Necessary and sufficient Karush–Kuhn–Tucker conditions for multiobjective Markov chains optimality

Abstract

The solution concepts proposed in this paper follow the Karush–Kuhn–Tucker (KKT) conditions for a Pareto optimal solution in finite-time, ergodic and controllable Markov chains multi-objective programming problems. In order to solve the problem we introduce the Tikhonov’s regularizator for ensuring the objective function is strict-convex. Then, we consider the c-variable method for introducing equality constraints that guarantee the result belongs to the simplex and satisfies ergodicity constraints. Lastly, we restrict the cost-functions allowing points in the Pareto front to have a small distance from one another. The computed image points give a continuous approximation of the whole Pareto surface. The constraints imposed by the c-variable method make the problem computationally tractable and, the restriction imposed by the small distance change ensures the continuation of the Pareto front. We transform the multi-objective nonlinear problem into an equivalent nonlinear programming problem by introducing the Lagrange function multipliers. As a result we obtain that the objective function is strict-convex, the inequality constraints are continuously differentiable and the equality constraint is an affine function. Under these settings, the KKT optimality necessary and sufficient conditions are elicited naturally. A numerical example is solved for providing the basic techniques to compute the Pareto optimal solutions by resorting to KKT conditions.