Optimality Functions and Lopsided Convergence
- 153 Downloads
Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems indeed are approximately optimal and stationary for an original problem. In this paper, we review the framework and illustrate its application to nonlinear programming and other areas. Moreover, we introduce lopsided convergence of bifunctions on metric spaces and show that this notion of convergence is instrumental in establishing consistency of approximations. Lopsided convergence also leads to further characterizations of stationary points under perturbations and approximations.
Keywordsepi-Convergence Lopsided convergence Consistent approximations Optimality functions Optimality conditions
Mathematics Subject Classification90C46 49J53
This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office under Grant Numbers 00101-80683, W911NF-10-1-0246 and W911NF-12-1-0273.
- 4.Foraker, J.C., Royset, J.O., Kaminer, I.: Search-trajectory optimization: part 1, formulation and theory. J. Optim. Theory Appl. (2015)Google Scholar
- 5.Phelps, C., Royset, J.O., Gong, Q.: Optimal control of uncertain systems using sample average approximations. SIAM J. Control Optim. (2015)Google Scholar
- 11.Polak, E.: On the use of models in the synthesis of optimization algorithms. In: Kuhn, H., Szego, G. (eds.) Differential Games and Related Topics, pp. 263–279. North Holland, Amsterdam (1971)Google Scholar
- 12.Polak, E.: Computational Methods in Optimization. A Unified Approach. Academic Press, New York (1971)Google Scholar
- 18.Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der Mathematischen Wissenschaft, vol. 317, 3rd printing-2009 edn. Springer (1998)Google Scholar
- 20.Beer, G.: Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, vol. 268. Kluwer, Dordrecht (1992)Google Scholar
- 22.Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities-III, pp. 103–113. Academic Press, New York (1972)Google Scholar