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 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 Lp norm. In this paper, we consider the inverse linear programming problem under L1 norm (where ||d – c||p = j j j j J w | d c |, ∈ − ∑ with J denoting the index set of variables xj, and wj denoting the weight of the variable j) and under L∞ norm (where ||d – c||p = maxj∈J{wj|dj – cj|}). 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 non-unit 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 non-unit 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 non-unit 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. 1 Industrial and Systems Engineering Department, University of Florida, Gainesville, FL 32611, USA. 2 Sloan School of Management, MIT, Cambridge, MA 02139, USA.