  We study geometric duality for convex vector optimization problems. For a
primal problem with a $q$-dimensional objective space, we formulate a dual
problem with a $(q+1)$-dimensional objective space. Consequently, different
from an existing approach, the geometric dual problem does not depend on a
fixed direction parameter and the resulting dual image is a convex cone. We
prove a one-to-one correspondence between certain faces of the primal and dual
images. In addition, we show that a polyhedral approximation for one image
gives rise to a polyhedral approximation for the other. Based on this, we
propose a geometric dual algorithm which solves the primal and dual problems
simultaneously and is free of direction-biasedness. We also modify an existing
direction-free primal algorithm in a way that it solves the dual problem as
well. We test the performance of the algorithms for randomly generated problem
instances by using the so-called primal error and hypervolume indicator as
performance measures.
