Exact finitary solution methods for nonlinear parabolic semi-infinite programs

Develop a finitary numerical method that guarantees exact solutions for the nonconvex semi-infinite programs arising from spatial semi-discretization of general nonlinear parabolic systems, and determine the required number of uncertainty-parameter realizations.

Background

The paper's exactness and finite-reduction guarantees rely on the fact that semi-discretization of the considered linear parabolic heat equation produces a linear ODE and hence a convex semi-infinite program. For general nonlinear parabolic systems, the corresponding semi-discretization produces a nonlinear ODE and a nonconvex semi-infinite program.

The authors note that existing numerical techniques for nonconvex semi-infinite programs do not, to their knowledge, provide exact solutions while remaining finitary. In particular, the number of uncertainty realizations needed by such methods is not established, apart from an asymptotic-optimality guarantee. The problem is identified in connection with extending the proposed architecture to nonlinear and nonconvex settings.

References

While there are numerical techniques available for solving nonconvex SIPs, to the best of our knowledge, no method guarantees exact solutions while remaining finitary, and the required number of realizations of the uncertainty parameters remains undetermined, except for the assurance that the algorithm is asymptotically optimal.

— Constrained minmax density transportation for linear parabolic PDEs: a numerical optimal control perspective  (2608.19170 - Ganguly et al., 19 Aug 2026) in Remark on nonconvex SIPs, Section 3 (Main results: theory and algorithm)