An approximation algorithm for convex multi-objective programming problems

Abstract

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson's outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

“…Thus, x u is an optimal ( -optimal) solution of (P X ). (1,9), (6,1) vert D(I k ) (1,1) 1 vert O k (0,1),(1,1),( 1 2 , 7 2 ),( 4 5 , 13 5 ) v e r tI k (0,1),(1,1),( 8 13 , 41 13 ) (1,9), (2,5), (6,1) vert D(I k ) (1, (1,9), (2,5), (4,2), (6,1) vert D(I k ) (1, 33 5 ),( 11 3 , 7 3 ),(4,2), (6,1) bounds on (P P ). Since the vertices of I k are known, it is easy to obtain the facets of D(I k ) using the function ϕ.…”

“…In our previous work, we have extended Benson's outer approximation algorithm for MOLPs to solve convex MOP problems [9] and further the algorithm has been adapted to be an objective space cut and bound algorithm to solve convex multiplicative programmes. [10] We call the cut and bound algorithm 'the primal algorithm'.…”

“…Theorem 3.1 [9,14] Let y k / ∈ P. Let (x k , z k ) be an optimal solution of linear programme (P 1 (y k )), and (u * , λ * ) be the corresponding dual optimal solution of (D 1 (y)). Let…”

“…Their corresponding facets (supporting hyperplanes) of P are y 2 = 1, y 1 + 2y 2 = 8, 3y 1 + 2y 2 = 16, 4y 1 + y 2 = 13, y 1 = 1. The four vertices of P, (1,9), (2,5), (4,2) and (6,1) correspond to the facets (supporting hyperplanes) of D, 8v 1 + v 2 = 9,3v 1 + v 2 = 5, −2v 1 + v 2 = 2 and −5v 1 + v 2 = 1, respectively. Proof The proof is the same as the proof for the finiteness of the dual variant of Benson's algorithm (see [11]) since the additional inner approximation can only possibly cause the algorithm to terminate earlier.…”

An approximation algorithm for convex multiplicative programming problems

“…Last, we show the correspondences between multiplicative programming problems and convex multiobjective programming problems. In Section III we review the approximation algorithm in [17] for convex multiobjective programming problems. The proposed algorithm for multiplicative programming problems as well as an illustrative example and some results are given in Section IV.…”