The Lagrange multiplier rule revisited

In this paper we give a short novel proof of the well-known Lagrange multiplier rule, discuss the sources of the power of this rule and consider several applications of this rule. The new proof does not use the implicit function theorem and combines the advantages of two of the most well-known proofs: it provides the useful geometric insight of the elimination approach based on differentiable curves and technically it is not more complicated than the simple penalty approach. Then we emphasize that the power of the rule is the reversal of order of the natural tasks, elimination and differentiation. This turns the hardest task,elimination, from a nonlinear problem into a linear one. This phenomenon is illustrated by several convincing examples of applications of the rule to various areas. Finally we give three hints on the use of the rule.Lagrange multiplier rule;compactness;optimization

Novel insights into the multiplier rule

We present the Lagrange multiplier rule, one of the basic optimization methods, in a new way. Novel features include:• Explanation of the true source of the power of the rule: reversal of tasks, but not the use of multipliers.• A natural proof based on a simple picture, but not the usual technical derivation from the implicit function theorem.• A practical method to avoid the cumbersome second order conditions.• Applications from various areas of mathematics, physics, economics.• Some hnts on the use of the rule.bargaining;dynamical systems;economics;finance;multiplier rule;second order condition

Billiards in confocal quadrics as a pluri-Lagrangian system

We illustrate the theory of one-dimensional pluri-Lagrangian systems with the example of commuting billiard maps in confocal quadrics.Comment: 7 p

Some sensitivity results in stochastic optimal control: A Lagrange multiplier point of view

In this work we provide a first order sensitivity analysis of some parameterized stochastic optimal control problems. The parameters can be given by random processes. The main tool is the one-to-one correspondence between the adjoint states appearing in a weak form of the stochastic Pontryagin principle and the Lagrange multipliers associated to the state equation

Elementary approach to closed billiard trajectories in asymmetric normed spaces

We apply the technique of K\'aroly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.Comment: 10 figures added. The title change