hybrid GA (Genetic Algorithm)–CA method.
We develop the FNDP formulation as bilevel stochastic mathematical programming with complementarity constraints (STOCH-MPEC) in which the bi-level formulation is converted to a single level using non-linear complementarity constraints conditions for user equilibrium (UE) problem.
In this study, this problem will be formulated as a mathematical programming (optimizattion with constraints) and solved by decomposing it into four hierarchiacal stages: (i) target selection, (ii) HEN design analysis, (iii) controllability analysis, and (iv) optimal selection and verification.
