Perturbed cones for analysis of uncertain multi-criteria optimization problems

AbstractPartial ordering of two quantities x and y (i.e., the ability to declare that x is better than y with respect to some decision criteria) can be stated mathematically as: x is better than y iff x−y∈K, where K is an ordering convex cone, not necessarily pointed. Cones can be very important in representing feasible domains (i.e., {Ax⩽b}=M+G, where M is a bounded convex hull of a finite number of points and G is a convex cone). We consider specific perturbations of the Cone of Feasible Directions, which lead to a better feasible solution with respect to some decision criteria. Such cones are introduced as a tool to mitigate and analyze the effects of input data uncertainty on the solution of a given problem. Properties of this cone provide a basis to prove necessary and sufficient conditions for stable/unstable unboundedness of the multi-criteria optimization problem

On the binary solitaire cone

The solitaire cone SB is the cone of all feasible fractional Solitaire Peg games. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. The link with the well studied dual metric cone and the similarities between their combinatorial structures- see (3)- leads to the study of a dual cut cone analogue; that is, the cone generated by the {0,1}-valued facets of the solitaire cone. This cone is called binary solitaire cone and denoted BSB. We give some results and conjectures on the combinatorial and geometric properties of the binary solitaire cone. In particular we prove that the extreme rays of SB are extreme rays of BSB strengthening the analogy with the dual metric cone whose extreme rays are extreme rays of the dual cut cone. Other related cones are also considered