Some Strongly Polynomially Solvable Convex Quadratic Programs with Bounded Variables

TL;DR

This paper identifies classes of convex quadratic programs with bounded variables that are solvable in strongly polynomial time using the parametric principal pivoting algorithm, extending previous results to broader Hessian structures.

Contribution

It introduces new strongly polynomial solvability results for convex QPs with bounded variables, including extensions to Hessians beyond tridiagonal matrices.

Findings

Solvable in O(n^3) time using parametric principal pivoting.

Extension to Hessian matrices beyond tridiagonal structures.

Includes special cases like weakly quasi-diagonally dominant problems.

Abstract

This paper begins with a class of convex quadratic programs (QPs) with bounded variables solvable by the parametric principal pivoting algorithm with $\mathcal{O}(n^3)$ strongly polynomial complexity, where $n$ is the number of variables of the problem. Extension of the Hessian class is also discussed. Our research is motivated by a recent reference [7] wherein the efficient solution of a quadratic program with a tridiagonal Hessian matrix in the quadratic objective is needed for the construction of a polynomial-time algorithm for solving an associated sparse variable selection problem. With the tridiagonal structure, the complexity of the QP algorithm reduces to $\mathcal{O}(n^2)$. Our strongly polynomiality results extend previous works of some strongly polynomially solvable linear complementarity problems with a P-matrix [9]; special cases of the extended results include weakly quasi-diagonally dominant problems in addition to the tridiagonal ones.