A parametric simplex algorithm for linear vector optimization problems

Abstract

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (the Evans-Steuer) algorithm [15]. Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Löhne [19], that is, it finds a subset of efficient solutions that allows to generate the whole efficient frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczyński and Vanderbei [27]. The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems.Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm (Benson's algorithm in [16]), that also provides a solution in the sense of [19], and with the Evans-Steuer algorithm [15]. The results show that for nondegenerate problems the proposed algorithm outperforms Benson's algorithm and is on par with the Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [16] outperforms the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than the Evans-Steuer algorithm.

Applying Simplex Algorithm for Ship’s Motion Simulation Optimization by Using Maneuvering Tests Data

“…The simulation results of the ship's trajectory in Turning Circle test and ship's yaw/rudder angles in the Zigzag test [1,13,14] are given in Figure 3 and Figure 4 respectively.…”

“…In this paper, we introduce another optimization algorithm carrying out by an effective way to calculate OHCs by applying the Simplex algorithm for ship trajectory simulation based on ship maneuvering test data. As the previous researches, this procedure is achieved efficiently through 3 steps as follows [14]:…”

Tractability of convex vector optimization problems in the sense of polyhedral approximations

“…Note that if the recession cone of the upper image of a two-dimensional CVOP is a halfspace, then the upper image itself is a halfspace by convexity. Then one could simplify the problem to a linear vector optimization problem and apply for instance, the parametric simplex algorithm from [19], which works even if the upper image is a halfspace.…”

“…Indeed, for a LVOP with ∅ = P = R q , the recession cone of the upper image recc P is polyhedral, it can be computed by solving the so called homogeneous problem, and problem (P) is bounded with respect to recc P, see, for instance, [15] for the details. Moreover, it is also known that for linear problems we have (recc P) + = {w ∈ C + | (P w ) is bounded}, see [19].…”

“…Accordingly, a solution consists of minimizers which, together with the extreme directions of the ordering cone, generate the Pareto frontier. Clearly, for linear vector optimization problems (LVOPs) a solution in this sense contains finitely many minimizers and there are algorithms to generate one, see, for instance, [2,3,4,7,10,12,19].…”