Local convergence analysis of augmented Lagrangian method for nonlinear semidefinite programming
Shiwei Wang, Chao Ding
TL;DR
This paper proves the local convergence rate of the augmented Lagrangian method for nonlinear semidefinite programming, showing it converges Q-linearly under certain conditions without requiring multiplier uniqueness.
Contribution
It establishes the local convergence rate of ALM for NLSDP without assuming multiplier uniqueness, under SOSC and semi-isolated calmness.
Findings
Q-linear convergence of primal-dual sequences is proven.
Convergence holds under SOSC and semi-isolated calmness.
No need for multiplier uniqueness assumption.
Abstract
The augmented Lagrangian method (ALM) has gained tremendous popularity for its elegant theory and impressive numerical performance since it was proposed by Hestenes and Powell in 1969. It has been widely used in numerous efficient solvers to improve numerical performance to solve many problems. In this paper, without requiring the uniqueness of multipliers, the local (asymptotic Q-superlinear) Q-linear convergence rate of the primal-dual sequences generated by ALM for the nonlinear semidefinite programming (NLSDP) is established by assuming the second-order sufficient condition (SOSC) and the semi-isolated calmness of the Karush-Kuhn-Tucker (KKT) solution under some mild conditions.
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Taxonomy
TopicsAdvanced Optimization Algorithms Research · Optimization and Variational Analysis · Sparse and Compressive Sensing Techniques