On a subproblem of trust region algorithms for constrained optimization
The author studies a subproblem which appears in some trust region algorithms for constrained optimization, where the original problem is to minimize a nonlinear function subject to equality constraints. It is the minimization of a general quadratic function with two special quadratic constraints. Some peculiarities of the problem, mainly with respect to the signs of the eigenvalues of the Hessian of the Lagrangian function at a solution are investigated. The main result is that the Hessian of the Lagrangian has at most one negative eigenvalue if the Lagrangian multipliers are unique. An example is presented to show that the Hessian may have a negative eigenvalue when one constraint is inactive at the solution.
scientific article; zbMATH DE number 4170649