A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming

An approximation algorithm for convex multi-objective programming problems

“…Proofs of 1., 2., and 3. can be found in [9] and [3]. To see 4. notice that P WN ⊂ bd P is always true and let y ∈ bd P. If there is y ∈ P with y < y it follows that y = y + d for some d ∈ int R p contradicting y ∈ bd P. Theorem 3.1 is the basis of Ehrgott et al's version of Benson's outer approximation algorithm [9]. It works on P to find all nondominated extreme points of Y.…”

“…In the next section we review Benson's outer approximation algorithm [2,3], which has been shown to construct a set of ε-nondominated points for multi-objective linear programmes in [9].…”

“…According to the following theorem, instead of directly finding Y N , the version of Ehrgott et al [9] is dedicated to finding P WN . Proofs of 1., 2., and 3. can be found in [9] and [3].…”

“…According to the following theorem, instead of directly finding Y N , the version of Ehrgott et al [9] is dedicated to finding P WN . Proofs of 1., 2., and 3. can be found in [9] and [3]. To see 4. notice that P WN ⊂ bd P is always true and let y ∈ bd P. If there is y ∈ P with y < y it follows that y = y + d for some d ∈ int R p contradicting y ∈ bd P. Theorem 3.1 is the basis of Ehrgott et al's version of Benson's outer approximation algorithm [9].…”

“…In Sect. 3 we review the version of Benson's algorithm given in [9] and prove a generalization of one of the results of [3]. Our algorithm for the convex case is given in Sect.…”