Valid Inequalities and Reformulation Techniques for Mixed Integer Nonlinear Programming

One of the most important breakthroughs in the area of Mixed Integer Linear Programming (MILP) is the characterization of the convex hull of specially structured non-convex polyhedral sets in order to develop valid inequalities or cutting planes. Development of strong valid inequalities such as Split cuts, Gomory Mixed Integer (GMI) cuts, and Mixed Integer Rounding (MIR) cuts has resulted in highly effective branch-and-cut algorithms. While such cuts are known to be equivalent, each of their characterizations provides different advantages and insights. The study of cutting planes for Mixed Integer Nonlinear Programming (MINLP) is still much more limited than that for MILP, since characterizing cuts for MINLP requires the study of the convex hull of a non-convex and non-polyhedral set, which has proven to be significantly harder than the polyhedral case. However, there has been significant work on the computational use of cuts in MINLP. Furthermore, there has recently been a significant interest in extending the associated theoretical results from MILP to the realm of MINLP. This dissertation is focused on the development of new cuts and extended formulations for Mixed Integer Nonlinear Programs. We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. We also study the relation between the introduced cuts and some known classes of cutting planes from MILP. Furthermore, we show how an aggregation technique can be easily extended to characterize the convex hull of sets defined by two quadratic or by a conic quadratic and a quadratic inequality. We also computationally evaluate the performance of the introduced cuts and extended formulations on two classes of MINLP problems

On Minimal Valid Inequalities for Mixed Integer Conic Programs

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone

Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion

Mixed-Integer Second-Order Cone Programs (MISOCPs) form a nice class of mixed-inter convex programs, which can be solved very efficiently due to the recent advances in optimization solvers. Our paper bridges the gap between modeling a class of optimization problems and using MISOCP solvers. It is shown how various performance metrics of M/G/1 queues can be molded by different MISOCPs. To motivate our method practically, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service networks. Four different MISOCPs are developed and compared on sets of benchmark test problems. The new formulations efficiently solve large-size test problems, which cannot be solved by the best existing method. Then, the general applicability of our method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality

Exploiting Structures in Mixed-Integer Second-Order Cone Optimization Problems for Branch-and-Conic-Cut Algorithms

This thesis studies computational approaches for mixed-integer second-order cone optimization (MISOCO) problems. MISOCO models appear in many real-world applications, so MISOCO has gained significant interest in recent years. However, despite recent advancements, there is a gap between the theoretical developments and computational practice. Three chapters of this thesis address three areas of computational methodology for an efficient branch-and-conic-cut (BCC) algorithm to solve MISOCO problems faster in practice. These chapters include a detailed discussion on practical work on adding cuts in a BCC algorithm, novel methodologies for warm-starting second-order cone optimization (SOCO) subproblems, and heuristics for MISOCO problems.The first part of this thesis concerns the development of a novel warm-starting method of interior-point methods (IPM) for SOCO problems. The method exploits the Jordan frames of an original instance and solves two auxiliary linear optimization problems. The solutions obtained from these problems are used to identify an ideal initial point of the IPM. Numerical results on public test sets indicate that the warm-start method works well in practice and reduces the number of iterations required to solve related SOCO problems by around 30-40%.The second part of this thesis presents novel heuristics for MISOCO problems. These heuristics use the Jordan frames from both continuous relaxations and penalty problems and present a way of finding feasible solutions for MISOCO problems. Numerical results on conic and quadratic test sets show significant performance in terms of finding a solution that has a small gap to optimality.The last part of this thesis presents application of disjunctive conic cuts (DCC) and disjunctive cylindrical cuts (DCyC) to asset allocation problems (AAP). To maximize the benefit from these powerful cuts, several decisions regarding the addition of these cuts are inspected in a practical setting. The analysis in this chapter gives insight about how these cuts can be added in case-specific settings