A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

Abstract We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but can only evaluate them know their global constants. The algorithm is successive linear relaxation in which alternate between solving master problem, of original subproblem, designed to tighten next problem by using information about respective functions. By doing so, follow ideas Schmidt, Sirvent, Wollner (Math Program 178(1):449–483 (2019) Optim Lett 16(5):1355-1372 (2022)) improve tackling constraints. Although nonlinearities obviously increase modeling capabilities, incorporation also significantly increases computational burden proposed algorithm. prove correctness our derive worst-case iteration bound. Finally, show generality addressed class illustrating both bilevel nonconvex quadratic lower levels as well models gas transport be tackled method. provide necessary theory applications briefly illustrate outcomes new when applied these two problems.