This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity and regularity concepts, and propose a flexible augmented Lagrangian scheme. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. It is demonstrated how the inner subproblems can be solved by off-the-shelf proximal methods, notwithstanding the possibility to adopt any solvers, insofar as they return approximate stationary points. Finally, we describe our matrix-free implementation of the proposed algorithm and test it numerically. Illustrative examples show the versatility of constrained composite programs as a modeling tool and expose difficulties arising in this vast problem class.

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