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In order to express a polyhedron as the (Minkowski) sum of a polytope and apolyhedral cone, Motzkin (1936) made a transition from the polyhedron to apolyhedral cone. Based on his excellent idea, we represent a set by acharacteristic cone. By using this representation, we then reach four mainresults: (i) expressing a closed convex set containing no line as the directsum of the convex hull of its extreme points and conical hull of its extremedirections, (ii) establishing a convex programming (CP) based framework fordetermining a maximal element-an element with the maximum number of positivecomponents-of a convex set, (iii) developing a linear programming problem forfinding a relative interior point of a polyhedron, and (iv) proposing twoprocedures for the identification of a strictly complementary solution inlinear programming.

Finding a maximal element of a convex set through its characteristic cone: An application to finding a strictly complementary solution