Automatic Reformulations for Convex Mixed-Integer Nonlinear Optimization: Perspective and Separability

Tight reformulations of combinatorial optimization problems like Convex Mixed-Integer Nonlinear Programs (MINLPs) enable one to solve these problems faster by obtaining tight bounds on the optimal value. We consider two techniques for reformulation: perspective reformulation and separability detection. We develop routines for the automatic detection of problem structures suitable for these reformulations and implement new … Read more

Linearization and Parallelization Schemes for Convex Mixed-Integer Nonlinear Optimization

We develop and test linearization and parallelization schemes for convex mixed-integer nonlinear programming. Several linearization approaches are proposed for LP/NLP based branch-and-bound. Some of these approaches strengthen the linear approximation to nonlinear constraints at the root node and some at the other branch-and-bound nodes. Two of the techniques are specifically applicable to commonly found univariate … Read more

Minotaur: A Mixed-Integer Nonlinear Optimization Toolkit

We present a flexible framework for general mixed-integer nonlinear programming (MINLP), called Minotaur, that enables both algorithm exploration and structure exploitation without compromising computational efficiency. This paper documents the concepts and classes in our framework and shows that our implementations of standard MINLP techniques are efficient compared with other state-of-the-art solvers. We then describe structure-exploiting … Read more

Facets of a mixed-integer bilinear covering set with bounds on variables

We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation … Read more

Mixed-Integer Nonlinear Optimization

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling … Read more

Solving Mixed-Integer Nonlinear Programs by QP-Diving

We present a new tree-search algorithm for solving mixed-integer nonlinear programs (MINLPs). Rather than relying on computationally expensive nonlinear solves at every node of the branch-and-bound tree, our algorithm solves a quadratic approximation at every node. We show that the resulting algorithm retains global convergence properties for convex MINLPs, and we present numerical results on … Read more

Exploiting Second-Order Cone Structure for Global Optimization

Identifying and exploiting classes of nonconvex constraints whose feasible region is convex after branching can reduce the time to compute global solutions for nonlinear optimization problems. We develop techniques for identifying quadratic and nonlinear constraints whose feasible region can be represented as the union of a finite number of second-order cones, and we provide necessary … Read more

On complexity of Selecting Branching Disjunctions in Integer Programming

Branching is an important component of branch-and-bound algorithms for solving mixed integer linear programs. We consider the problem of selecting, at each iteration of the branch-and-bound algorithm, a general branching disjunction of the form $“\pi x \leq \pi_0 \vee \pi x \geq \pi_0 + 1”$, where $\pi, \pi_0$ are integral. We show that the problem … Read more

Experiments with Branching using General Disjunctions

Branching is an important component of the branch-and-cut algorithm for solving mixed integer linear programs. Most solvers branch by imposing a disjunction of the form“$x_i \leq k \vee x_i \geq k+1$” for some integer $k$ and some integer-constrained variable $x_i$. A generalization of this branching scheme is to branch by imposing a more general disjunction … Read more