Abstract
In this paper, we consider a class of nonsmooth and nonconvex optimization problem with an abstract constraint. We propose an augmented Lagrangian method for solving the problem and construct global convergence under a weakly nonsmooth Mangasarian-Fromovitz constraint qualification. We show that any accumulation point of the iteration sequence generated by the algorithm is a feasible point which satisfies the first order necessary optimality condition provided that the penalty parameters are bounded and the upper bound of the augmented Lagrangian functions along the approximated solution sequence exists. Numerical experiments show that the algorithm is efficient for obtaining stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the nonsmooth Mangasarian-Fromovitz constraint qualification.
Highlights
In this paper, we consider a nonsmooth constrained optimization problem:(P) min f (x) s.t. gi(x) ≤ 0, i = 1, · · ·, p, hj(x) = 0, j = p + 1, · · ·, q, x ∈ Ω, where Ω ⊂ Rn is a closed convex set and f, gi, i = 1, · · ·, p, hj, j = p + 1, · · ·, q : Rn → R are Lipschitz continuous functions.There are various methods to deal with nonsmooth programs
We extend the concept of weakly generalized Mangasarian Fromovitz constraint qualification (WGMFCQ) to the case where abstract constraint is involved
For the general nonsmooth and nonconvex problem (P), we propose a smoothing augmented Lagrange algorithm and show that any accumulation point is a stationary point of the original problem (P) under the extension version of WGMFCQ
Summary
We consider a nonsmooth constrained optimization problem:. (P) min f (x) s.t. gi(x) ≤ 0, i = 1, · · · , p, hj(x) = 0, j = p + 1, · · · , q, x ∈ Ω, where Ω ⊂ Rn is a closed convex set and f, gi, i = 1, · · · , p, hj, j = p + 1, · · · , q : Rn → R are Lipschitz continuous functions. We assume at least one feasible point of problem (P) exists, denoted by xfeas, and the upper bound of the augmented Lagrangian functions along the approximated solution sequence exists. In this case, any accumulation point will be a feasible point and the WNNAMCQ guarantees a stationary point. We discuss the situation if a feasible point of problem (P) is known, denotes by xfeas ∈ Ω, the values of the augmented Lagrangian functions along the approximated solution sequence are bounded above, i.e., for any k, Gλk ,ck ρk.
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