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Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint function at a time. In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method that relies on determining the convex envelope of linear combinations of the constraint functions and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.
Exploiting complete linear descriptions for decentralized power market problems with integralities
(2019)
It is well known that linear prices supporting a competitive equilibrium exist in the case of convex markets, however, in the presence of integralities this is open and hard to decide in general. We present necessary and sufficient conditions for the existence of such prices for decentralized market problems where market participants have integral decision variables and their feasible sets are given in complete linear description. We utilize total unimodularity and the aforementioned conditions to show that such linear prices exist and present some applications. Furthermore, we compute competitive equilibria for two classes of decentralized market problems arising in energy markets and show that competitive equilibria may exist regardless of integralities.
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.