Optimization and Equilibrium Seminar “Perturbation as a Decision Variable: Rockafellian Relaxations in Optimization, Valuation, and Equilibrium”

Optimization models are solved as if their integrands, distributions, and constraints were exact. When they are not, that is, when expectations are approximated, probabilities estimated, or market-clearing conditions inexact, minimizers may shift discontinuously or fail to exist altogether. This talk develops an approach based on Rockafellian functions: rather than solving the nominal problem, one solves a substitute problem in which the perturbation is itself a decision variable, penalized rather than pinned at zero.

I first present stability results for the family of substitute problems generated by a Rockafellian: the relaxation, its Lagrangian, and its dual function, in the nonconvex
setting. The analysis rests on epi- and hypo-convergence rather than local sensitivity, and yields rates that give Lipschitz-type stability. Two applications follow. For discrete-time contingent claims, perturbing the probability distribution, the claim, or both produces epi-convergent primal and hypo-convergent dual approximations; dual variables converge as shadow prices, the duality gap connects to the value of perfect information, and explicit examples locate the failure boundary, where critical scenarios combine vanishing probability with unbounded impact. For exchange economies admitting no Walrasian equilibrium, slackening market clearing by a penalized nonnegative variable is well posed throughout, and as the penalty grows the residual converges to a minimum-norm vector measuring the distance to the nearest equilibrium-admitting economy.

Based on joint work with J. O. Royset, W. Breytmann, and N. Hernández

Speaker: Julio Deride  (UAI)