On the asymptotic convergence and acceleration of gradient methods

We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate … Read more

Successive Quadratic Upper-Bounding for Discrete Mean-Risk Minimization and Network Interdiction

The advances in conic optimization have led to its increased utilization for modeling data uncertainty. In particular, conic mean-risk optimization gained prominence in probabilistic and robust optimization. Whereas the corresponding conic models are solved efficiently over convex sets, their discrete counterparts are intractable. In this paper, we give a highly effective successive quadratic upper-bounding procedure … Read more

Scoring positive semidefinite cutting planes for quadratic optimization via trained neural networks

Semidefinite programming relaxations complement polyhedral relaxations for quadratic optimization, but global optimization solvers built on polyhedral relaxations cannot fully exploit this advantage. This paper develops linear outer-approximations of semidefinite constraints that can be effectively integrated into global solvers. The difference from previous work is that our proposed cuts are (i) sparser with respect to the … Read more

Global Solutions of Nonconvex Standard Quadratic Programs via Mixed Integer Linear Programming Reformulations

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program … Read more

Minimizing convex quadratics with variable precision Krylov methods

Iterative algorithms for the solution of convex quadratic optimization problems are investigated, which exploit inaccurate matrix-vector products. Theoretical bounds on the performance of a Conjugate Gradients and a Full-Orthormalization methods are derived, the necessary quantities occurring in the theoretical bounds estimated and new practical algorithms derived. Numerical experiments suggest that the new methods have significant … Read more

Quasi-Newton approaches to Interior Point Methods for quadratic problems

Interior Point Methods (IPM) rely on the Newton method for solving systems of nonlinear equations. Solving the linear systems which arise from this approach is the most computationally expensive task of an interior point iteration. If, due to problem’s inner structure, there are special techniques for efficiently solving linear systems, IPMs enjoy fast convergence and … Read more

Stable interior point method for convex quadratic programming with strict error bounds

We present a short step interior point method for solving a class of nonlinear programming problems with quadratic objective function. Convex quadratic programming problems can be reformulated as problems in this class. The method is shown to have weak polynomial time complexity. A complete proof of the numerical stability of the method is provided. No … Read more

On a Frank-Wolfe Type Theorem in Cubic Optimization

A classical result due to Frank and Wolfe (1956) says that a quadratic function $f$ attains its supremum on a nonempty polyhedron $M$ if $f$ is bounded from above on $M$. In this note, we present a stringent proof of the extension of this result to cubic optimization (known from Andronov, Belousov and Shironin (1982)). … Read more

Trust your data or not – StQP remains StQP: Community Detection via Robust Standard Quadratic Optimization

We consider the Robust Standard Quadratic Optimization Problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is extremized over the standard simplex. Following most approaches, we model the uncertainty sets by ellipsoids, polyhedra, or spectrahedra, more precisely, by intersections of sub-cones of the copositive matrix cone. We show that the copositive relaxation gap of … Read more

Solving Quadratic Programs to High Precision using Scaled Iterative Refinement

Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause inconveniences when solutions are used for rigorous reasoning. We contribute on three levels to overcome this issue. First, we present a novel refinement … Read more