Trust Region Subproblem with a Fixed Number of Additional Linear Inequality Constraints has Polynomial Complexity

TL;DR

This paper investigates the computational complexity of the trust region subproblem with a fixed number of linear inequality constraints, showing polynomial solvability for fixed m and NP-hardness when constraints exceed problem dimension.

Contribution

It introduces an inductive reduction scheme proving polynomial complexity for fixed m and provides complexity results contrasting with NP-hard cases when constraints are large.

Findings

Polynomial complexity for fixed number of constraints m

NP-hardness when constraints exceed problem dimension

Improved SDP relaxation conditions for (Tm)

Abstract

The trust region subproblem with a fixed number m additional linear inequality constraints, denoted by (Tm), have drawn much attention recently. The question as to whether Problem (Tm) is in Class P or Class NP remains open. So far, the only affirmative general result is that (T1) has an exact SOCP/SDP reformulation and thus is polynomially solvable. By adopting an early result of Martinez on local non-global minimum of the trust region subproblem, we can inductively reduce any instance in (Tm) to a sequence of trust region subproblems (T0). Although the total number of (T0) to be solved takes an exponential order of m, the reduction scheme still provides an argument that the class (Tm) has polynomial complexity for each fixed m. In contrast, we show by a simple example that, solving the class of extended trust region subproblems which contains more linear inequality constraints than the problem dimension; or the class of instances consisting of an arbitrarily number of linear constraints is NP-hard. When m is small such as m = 1,2, our inductive algorithm should be more efficient than the SOCP/SDP reformulation since at most 2 or 5 subproblems of (T0), respectively, are to be handled. In the end of the paper, we improve a very recent dimension condition by Jeyakumar and Li under which (Tm) admits an exact SDP relaxation. Examples show that such an improvement can be strict indeed.