TY - INPR A1 - Göß, Adrian A1 - Martin, Alexander A1 - Pokutta, Sebastian A1 - Sharma, Kartikey T1 - Norm-induced Cuts: Optimization with Lipschitzian Black-box Functions N2 - Optimal control problems usually involve constraints which model physical states and their possible transitions. These are represented by ordinary or partial differential equations (ODEs/PDEs) which add a component of infinite dimension to the problem. In recent literature, one method to simulate such ODEs/PDEs are physics-informed neural networks. Typically, neural networks are highly non-linear which makes their addition to optimization problems challenging. Hence, we leverage their often available Lipschitz property on a compact domain. The respective Lipschitz constants have to be computed only once and are accessible thereafter. We present a method that, based on this property, iteratively adds cuts involving the violation of the constraints by the current incumbent and the Lipschitz constant. Hereby, the “shape” of a cut depends on the norm used. We prove the correctness of the method by showing that it either returns an optimal solution when terminating or creates a sequence with optimal accumulation points. This is complemented by a discussion about the termination in the infeasible case, as well as an analysis of the problem complexity. For the analysis, we show that the lower and upper iteration bound asymptotically coincide when the relative approximation error goes to zero. In the end, we visualize the method on a small example based on a two-dimensional non-convex optimization problem, as well as stress the necessity of having a globally optimal oracle for the sub-problems by another example. KW - Global Optimization KW - Lipschitz Optimization KW - Black-box Optimization KW - Derivative-free Optimization Y1 - ER - TY - INPR A1 - Grübel, Julia A1 - Krug, Richard A1 - Schmidt, Martin A1 - Wollner, Winnifried T1 - A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities N2 - 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 that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems. KW - Mixed-Integer Nonlinear Optimization KW - Global Optimization KW - Lipschitz Optimization KW - Bilevel Optimization KW - Gas Networks Y1 - 2022 ER - TY - JOUR A1 - Habeck, Oliver A1 - Pfetsch, Marc E. A1 - Ulbrich, Stefan T1 - Global optimization of mixed-integer ODE constrained network problems using the example of stationary gas transport N2 - In this paper we propose a new approach for finding global solutions of mixed-integer nonlinear optimization problems with ordinary differential equation constraints on networks. Instead of using a first discretize then optimize approach, we combine spatial and variable branching with appropriate discretizations of the differential equations to derive relaxations of the original problem. To construct the relaxations we derive convex under- and concave over-estimators for the ODE solution operators using numerical discretization schemes. Thereby, we make use of the underlying network structure, where the solutions of the ODEs only need to be known at a finite number of points. This property enables us to adaptively refine the discretization and relaxation without introducing new variables. The incorporation into a spatial branch-and-bound process allows to compute global epsilon-optimal solutions or decide infeasibility. We prove that this algorithm terminates finitely under some natural assumptions. We then show how this approach works for the example of stationary gas transport and provide some illustrative computational examples. KW - Global Optimization KW - Mixed-Integer Nonlinear Optimization Y1 - 2017 U6 - https://doi.org/10.1137/17M1152668 VL - 29 IS - 4 SP - 2949 EP - 2985 ET - SIAM Journal of Optimization ER -