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A Stabilized SQP Method: Superlinear Convergence
by Philip E. Gill, Vyacheslav Kungurtsev, Daniel P. Robinson
Abstract:
Stabilized sequential quadratic programming (sSQP) methods for nonlinear optimization generate a sequence of iterates with fast local convergence regardless of whether or not the active-constraint gradients are linearly dependent. This paper concerns the local convergence analysis of a sSQP method that uses a line search with a primal-dual augmented Lagrangian merit function to enforce global convergence (Report CCoM 13-04, Center for Computational Mathematics, University of California, San Diego, 2013). The method is provably well-defined and is based on solving a strictly convex quadratic programming subproblem at each iteration. It is shown that the method has superlinear local convergence under assumptions that are no stronger than those required by conventional stabilized SQP methods. The fast local convergence is obtained by allowing a small relaxation of the optimality conditions for the quadratic programming subproblem in the neighborhood of a solution. In the limit, the line search selects the unit step length, which implies that the method does not suffer from the Maratos effect. The analysis implies that the method has the same strong first- and second-order global convergence properties that have been established for augmented Lagrangian methods, yet is able to transition seamlessly to sSQP with fast local convergence in the neighborhood of a solution. Numerical results on some degenerate problems are reported.
UCSD-CCoM-14-01.pdf June 2014
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Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
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by Philip E. Gill, Vyacheslav Kungurtsev, Daniel P. Robinson
Abstract:
Stabilized sequential quadratic programming (sSQP) methods for nonlinear optimization generate a sequence of iterates with fast local convergence regardless of whether or not the active-constraint gradients are linearly dependent. This paper concerns the local convergence analysis of a sSQP method that uses a line search with a primal-dual augmented Lagrangian merit function to enforce global convergence (Report CCoM 13-04, Center for Computational Mathematics, University of California, San Diego, 2013). The method is provably well-defined and is based on solving a strictly convex quadratic programming subproblem at each iteration. It is shown that the method has superlinear local convergence under assumptions that are no stronger than those required by conventional stabilized SQP methods. The fast local convergence is obtained by allowing a small relaxation of the optimality conditions for the quadratic programming subproblem in the neighborhood of a solution. In the limit, the line search selects the unit step length, which implies that the method does not suffer from the Maratos effect. The analysis implies that the method has the same strong first- and second-order global convergence properties that have been established for augmented Lagrangian methods, yet is able to transition seamlessly to sSQP with fast local convergence in the neighborhood of a solution. Numerical results on some degenerate problems are reported.
UCSD-CCoM-14-01.pdf June 2014