The Generalized Trust Region Subproblem: solution complexity and convex hull results

07/20/2019
by   Alex L. Wang, et al.
0

We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two nonconvex quadratic constraints. Our first main contribution is structural: we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil. This result may be of interest in building relaxations for nonconvex quadratic programs. Moreover, this result allows us to reformulate the GTRS as the minimization of two convex quadratic functions in the original space. Our next set of contributions is algorithmic: we present an algorithm for solving the GTRS up to an epsilon additive error based on this reformulation. We carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms. Notably, our algorithms run in linear (in the size of the input) time. Furthermore, our algorithm for computing an epsilon-optimal solution has a slightly-improved running time dependence on epsilon over the state-of-the-art algorithm. Our analysis shows that the dominant cost in solving the GTRS lies in solving a generalized eigenvalue problem -- establishing a natural connection between these problems. Finally, generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality-, interval-, and hollow- constrained variants of the GTRS. This gives the first linear-time algorithm in the literature for these variants of the GTRS.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

07/15/2014

Fast matrix completion without the condition number

We give the first algorithm for Matrix Completion whose running time and...
02/24/2020

Upper Tail Analysis of Bucket Sort and Random Tries

Bucket Sort is known to run in expected linear time when the input keys ...
12/27/2021

Implicit regularity and linear convergence rates for the generalized trust-region subproblem

In this paper we develop efficient first-order algorithms for the genera...
02/19/2021

A comparison of eigenvalue-based algorithms and the generalized Lanczos trust-region algorithm for Solving the trust-region subproblem

Solving the trust-region subproblem (TRS) plays a key role in numerical ...
05/19/2015

Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems

We consider the fundamental problem of solving quadratic systems of equa...
04/22/2020

Eigendecomposition of Q in Equally Constrained Quadratic Programming

When applying eigenvalue decomposition on the quadratic term matrix in a...
05/23/2018

Incomplete Nested Dissection

We present an asymptotically faster algorithm for solving linear systems...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.