Solving a class of constrained ‘black-box’ inverse variational inequalities

It is well known that a general network economic equilibrium problem can be formulated as a variational inequality (VI) and solving the VI will result in a description of network equilibrium state. In this paper, however, we discuss a class of normative control problem that requires the network equilibrium state to be in a linearly constrained set. We formulate the problem as an inverse variational inequality (IVI) because the variables and the mappings in the IVI are in the opposite positions of a classical VI. In addition, the mappings in IVI usually do not have any explicit forms and only implicit information on the functional value is available through exogenous evaluation or direct observation. For such class of network equilibrium control problem, we present a linearly constrained implicit IVI formulation and a solution method based on proximal point algorithm (PPA) that only needs functional values for given variables in the solution process.