A relaxation algorithm for the minimization of a quasiconcave function on a convex polyhedron

Multiple‐objective weighting factor auxiliary optimization problems

“…If X is a polytope, then the auxiliary optimization problem AP(q, A, 8 ( * ) ) takes the form of minimizing a concave function over a polytope, for which algorithms do exist (e.g. Glover and Klingman, 1973;Falk and Hoffman, 1976;Carrillo, 1977). This is still a hard problem, but somewhat more satisfying than trying to find global optima for problems which have neither convex nor concave auxiliary optimization functions.…”

“…When X is a polytope, several algorithms are relevant (e.g. Glover and Klingman, 1973;Falk and Hoffman, 1976;Carrillo, 1977).…”

“…When X is a polytope, so is X, and algorithms exist for solving such problems (e.g. Glover and Klingman, 1973;Falk and Hoffman, 1976;Carrillo, 1977). We note that in this case step (vi)B adds cutting constraints which exclude the current solution and for which the dual simplex method is useful (Hadley, 1973).…”

Convergence of a Tuy-type algorithm for concave minimization subject to linear inequality constraints

“…Other recent research on the minimization of a concave function subject to linear constraints includes CarriUo [4], Carvajal-Moreno [5], and Falk and Hoffman [7]. Also, Bali and Jacobsen [2] produced partial results with respect to convergence.…”

A finite algorithm for concave minimization over a polyhedron

“…Taken together, the last two sentences imply that f ( x ) 2 f(xc). Since x was an arbitrary element of X , this completes the proof.Note that if the algorithm stops, it finds an extreme point xc of X that is an optimal on both sides(4) of +ij, since solution for problem (P). From the next theorem, the algorithm is guaranteed to terminate eventually.…”