Inverse Optimization

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 x 0 be a given feasible solution. The solution x 0 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 x 0 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 L 1 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 L 1 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 L 1 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 L 1 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 L 1 and L ∞ norms are also polynomially solvable.