A method of projection onto an acute cone with level control in convex minimization

Selection strategies in projection methods for convex minimization problems

“…In this case, the approximation α k of f * can be updated (see [7,10,2] for details). Now we go to the problem of selection of…”

“…Various models of selections of L ⊂ J with such properties were employed for convex feasibility problems or for convex minimization problems in [15,1,10,2]. In these papers, a preselection J ⊂ J such that G J x ≥ b J for a current approximation x of a solution was made, and then a subset L ⊂ J was selected such that G L has a full column rank and ( …”

“…An application of this version of Theorem 1 to a sequential selections of a subset L ⊂ J and, actually, to a construction of an obtuse cone was employed in [15,1,9,2]. We call such a selection a regular obtuse cone selection.…”

“…We will seek in this paper methods which have this property. The methods constructed in the next part of the paper have the form (2), where…”

“…Cegielski and R. Dylewski where r 1 ∈ R m−1 and ρ ∈ R. The following Theorem renders it possible to construct sequentially a full column rank submatrix A of G for which y := (A A) −1 r ≥ 0. Contrary to the obtuse cone selection model [15,1,10,2] the residual vector r is not necessarily nonnegative. Therefore we will call a model obtained by such a construction a residual selection model.…”