Sensitivity Analysis for Convex Separable Optimization over Integral Polymatroids

TL;DR

This paper investigates how optimal solutions to convex separable optimization problems over integral polymatroids change with parameters, providing bounds on solution differences, and applies these results to polymatroid games to establish equilibrium existence.

Contribution

It introduces sensitivity analysis methods for convex optimization over integral polymatroids and demonstrates their application to game theory, showing the special role of polymatroids in these results.

Findings

Reoptimization can be achieved through elementary local operations.

Optimal solutions differ by at most twice the parameter change in L1-norm.

Pure Nash equilibria exist in polymatroid-based games.

Abstract

We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution there is a new optimal solution to new parameters such that the L1-norm of the difference of the two solutions is at most two times the L1 norm of the difference of the parameters. We apply these sensitivity results to a class of non-cooperative polymatroid games and derive the existence of pure Nash equilibria. We complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any non-polymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium.