Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints

Abstract

We discuss possible scenarios of behaviour of the dual part of sequences generated by primal-dual Newton-type methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so-called critical multiplier (which, in particular, violates some second-order sufficient conditions for optimality), the latter appearing to be a typical scenario when critical multipliers exist; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate those scenarios with examples, and discuss consequences for the speed of convergence. We also put together a collection of examples of optimization problems with constraints violating some standard constraint qualifications, intended for preliminary testing of existing algorithms on degenerate problems, or for developing special new algorithms designed to deal with constraints degeneracy.

Combining stabilized SQP with the augmented Lagrangian algorithm

“…As already mentioned, in the case when the inverse penalty parameter is not separated from zero, the presented result allows the possibility of convergence to infeasible accumulation points satisfying (27). However, any subsequence with this property must be generated by Aug-L iterations (at least from some point on), and in particular, the iterative process then reduces to the augmented Lagrangian algorithm.…”

“…for all k ∈ K. Taking into account the boundedness of {(λ k ,μ k ) | k ∈ K}, and passing onto the limit in the last relation above along the corresponding subsequence, we obtain the second equality in (27).…”

“…Using (29) and passing onto the limit along the corresponding subsequence, we obtain the first equality in (27). If it also holds that ψ(x,μ) = 0, then ρ(x,λ,μ) = 0.…”

“…Then for every accumulation point (x,λ,μ) of {(x k , λ k , µ k )}, the following holds true: (27) and eitherx is a stationary point of (1) with the associated Lagrange multiplier (λ,μ), or for any subsequence of {(x k , λ k , µ k )} converging to (x,λ,μ), all its elements with sufficiently large indices are generated by the Aug-L iterations of Algorithm 1. Moreover, if the sequence {σ k } generated by Algorithm 1 is separated from zero, thenx is a stationary point of (1) with the associated Lagrange multiplier (λ,μ).…”

“…If for k ∈ K the condition (28) holds infinitely often, then r k → 0, since r k = qr k−1 for all such k and since {r k } is nonincreasing. But then (28) implies ρ(x,λ,μ) = 0, i.e.,x is a stationary point of (1) with associated Lagrange multipliers (λ,μ) (in particular, (27) …”