We study the generalization of 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 of two convex sets with specific geometric structures. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split...

We consider the constrained assortment optimization problem under the mixed multinomial logit model. Even moderately sized instances of this problem are challenging to solve directly using standard mixed-integer linear optimization formulations. This has motivated recent research exploring customized optimization strategies and approximation techniques. In contrast, we develop a novel conic qua...

Abstract This paper concerns a method of selecting a subset of features for a sequential logit model. Tanaka and Nakagawa (2014) proposed a mixed integer quadratic optimization formulation for solving the problem based on a quadratic approximation of the logistic loss function. However, since there is a significant gap between the logistic loss function and its quadratic approximation, their fo...

In the literature on the quadratic 0-1 knapsack problem, several alternative ways have been given to represent the knapsack constraint in the quadratic space. We extend this work by constructing analogous representations for arbitrary linear inequalities for arbitrary nonconvex mixed-integer quadratic programs with bounded variables.

In a number of situations the derivative of the objective function of an optimization problem is not available. This thesis presents a novel algorithm for solving mixed integer programs when this is the case. The algorithm is the first developed for problems of this type which uses a trust region methodology. Three implementations of the algorithm are developed and deterministic proofs of conve...

In this work we focus on solving quadratically constrained pseudoboolean optimization problems with quadratic objective as mixed integer linear programs. The standard mixed integer linear formulation of such problems is strengthened using valid inequalities derived from solving Reformulation-Linearization relaxation called partial DRL* relaxation. The proposed PDRL* relaxation features block-de...

The Quadratic Convex Reformulation (QCR) method is used to solve quadratic unconstrained binary optimization problems. In this method, the semidefinite relaxation is used to reformulate it to a convex binary quadratic program which is solved using mixed integer quadratic programming solvers. We extend this method to random quadratic unconstrained binary optimization problems. We develop a Penal...