A hypoquadratic convergence method for Lagrange multipliers

Convergence of a Substructuring Methodwith Lagrange Multipliers 1

Framework The main idea of our convergence analysis is summarized in the following lemma, which we will apply to F and M from (18) and (19). Lemma 1 Let U be a nite dimensional linear space with the inner product h ; i, k kU a norm on U , and the dual norm de ned by kukU 0 = sup~ u2U hu; ~ ui=k~ ukU . Let PV : U ! V be the h ; i orthogonal projection onto V , and F;M : U ! U symmetric linear op...

On the Method of Lagrange Multipliers

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...

Local Stability Conditions for the Babuska Method of Lagrange Multipliers

We consider the so-called Babuska method of finite elements with Lagrange multipliers for numerically solving the problem Au = f in il, u = g on 3Í2, iî C Rn, 7i > 2. We state a number of local conditions from which we prove the uniform stability of the Lagrange multiplier method in terms of a weighted, mesh-dependent norm. The stability conditions given weaken the conditions known so far and a...

Substructuring by Lagrange Multipliers

Lagrange Multipliers and Optimality

Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger t...