Inverse Optimization

Abstract

In this paper, we study inverse optimization problems defined as follows. Let S denote the set of feasible solutions of an optimization problem P, let c be a specified cost vector, and x⁰ be a given feasible solution. The solution x⁰ may or may not be an optimal solution of P with respect to the cost vector c. The inverse optimization problem is to perturb the cost vector c to d so that x⁰ is an optimal solution of P with respect to d and ‖d−c‖_p is minimum, where ‖d−c‖_p is some selected L_p norm. In this paper, we consider the inverse linear programming problem under L₁ norm (where ‖d−c‖_p = Σ_i∈J w_j|d_j−c_j|, with J denoting the index set of variables x_j and w_j denoting the weight of the variable j) and under L_∞ norm (where ‖d−c‖_p = max_j∈J{w_j|d_j−c_j|}). We prove the following results: (i) If the problem P is a linear programming problem, then its inverse problem under the L₁ as well as L_∞ norm is also a linear programming problem. (ii) If the problem P is a shortest path, assignment or minimum cut problem, then its inverse problem under the L₁ norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problem P is a minimum cost flow problem, then its inverse problem under the L₁ norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problem P is a minimum cost flow problem, then its inverse problem under the L_∞ norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problem P is polynomially solvable for linear cost functions, then inverse versions of P under the L₁ and L_∞ norms are also polynomially solvable.