Proximity control in bundle methods for convex nondifferentiable minimization

An inexact spectral bundle method for convex quadratic semidefinite programming

“…The choice of weight u k is somewhat of an art, which impacts significantly both global convergence in theory and efficiency in practice, for instance, as indicated in [16], a suitable adjustment of u k shall guide f W k to the area where it is a reliable approximation of f . There are several wise update rules published in the literature; see, for instance, [5,24,44]. The idea of the updating rule here is the same as that in [24], we outline the main results here.…”

“…There are several wise update rules published in the literature; see, for instance, [5,24,44]. The idea of the updating rule here is the same as that in [24], we outline the main results here.…”

“…In this subsection, we present the inexact spectral bundle method for the eigenvalue optimization problem (Eigform), combining ideas from the algorithms in [24], [34] and [14]. In what follows, the expression [λ ap , v ap , ϵ] = eigest(X, y, δ) is the routine that produces a Ritz triplet (λ ap , v ap , ϵ) for λ 1 (−C − A * (y) − Q(X)) such that aϵ ≤ δ.…”

“…This updating rule of the metric originates from [27], and it is employed in [23]. At the same time, to the best of our knowledge, most of the literature about bundle methods exploit the special matrix M y,k = I and exclude the matrix variable X, for instance, see [2,4,13,14,16,24,34,44]. However, in order to ensure convergence, we should use variable metric at both descent steps and null steps in Algorithm 3.1.…”

“…The spectral bundle method is a specialization of the proximal bundle method of Kiwiel [24] to the largest eigenvalue optimization problem, which is one of three popular types of first-order bundle methods. We refer the readers to [47] for realizing the classification of bundle methods.…”