## Exact and Fast Algorithms for Mixed-Integer Nonlinear Programming

• Mixed-integer nonlinear programming (MINLP) comprises the broad class of finite-dimensional mathematical optimization problems from mixed-integer linear programming and global optimization. The combination of the two disciplines allows us to construct more accurate models of real-world systems, while at the same time it increases the algorithmic challenges that come with solving them. This thesis presents new methods that improve the numerical reliability and the computational performance of global MINLP solvers. Since state-of-the-art algorithms for nonconvex MINLP fundamentally rely on solving linear programming (LP) relaxations, we address numerical accuracy directly for LP by means of LP iterative refinement: a new algorithm to solve linear programs to arbitrarily high levels of precision. The thesis is supplemented by an exact extension of the LP solver SoPlex, which proves on average 1.85 to 3 times faster than current state-of-the-art software for solving general linear programs exactly over the rational numbers. These methods can be generalized to quadratic programming. We study their application to numerically difficult multiscale LP models for metabolic networks in systems biology. To improve the computational performance of LP-based MINLP solvers, we show how the expensive, but effective, bound-tightening technique called optimization-based bound tightening can be approximated more efficiently via feasibility-based bound tightening. The resulting implementation increases the number of instances that can be solved and reduces the average running time of the MINLP solver SCIP by 17-19% on hard mixed-integer nonlinear programs. Last, we present branching rules that exploit the presence of nonlinear integer variables, i.e., variables both contained in nonlinear terms and required to be integral. The new branching rules prefer integer variables when performing spatial branching, and favor variables in nonlinear terms when resolving integer infeasibility. They reduce the average running time of SCIP by 17% on affected instances. Most importantly, all of the new methods enable us to solve problems which could not be solved before, either due to their numerical complexity or because of limited computing resources.

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