A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds.
Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

Comment: 36 pages

• 49J52 - Nonsmooth analysis
• 49J53 - Set-valued and variational analysis
• 65K10 - Numerical optimization and variational techniques
• 90C26 - Nonconvex programming, global optimization
• 90C30 - Nonlinear programming
• 90C33 - Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)