  Most inverse optimization models impute unspecified parameters of an
objective function to make an observed solution optimal for a given
optimization problem with a fixed feasible set. We propose two approaches to
impute unspecified left-hand-side constraint coefficients in addition to a cost
vector for a given linear optimization problem. The first approach identifies
parameters minimizing the duality gap, while the second minimally perturbs
prior estimates of the unspecified parameters to satisfy strong duality, if it
is possible to satisfy the optimality conditions exactly. We apply these two
approaches to the general linear optimization problem. We also use them to
impute unspecified parameters of the uncertainty set for robust linear
optimization problems under interval and cardinality constrained uncertainty.
Each inverse optimization model we propose is nonconvex, but we show that a
globally optimal solution can be obtained either in closed form or by solving a
linear number of linear or convex optimization problems.
