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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Abstract	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.
AbstractList	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.
Author	Uskov, E. I. Solodov, M. V. Izmailov, A. F.
Author_xml	– sequence: 1 givenname: A. F. surname: Izmailov fullname: Izmailov, A. F. organization: VMK Faculty, OR Department, Lomonosov Moscow State University (MSU), Peoples’ Friendship University of Russia – sequence: 2 givenname: M. V. surname: Solodov fullname: Solodov, M. V. email: solodov@impa.br organization: Instituto de Matemática Pura e Aplicada (IMPA) – sequence: 3 givenname: E. I. surname: Uskov fullname: Uskov, E. I. organization: Tambov State University
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CitedBy_id	crossref_primary_10_1007_s10589_016_9890_5 crossref_primary_10_1080_02331934_2021_1981893 crossref_primary_10_1007_s10589_019_00096_8 crossref_primary_10_1007_s10107_019_01384_1 crossref_primary_10_1016_j_cam_2020_113368 crossref_primary_10_1080_10556788_2021_1908283 crossref_primary_10_1137_16M1088570 crossref_primary_10_1007_s11228_017_0461_4 crossref_primary_10_1007_s10107_023_01980_2 crossref_primary_10_1007_s10107_016_0994_6 crossref_primary_10_1007_s10589_020_00201_2 crossref_primary_10_1007_s10898_019_00867_x crossref_primary_10_3390_en10070936 crossref_primary_10_1155_2017_7076583 crossref_primary_10_1007_s10589_018_0038_7 crossref_primary_10_1007_s10589_022_00395_7 crossref_primary_10_1016_j_mechatronics_2018_07_002
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Keywords	Global convergence 65K15 65K05 90C30 Stabilized sequential quadratic programming Exact penalty function Second-order sufficiency Superlinear convergence Noncritical Lagrange multiplier
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Snippet	An iteration of the stabilized sequential quadratic programming method consists in solving a certain quadratic program in the primal-dual space, regularized in...
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SubjectTerms	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
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Title	Globalizing Stabilized Sequential Quadratic Programming Method by Smooth Primal-Dual Exact Penalty Function
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