A Gentle, Geometric Introduction to Copositive Optimization

This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We intend the paper for readers new to the area, and hence the exposition is largely self-contained. We focus on examples having just a few variables … Read more

A Branch-and-Bound Algorithm for Instrumental Variable Quantile Regression

This paper studies a statistical problem called instrumental variable quantile regres- sion (IVQR). We model IVQR as a convex quadratic program with complementarity constraints and—although this type of program is generally NP-hard—we develop a branch-and-bound algorithm to solve it globally. We also derive bounds on key vari- ables in the problem, which are valid asymptotically … Read more

How to Convexify the Intersection of a Second Order Cone and a Nonconvex Quadratic

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, $\E$, and a split disjunction, $(l – x_j)(x_j – u) \le 0$ with $l < u$, equals the intersection ... Read more

A Two-Variable Approach to the Two-Trust-Region Subproblem

The trust-region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite-programming (SDP) relaxation. Intersecting the feasible set with a second ellipsoid results in the two-trust-region subproblem (TTRS). Even though TTRS can also be solved in polynomial-time, existing algorithms do not use SDP. In this paper, … Read more

Faster, but Weaker, Relaxations for Quadratically Constrained Quadratic Programs

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than … Read more

The Trust Region Subproblem with Non-Intersecting Linear Constraints

This paper studies an extended trust region subproblem (eTRS)in which the trust region intersects the unit ball with m linear inequality constraints. When m=0, m=1, or m=2 and the linear constraints are parallel, it is known that the eTRS optimal value equals the optimal value of a particular convex relaxation, which is solvable in polynomial … Read more

Non-Convex Mixed-Integer Nonlinear Programming: A Survey

A wide range of problems arising in practical applications can be formulated as Mixed-Integer Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When non-convexities are present, however, things become much more difficult, since then even the continuous relaxation is … Read more

A First-Order Smoothing Technique for a Class of Large-Scale Linear Programs

We study a class of linear programming (LP) problems motivated by large-scale machine learning applications. After reformulating the LP as a convex nonsmooth problem, we apply Nesterov’s primal-dual smoothing technique. It turns out that the iteration complexity of the smoothing technique depends on a parameter $\th$ that arises because we need to bound the originally … Read more

Robust Rankings for College Football

We investigate the sensitivity of the Colley Matrix (CM) rankings—one of six computer rankings used by the Bowl Championship Series—to (hypothetical) changes in the outcomes of (actual) games. Specifically, we measure the shift in the rankings of the top 25 teams when the win-loss outcome of, say, a single game between two teams, each with … Read more

Unbounded Convex Sets for Non-Convex Mixed-Integer Quadratic Programming

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are … Read more