Combining stabilized SQP with the augmented Lagrangian algorithm

For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavi...

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Bibliographic Details
Published in:	Computational optimization and applications Vol. 62; no. 2; pp. 405 - 429
Main Authors:	Izmailov, A. F., Solodov, M. V., Uskov, E. I.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01-11-2015 Springer Nature B.V
Subjects:	Algorithms Computation Convergence Convex and Discrete Geometry Estimates Inequalities Lagrange multiplier Management Science Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Multipliers Operations Research Operations Research/Decision Theory Optimization Quadratic programming Statistics Studies Superlinear convergence Global convergence Stabilized sequential quadratic programming Augmented Lagrangian
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Summary:	For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit stepsize) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direction is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported.
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23
ISSN:	0926-6003 1573-2894
DOI:	10.1007/s10589-015-9744-6