Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming

Abstract

Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalent to multiple objective linear programming. The number of objectives of the multiple objective linear program is by one higher than the dimension of the projected polyhedron. The result implies that an arbitrary vector linear program (with arbitrary polyhedral ordering cone) can be solved by solving a multiple objective linear program (i.e. a vector linear program with the standard ordering cone) with one additional objective space dimension.

Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

“…Solving (PP) essentially means finding a V-representation of Y , see [29] for details. Using the terminology of Section 2, (PP) provides a P-representation of Y .…”

“…In [29], the equivalence between the projection problem (PP) and multiple objective linear programming (MOLP) is shown. In particular, [29,Theorem 3] yields that a V-representation of Y can be obtained by solving the multiple objective linear program…”

“…However, the first part of this procedure (Fourier-Motzkin elimination) is not tractable if k is large. It has been shown in [29] that both an H-representation and a V-representation can be obtained from a P-representation by solving a multiple objective linear program (MOLP). This follows from the equivalence between a polyhedral projection problem (which is roughly speaking the problem to obtain a V-representation from a P-representation (3)) and multiple objective linear programming, as shown in [29].…”

“…It has been shown in [29] that both an H-representation and a V-representation can be obtained from a P-representation by solving a multiple objective linear program (MOLP). This follows from the equivalence between a polyhedral projection problem (which is roughly speaking the problem to obtain a V-representation from a P-representation (3)) and multiple objective linear programming, as shown in [29]. By equivalence the authors of [29] understand that a solution of the one problem can be "easily" (for details see [29] or Section 6 below) obtained from a solution of the other problem.…”

“…This follows from the equivalence between a polyhedral projection problem (which is roughly speaking the problem to obtain a V-representation from a P-representation (3)) and multiple objective linear programming, as shown in [29]. By equivalence the authors of [29] understand that a solution of the one problem can be "easily" (for details see [29] or Section 6 below) obtained from a solution of the other problem. In order to compute a V-representation of (3), a multiple objective linear program with n + 1 objective functions needs to be solved.…”