Martin Schmidt Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test … Read more
In entry-exit gas markets as they are currently implemented in Europe, network constraints do not affect market interaction beyond the technical capacities determined by the TSO that restrict the quantities individual firms can trade at the market. It is an up to now unanswered question to what extent existing network capacity remains unused in an … Read more
We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations … Read more
Robust model predictive control approaches and other applications lead to nonlinear optimization problems defined on (scenario) trees. We present structure-preserving Quasi-Newton update formulas as well as structured inertia correction techniques that allow to solve these problems by interior-point methods with specialized KKT solvers for tree-structured optimization problems. The same type of KKT solvers could be … Read more
Mathematical modeling of market design issues in liberalized electricity markets often leads to mixed-integer nonlinear multilevel optimization problems for which no general-purpose solvers exist and which are intractable in general. In this work, we consider the problem of splitting a market area into a given number of price zones such that the resulting market design … Read more
Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use … Read more
We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinearity. We prove that … Read more
Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network as well as accurate physical and technical modeling is required in order to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for … Read more
Feasibility pumps are highly effective primal heuristics for mixed-integer linear and nonlinear optimization. However, despite their success in practice there are only few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. … Read more
Multistage stochastic optimization leads to NLPs over scenario trees that become extremely large when many time stages or fine discretizations of the probability space are required. Interior-point methods are well suited for these problems if the arising huge, structured KKT systems can be solved efficiently, for instance, with a large scenario tree but a moderate … Read more