AbstractOptimal vertices of multiparametric linear-programming problems can be found by a local search procedure which involves testing only neighbouring vertices for optimality. When degeneracy is present, vertices and bases will not uniquely correspond, but a similar basis exploration procedure can be used. It is shown that, to within closure, local search applied to bases generates all optimal vertices (but not necessarily all optimal bases) for any constraint set if and only if the cone of permitted objective functions is convex. This implies that the procedure is successful if and only if the problem is essentially one of vector optimization.