On Augmented Lagrangian Methods with General Lower-Level Constraints

Abstract

Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lower-level type. Two methods of this class are introduced and analyzed. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot . All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.

Semismooth SQP method for equality-constrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints

“…The advantage is that the problem (3) has better regularity properties than when the power s = 3 is used; see the discussion in [12] and also Remark 4 below. In this paper, we consider the semismooth version of the sequential quadratic programming method (SQP) [2] for problem (1), and its modification tailored specifically to the structure of lifted MPCC (3). It should be noted that, in local analysis, semismooth SQP for (1) is just the semismooth Newton method (SNM) applied to the Lagrange optimality system of (1).…”

“…For this reason, instead of using standard quasi-Newton updates to make the matrices positive definite, we suggest a special modification directly linked to the structure of the problem at hand (the two approaches are also compared numerically in Section 4, and our results confirm that the proposed special approach does work better). To show that this modification can be expected to preserve high convergence rate, we need some general quasi-Newton type results for problem (1). Since there appears to be no quasi-Newton theory for semismooth problems that suits our specific needs, our first contribution is developing such theory.…”

“…Once but not twice differentiable functions arise also when reformulating complementarity constraints as in [14] or in the lifting approach [12,32]. Other possible sources are subproblems in augmented Lagrangian methods with lowerlevel equality constraints (treated directly) and upper-level inequality constraints (treated via Lagrangian penalization, which gives certain terms that are not twice differentiable in general); see, e.g., [1]. Our main application in this paper, considered in Section 3, is concerned with reformulations of complementarity conditions via the lifting approach [12,32].…”

Combining stabilized SQP with the augmented Lagrangian algorithm

“…Our algorithm uses the boundsλ min <λ max andμ max > 0 to safeguard the dual iterates, as does the ALGENCAN solver [1], for example (see [2,Algorithm 3.1] and also [5]). …”

“…Taking this point of view, it appears very natural to combine sSQP with the usual augmented Lagrangian algorithm. One reason is that Aug-L methods are very robust and have good convergence properties [2,3,7], including when applied to degenerate problems [13,33]. Moreover, it is known that sSQP and Aug-L methods are related: in a sense, the former can be considered as linearization of the iterative subproblems of the latter.…”