Perturbed cones for analysis of uncertain multi-criteria optimization problems

Abstract

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.

Stability analysis and regularization of uncertain linear multi-objective integer optimization problems

“…Here, the post-optimality approach, which determines stability information after solving the optimization problem, is used. Some previously published articles based on this approach are Kozeratskaya et al (1988Kozeratskaya et al ( , 1993Kozeratskaya et al ( , 2004, Emelichev and Podkopaev (1998) and Emelichev et al (2002).…”

Properties of Perturbed Cones Ordering the Set of Feasible Solutions of Vector Optimization Problem1

“…In the present paper, we will continue (from [5][6][7][8][9][10]) the analysis of properties of ordering cones perturbed specially according to formulas (3) and (4). Based on some of these properties, we will develop an approach [6,8] to the regularization of problem (1) with integer variables, which is possibly unstable against perturbations of initial data.…”