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

• Published in: Journal of optimization theory and applications Vol. 169; no. 1; pp. 148 - 178
• Main Authors: Izmailov, A. F., Solodov, M. V., Uskov, E. I.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-04-2016; Springer Nature B.V
• Subjects: 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
• DOI: 10.1007/s10957-016-0889-y

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.