... ITERATIVE METHODS FOR OPTIMIZATIONProof. By definition of Px(λ) −x + λ∇f(x)≤y −x + λ∇f(x) for all y ∈ Ω. Hence t =0is a local minimum for φ(t)=(1 −t)x(λ)+ty − x + λ∇f(x)2/2and, therefore,0 ... on linesearch methods for bound constrained problems can be found in [47], [194], and [42].Very general theories have been developed for convergence of trust region methods for bound constrained ... performed analytically and impressiveresults for special-purpose filtering algorithms for computational chemistry have been reportedin, for example, [196] and [277]. Nor do we discuss analog methods...