Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

“…Similarly, if A 0, takingλ = 0 shows the pencil is definite, so such QCQP is definite feasible as long as it is strictly feasible. Indeed a number of studies focus on such cases [8,9].…”

“…Since γ (λ) ≤ 0 as λ → +0, letting x 0 be the limit (which exists [9]) we have g(x 0 ) ≤ 0 and Ax 0 = −a, so taking λ * = 0, x * = x 0 we see that the conditions (8) are satisfied.…”

“…While this work appears to be the first to use the canonical form in the analysis of QCQP, related results have been presented in the literature. The paper [9] also shows that if the matrices A, B are diagonalizable by congruence and this congruence transformation is known, the dual problem can be solved by linear programming. The preprint [16] also investigates the matrix pencil and illustrates why QCQP is nontrivial when the pencil is not simultaneously diagonalizable under congruence.…”

“…[13,25,29,31]. The QCQP (2) is sometimes called the generalized TRS [24], and has applications in double well potential problems [9] and compressed sensing for geological data, in which A is positive semidefinite and B is indefinite [18]. In this paper we refer to QCQP with one quadratic constraint (2) simply as QCQP, unless otherwise mentioned.…”

“…Many other iterative algorithms have been proposed for TRS, but as indicated in the experiments in [1] for TRS, a one-step algorithm based on eigenvalues can significantly outperform such algorithms. Another approach [9,16] is to note the Lagrange dual problem can be expressed equivalently as …”