Lagrange multipliers offer a way to find the extrema of an objective function f(x1, x2, . . . , xn) subject to a constraint function g(x1, x2, . . . , xn) = 0; for example, we may wish to maximize a+ b+ c subject to a2 + b2 + c2 = 1. However, there are lots of tiny details that need to be checked in order to completely solve a problem with Lagrange multipliers. On an olympiad the use of Lagrang...

A 3D slip-dependent frictional contact problem in elastostatics is discussed. We deliver a variational formulation as a mixed variational problem whose Lagrange multipliers set is solutiondependent. Then, the existence and the boundedness of the solutions is investigated. The proof is based on a recent result for an abstract mixed variational problem with solution-dependent set of Lagrange mult...

This paper provides the explicit expression of investment facing a binding debt ceiling and the explicit expression of the Lagrange multipliers related to the binding debt ceiling constraint. This result allows us to check for misspecification of the parameterizations of these Lagrange multipliers used in Euler investment equations tests during the 1990’s.  2000 Elsevier Science S.A. All right...

We consider optimization problems with equality, inequality, and abstract set constraints. We investigate the relations between various characteristics of the constraint set related to the existence of Lagrange multipliers. For problems with no abstract set constraint, the classical condition of quasiregularity provides the connecting link between the most common constraint qualifications and e...

When implementing a nonlinear constraint in quantum field theory by means of Lagrange multiplier, $\ensuremath{\lambda}(x)$, it is often the case that dynamics induce quadratic and even higher-order terms which then does not enforce anymore. This illustrated unimodular gravity, where metric tensor has to be ($g(x)\ensuremath{\equiv}\mathrm{det}\text{ }{g}_{\ensuremath{\mu}\ensuremath{\nu}}(x)=\...

‎In this paper, using the idea of convexificators, we study boundedness and nonemptiness of Lagrange multipliers satisfying the first order necessary conditions. We consider a class of nons- mooth fractional programming problems with equality, inequality constraints and an arbitrary set constraint. Within this context, define generalized Mangasarian-Fromovitz constraint qualification and sh...