Economic and physical considerations often lead to equilibrium problems in multidisciplinary design optimization (MDO). We introduce MDO problems with complementarity constraints to model such equilibrium problems in this paper. Also introduced is a Stackelberg game to model design decision making process in MDO with shared variables. We provide solution schemes based on the inexact penalty and augmented Lagrangian techniques. Two classes of problems are solved in the paper: a regular MDO modeled as a Stackelberg game and a new MDO with complementarity constraints. The proposed method was tested for Golinski's speed reducer design and shows promising results with convergence to the all-in-one solution.