Schaible, S.

problem program algorithm function condition lemma programming solution result dinkelbach-type algorithm optimization theorem dinkelbach-type sequence proof optimization problem ratio min-max approach nonempty application single-ratio relation point dinkelbach-type -2 algorithm x 0 b section value research condition 17 condition 1 condition 29 method yield bifunction journal barro limk k theorem 6 converge analysis subgradient inequality max-min model xk sp 2 management frenk function p 2 objective duality system theory finite-valued objective function value schaible j.b.g lemma 20 functions f function g sequence k limk xk execution time 147-175 vector iteration affine /affine bifunction 0 p 2 x sp 2 /convex bifunction supya condition 48 lemma 5 y ~ v mapping affine lemma 14 subsequence condition 2 sum-of-ratio