### A Stabilized SQP Method: Global Convergence

Stabilized sequential quadratic programming (SQP) methods for nonlinear
optimization are designed to provide a sequence of iterates with fast local
convergence regardless of whether or not the active-constraint gradients are
linearly dependent. This paper concerns the global convergence properties
of a stabilized SQP method with a primal-dual augmented Lagrangian merit
function.

The proposed method incorporates several novel features. (i) A flexible
line search is used based on a direction formed from an approximate
solution of a strictly convex QP subproblem and, when one exists, a
direction of negative curvature for the primal-dual merit function. (ii)
When certain conditions hold, an approximate QP solution is computed by
solving a single linear system defined in terms of an estimate of the
optimal active set. The conditions exploit the formal equivalence between
the conventional stabilized SQP subproblem and a bound-constrained QP
associated with minimizing a quadratic model of the merit function. (iii)
It is shown that with an appropriate choice of termination condition, the
method terminates in a finite number of iterations without the assumption
of a constraint qualification. The method may be interpreted as an SQP
method with an augmented Lagrangian safeguarding strategy. This
safeguarding becomes relevant only when the iterates are converging to an
infeasible stationary point of the norm of the constraint violations.
Otherwise, the method terminates with a point that approximately satisfies
certain second-order necessary conditions for optimality. In this
situation, if all termination conditions are removed, then the limit points
either satisfy the same second-order necessary conditions exactly or fail
to satisfy a weak second-order constraint qualification. (iv) The global
convergence analysis concerns a specific algorithm that estimates the least
curvature of the merit function at each step. If negative curvature
directions are omitted, the analysis still applies and establishes
convergence to either first-order solutions or infeasible stationary
points.

The superlinear convergence of
the iterates and the formal local equivalence to stabilized SQP is
established in a companion paper (Report CCoM 14-01,
*Center for Computational Mathematics*, University of California, San Diego, 2014).

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