We present an elementary self-contained proof for the Lagrange multiplier rule. It does not refer to any preliminary material and it is only based on the observation that a certain limit is positive. At the end of this note, the power of the Lagrange multiplier rule is analyzed.
We consider reformulations of fractional (hyperbolic) 0-1 programming problems as equivalent mixed-integer linear programs (MILP). The key idea of the proposed technique is to exploit binary representations of certain linear combinations of the 0-1 decision variables. Consequently, under some mild conditions, the number of product terms that need to be linearized can be greatly decreased. We perform numerical experiments comparing the proposed approach against the previous MILP reformulations used in the literature.
The paper suggests a new --- to the best of the author's knowledge --- characterization of Pareto-optimal decisions for the case of two-dimensional utility space which is not supposed to be convex. The main idea is to use the angle distances between the bisector of the first quadrant and points of utility space. A necessary and sufficient condition for Pareto optimality in the form of an equation is derived. The first-order necessary condition for optimality in the form of a pair of equations is also obtained.
To ensure sustainable cooperation in multistage games with vector payoffs we use the payment schedule based approach. The main dynamic properties of cooperative solutions used in single-criterion multistage games are extended to multicriteria games.
We design two recurrent payment schedules that satisfy such advantageous properties as the efficiency and the time consistency conditions, non-negativity and irrational behavior proofness.