 
Summary: THE INVERSE OPTIMAL VALUE PROBLEM
SHABBIR AHMED AND YONGPEI GUAN
Abstract. This paper considers the following inverse optimization problem:
given a linear program, a desired optimal objective value, and a set of feasible
cost vectors, determine a cost vector such that the corresponding optimal ob
jective value of the linear program is closest to the desired value. The above
problem, referred here as the inverse optimal value problem, is significantly
different from standard inverse optimization problems that involve determin
ing a cost vector for a linear program such that a prespecified solution vector
is optimal. In this paper, we show that the inverse optimal value problem is
NPhard in general. We identify conditions under which the problem reduces
to a concave maximization or a concave minimization problem. We provide
sufficient conditions under which the associated concave minimization prob
lem and, correspondingly, the inverse optimal value problem is polynomially
solvable. For the case when the set of feasible cost vectors is polyhedral, we
describe an algorithm for the inverse optimal value problem based on solving
linear and bilinear programming problems. Some preliminary computational
experience is reported.
Keywords. Inverse optimization, Complexity, Linear programming, Bilinear
programming.
