Manyeconomicmodels andoptimizationproblemsgenerate (endogenous) shadow prices—alias dual variables or Lagrange multipliers. Frequently the “slopes” of resulting price curves—that is, multiplier derivatives—are of great interest. These objects relate to the Jacobian of the optimality conditions. That particular matrix often has block structure. So, we derive explicit formulas for the inverse of ...

In this paper we introduce a variant of the three-field formulation where we use only two sets of variables. Considering, to fix the ideas, the homogeneous Dirichlet problem for −∆u = g in Ω, our variables are i) an approximation ψh of u on the skeleton (the union of the interfaces of the sub-domains) on an independent grid (that could often be uniform), and ii) the approximations uh of u in ea...

are called stationary Stokes equations, where u : Ω→ R denotes the velocity of the uid, p : Ω→ R denotes the pressure and f : Ω → R is the density of forces acting on the uid (e.g. gravitational force). The Stokes equations govern a ow of a steady, viscous, incompresible uid. We note that (1) is called the momentum equation and (2) is called the incompressibility equation. We supplement the sys...

In this paper, the Babuška’s theory of Lagrange multipliers is extended to higher order elliptic Dirichlet problems. The resulting variational formulation provides an efficient numerical squeme in meshless methods for the approximation of elliptic problems with essential boundary conditions. Mathematics Subject Classification. 41A10, 41A17, 65N15, 65N30. Received: April 5, 2004.

and there are no inequality constraints (i.e. there are no fi(x) i = 1, . . . , m). We simply write the p equality constraints in the matrix form as Cx− d = 0. The basic idea in Lagrangian duality is to take the constraints in (1) into account by augmenting the objective function with a weighted sum of the constraint functions. We define the Lagrangian L : R ×R ×R → R associated with the proble...

usual theorem of Lagrange multipliers says that a = (a1, . . . , an) ∈ Y is a critical point of f |Y if and only if there exists b = (b1, . . . , br) ∈ R such that (a;b) ∈ U×R is a critical point of the auxiliary function F = f+ ∑r i=1 yifi : U×R r → R. The point b is unique when it exists. We establish a closer relation between f and F for algebraic varieties over an arbitrary field K. Let X =...

The current paper has presented the study on theoretical dependences of optimization parameter (the degree transmission transparency) design factors elastic-damping mechanism (EDM) in for tractors small traction class (14 kN). purpose was to obtain a unified functional relationship that integrates early results. determination function based reaching compromise by method indefinite Lagrange mult...

Constrained extremal points of a potential can be represented as saddle Lagrangian. We prove existence such for nonquadratic potentials in Banach spaces under nonlinear constraints. then apply these results to the so-called mixed formulation elliptic partial differential equations.