Presolving and regularization in mixed-integer second-order cone optimization

Mixed-integer second-order cone optimization is a powerful mathematical framework capable of representing both logical conditions and nonlinear relationships in mathematical models of industrial optimization problems. What is more, solution methods are already part of many major commercial solvers including that of MOSEK [72] as well as XPRESS [31], GUROBI [46] and CPLEX [50]. This thesis concerns the performance and reliability of these solvers and makes two contributions; a theoretical one and a practical one. In the theoretical part of the thesis a fundamental issue with reliability, affecting both continuous and mixed-integer conic optimization in general, is discovered and treated. This part of the thesis continues the studies of facial reduction preceding the work of Borwein and Wolkowicz [17] in 1981, when the first algorithmic cure for these kinds of reliability issues were formulated. An important distinction to make between continuous and mixed-integer optimization, however, is that the reliability issues occurring in mixed-integer optimization cannot be blamed on the practitioner’s formulation of the problem. Specifically, as shown, the causes for these issues may well lie within the modifications to the formulation performed by the solution method itself. Hence, this calls for native support of facial reduction mechanisms within the commercial solvers to function reliably. In pursuit of such mechanisms, many fast and accurate heuristics are explored, supplementing the main discovery of this thesis that facial reduction can be interleaved with common optimization methods of high efficiency. Finally, a branch-and-bound method utilizing these mechanisms is established. In the practical part of the thesis, a lack of consensus regarding basic definitions, representations and file formats is found, thereby increasing barriers for benchmarking with decreased market transparency as result. These differences are explored and results in the design of a new file format called The Conic Benchmark Format (CBF). Unlike any other file format for conic optimization, this one is both cross-platform compatible, high performant and future-proof by encompassing other conic extensions. Scripts and tools have moreover been developed to aid parsing (resp. conversion) of the file format in service of software developers (resp. optimization practitioners), and are actively distributed. The functionality of all of this is proven not only by first-class citizenship in the modeling language PICOS [87], but also by The Conic Benchmark Library (CBLIB) where the conversion tools have been used to test its more than a thousand instances with MOSEK and CPLEX. This benchmark library was compiled as part of this thesis in support of studies in performance and reliability, but has yet to be used for the theoretical subjects of this thesis.