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We present a novel method for mixed-integer optimization problems withmultivariate and Lipschitz continuous nonlinearities. In particular, we do notassume that the nonlinear constraints are explicitly given but that we can onlyevaluate them and that we know their global Lipschitz constants. The algorithmis a successive linear relaxation method in which we alternate between solvinga master problem, which is a mixed-integer linear relaxation of the originalproblem, and a subproblem, which is designed to tighten the linear relaxationof the next master problem by using the Lipschitz information about therespective functions. By doing so, we follow the ideas of Schmidt et al. (2018,2021) and improve the tackling of multivariate constraints. Althoughmultivariate nonlinearities obviously increase modeling capabilities, theirincorporation also significantly increases the computational burden of theproposed algorithm. We prove the correctness of our method and also derive aworst-case iteration bound. Finally, we show the generality of the addressedproblem class and the proposed method by illustrating that both bileveloptimization problems with nonconvex and quadratic lower levels as well asnonlinear and mixed-integer models of gas transport can be tackled by ourmethod. We provide the necessary theory for both applications and brieflyillustrate the outcomes of the new method when applied to these two problems.

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