Perturbed cones for analysis of uncertain multi-criteria optimization problems

Partial 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.