Abstract
A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré’s algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists.
Highlights
A quadratically constrained quadratic programming (QCQP) is an optimization problem of the form [4, Sec. 4.4]minimize f (x) := x Ax + 2a x, x ∈Rn subject to gi (x) := x Bi x + 2bi x + βi ≤ 0 (i = 1, . . . , k), (1)where A and Bi are n × n symmetric matrices and a, bi ∈ Rn, βi ∈ R
In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the iterations for computing the eigenpair
The main contribution of this paper is the development of an efficient algorithm for QCQP (2) that is strictly feasible and (A, B) is a definite pair with A + λB 0 for some λ ≥ 0, which we argue is a generic condition for QCQP to be bounded
Summary
A quadratically constrained quadratic programming (QCQP) is an optimization problem of the form [4, Sec. 4.4]. 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 maximize d(σ ) := inf x (A + σ B)x + 2(a + σ b) x + σβ. To our knowledge the earliest reference is Gander, Golub and von Matt [11] for TRS This algorithm was revisited and further developed recently in [1] to illustrate its high efficiency, which was extended in [34] to deal with an additional linear constraint. The Moore-Penrose pseudoinverse of a matrix A is denoted by A†. x∗ denotes a QCQP solution with associated Lagrange multiplier λ∗
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