Inverse Variational Inequality Research Articles

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.

Dynamic power price problem: An inverse variational inequality approach

This paper considers an optimal control perspective on dynamic power price problem where the load on the power-grid is controlled via price. The optimal regulatory price is characterized by inverse variational inequality in which the function value and the control variable are in the opposite positions of the classical variational inequality. Discrete and continuum models with load constraints are developed and existence theorems are established under quite reasonable assumptions. Preliminary numerical results also show the feasibility of the proposed models.