@phdthesis{Berthold, author = {Berthold, Timo}, title = {Heuristic algorithms in global MINLP solvers}, publisher = {Dr. Hut Verlag}, isbn = {978-3-8439-1931-9}, pages = {366}, abstract = {In the literature for mixed integer programming, heuristic algorithms (particularly primal heuristics) are often considered as stand-alone procedures; in that context, heuristics are treated as an alternative to solving a problem to proven optimality. This conceals the fact that heuristic algorithms are a fundamental component of state-of-the-art global solvers for mixed integer linear programming (MIP) and mixed integer nonlinear programming (MINLP). In the present thesis, we focus on this latter aspect; we study heuristic algorithms that are tightly integrated within global MINLP solvers and analyze their impact on the overall solution process. Our contributions comprise generalizations of primal heuristics for MIP towards MINLP as well as novel ideas for MINLP primal heuristics and for heuristic algorithms to take branching decisions and to collect global information in MIP. These are: - Shift-and-Propagate, a novel propagation heuristic for MIP that does not require the solution to an LP relaxation, - a generic way to generalize large neighborhood search (LNS) heuristics from MIP to MINLP, - an Objective Feasibility Pump heuristic for nonconvex MINLP that uses second-order information and a dynamic selection of rounding procedures, - RENS, an LNS start heuristic for MINLP that optimizes over the set of feasible roundings of an LP solution, - Undercover, an LNS start heuristic for MINLP that solves a largest sub-MIP of a given MINLP, - Rapid Learning, a heuristic algorithm to generate globally valid conflict constraints for MIPs, - Cloud Branching, a heuristic algorithm that exploits dual degeneracy to reduce the number of candidates for branching variable selection. Additionally, we propose a new performance measure, the primal integral, that captures the benefits of primal heuristics better than traditional methods. In our computational study, we compare the performance of the MIP and MINLP solver SCIP with and without primal heuristics on six test sets with altogether 983 instances from academic and industrial sources, including our project partners ForNe, SAP, and Siemens. We observe that heuristics improve the solver performance regarding all measures that we used - by different orders of magnitude. We further see that the harder a problem is to solve to global optimality, the more important the deployment of primal heuristics becomes. The algorithms presented in this thesis are available in source code as part of the solver SCIP, of which the author has been a main developer for the last years. Methods described in this thesis have also been re-implemented within several commercial and noncommercial MIP and MINLP software packages, including Bonmin, CBC, Cplex, Gams, Sulum, and Xpress.}, language = {en} } @phdthesis{Schweiger, author = {Schweiger, Jonas}, title = {Exploiting structure in non-convex quadratic optimization and gas network planning under uncertainty}, pages = {411}, abstract = {The amazing success of computational mathematical optimization over the last decades has been driven more by insights into mathematical structures than by the advance of computing technology. In this vein, we address applications, where nonconvexity in the model and uncertainty in the data pose principal difficulties. The first part of the thesis deals with non-convex quadratic programs. Branch\&Bound methods for this problem class depend on tight relaxations. We contribute in several ways: First, we establish a new way to handle missing linearization variables in the well-known Reformulation-Linearization-Technique (RLT). This is implemented into the commercial software CPLEX. Second, we study the optimization of a quadratic objective over the standard simplex or a knapsack constraint. These basic structures appear as part of many complex models. Exploiting connections to the maximum clique problem and RLT, we derive new valid inequalities. Using exact and heuristic separation methods, we demonstrate the impact of the new inequalities on the relaxation and the global optimization of these problems. Third, we strengthen the state-of-the-art relaxation for the pooling problem, a well-known non-convex quadratic problem, which is, for example, relevant in the petrochemical industry. We propose a novel relaxation that captures the essential non-convex structure of the problem but is small enough for an in-depth study. We provide a complete inner description in terms of the extreme points as well as an outer description in terms of inequalities defining its convex hull (which is not a polyhedron). We show that the resulting valid convex inequalities significantly strengthen the standard relaxation of the pooling problem. The second part of this thesis focuses on a common challenge in real world applications, namely, the uncertainty entailed in the input data. We study the extension of a gas transport network, e.g., from our project partner Open Grid Europe GmbH. For a single scenario this maps to a challenging non-convex MINLP. As the future transport patterns are highly uncertain, we propose a robust model to best prepare the network operator for an array of scenarios. We develop a custom decomposition approach that makes use of the hierarchical structure of network extensions and the loose coupling between the scenarios. The algorithm used the single-scenario problem as black-box subproblem allowing the generalization of our approach to problems with the same structure. The scenario-expanded version of this problem is out of reach for today's general-purpose MINLP solvers. Yet our approach provides primal and dual bounds for instances with up to 256 scenarios and solves many of them to optimality. Extensive computational studies show the impact of our work.}, language = {en} }