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On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming
problems
We consider an inverse problem raised from the semi-definite
quadratic programming (SDQP) problem. In the inverse problem, the
parameters in the objective function of a given SDQP problem are
adjusted as little as possible so that a known feasible solution
becomes the optimal one. We formulate this problem as a minimization
problem with a positive semi-definite cone constraint and its dual
is a linearly positive semi-definite cone constrained semismoothly
differentiable ($\mbox{SC}^1$) convex programming problem with fewer
variables than the original one. We demonstrate the global
convergence of the augmented Lagrangian method for the dual problem
and prove that the convergence rate of primal iterates, generated
by the augmented Lagrange method, is proportional to $1/t$, and the
rate of multiplier iterates is proportional to $1/\sqrt{t}$, where
$t$ is the penalty parameter in the augmented Lagrangian. The
numerical results are reported to show the effectiveness of the
augmented Lagrangian method for solving the inverse semi-definite
quadratic programming problem.