An augmented Lagrangian method for optimization problems with structured geometric constraints
研究了一种求解带几何约束优化问题的增广拉格朗日方法，将复杂非凸约束保留在约束中，仅对剩余约束进行惩罚，并利用非单调投影梯度法求解子问题，适用于互补约束、基数约束及矩阵秩约束等问题。
Abstract 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.