Globalizing Stabilized Sequential Quadratic Programming Method by Smooth Primal-Dual Exact Penalty Function

An iteration of the stabilized sequential quadratic programming method consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming is that no constraint qualifications are re...

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Veröffentlicht in:Journal of optimization theory and applications Jg. 169; H. 1; S. 148 - 178
Hauptverfasser: Izmailov, A. F., Solodov, M. V., Uskov, E. I.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.04.2016
Springer Nature B.V
Schlagworte:
Algorithms
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Convergence
Engineering
Globalization
Lagrange multiplier
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Penalty function
Preserves
Quadratic programming
Studies
Tasks
Theory of Computation
Global convergence
65K15
65K05
90C30
Stabilized sequential quadratic programming
Exact penalty function
Second-order sufficiency
Superlinear convergence
Noncritical Lagrange multiplier
ISSN:0022-3239, 1573-2878
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Zusammenfassung:An iteration of the stabilized sequential quadratic programming method consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming is that no constraint qualifications are required for fast local convergence (i.e., the problem can be degenerate). In particular, for equality-constrained problems, the superlinear rate of convergence is guaranteed under the only assumption that the primal-dual starting point is close enough to a stationary point and a noncritical Lagrange multiplier (the latter being weaker than the second-order sufficient optimality condition). However, unlike for the usual sequential quadratic programming method, designing natural globally convergent algorithms based on the stabilized version proved quite a challenge and, currently, there are very few proposals in this direction. For equality-constrained problems, we suggest to use for the task linesearch for the smooth two-parameter exact penalty function, which is the sum of the Lagrangian with squared penalizations of the violation of the constraints and of the violation of the Lagrangian stationarity with respect to primal variables. Reasonable global convergence properties are established. Moreover, we show that the globalized algorithm preserves the superlinear rate of the stabilized sequential quadratic programming method under the weak conditions mentioned above. We also present some numerical experiments on a set of degenerate test problems.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-016-0889-y