  We consider the problem of learning optimal solutions of a partially known
linear optimization problem and recovering its underlying cost function where a
set of past decisions and the feasible set are known. We develop a new
framework, denoted as Inverse Learning, that extends the inverse optimization
literature to (1) learn the optimal solution of the underlying problem, (2)
integrate additional information on constraints and their importance, and (3)
control the balance between mimicking past behaviors and reaching new goals and
rules for the learned solution. We pose inverse learning as an optimization
problem that maps given (feasible and infeasible) observations to a single
optimal solution with minimum perturbation, hence, not only recovering the
missing cost vector but also providing an optimal solution simultaneously. The
framework provides insights into an essential tradeoff in recovering linear
optimization problems with regard to preserving observed behaviors and binding
constraints of the known feasible set at optimality. We propose a series of
mixed integer linear programming models to capture the effects of this tradeoff
and validate it using a two-dimensional example. We then demonstrate the
framework's applicability to a diet recommendation problem for a population of
hypertension and prediabetic patients. The goal is to balance dietary
constraints to achieve the necessary nutritional goals with the dietary habits
of the users to encourage adherence to the diet. Results indicate that our
models recommend daily food intakes that preserve the original data trends
while providing a range of options to patients and providers based on the
tradeoff.
