A Global Convergence Theory for General Trust-Region-Based Algorithms for Equality Constrained Optimization
Dennis, J.E. Jr.
Maciel, Maria C.
This work presents a global convergence theory for a broad class of trust-region algorithms for the smooth nonlinear programming problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trust-region algorithms. The trial step is characterized by very mild conditions on its normal and tangential components. The normal component must satisfy a fraction of Cauchy decrease condition on the quadratic model of the linearized constraints. The tangential component then must satisfy a fraction of Cauchy decrease condition on a quadratic model of the Lagrangian function in the translated tangent space of the constraints determined by the normal component. The Lagrange multipliers and the Hessians are assumed only to be bounded. The other main characteristic of this class of algorithms is that the step is evaluated by using the augmented Lagrangian as a merit function and the penalty parameter is updated using the El-Alem scheme. The properties of the step together with the way that the penalty parameter is chosen are sufficient to establish global convergence. As an example, an algorithm is presented which can be viewed as a generalization of the Steihaug-Toint dogleg algorithm for the unconstrained case. It is based on a quadratic programming algorithm that uses as a feasible point a step in the normal direction to the tangent space of the constraints and then does feasible conjugate reduced-gradient steps to solve the quadratic program. This algorithm should cope quite well with large problems for which effective preconditions are known.
Citable link to this pagehttps://hdl.handle.net/1911/101768
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