Solving Generalized CDT Problems via Two-Parameter Eigenvalues

Abstract

We consider solving a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. This problem is a generalization of the Celis-Denis-Tapia (CDT) problem and thus we refer to it as GCDT (Generalized CDT). The CDT problem has been widely studied, but no polynomial-time algorithm was known until Bienstock's recent work. His algorithm solves the CDT problem in polynomial time with respect to the number of bits in data and log ϵ −1 by admitting an ϵ error in the constraints. The algorithm, however, appears to be difficult to implement. In this paper, we present another algorithm for GCDT, which is guaranteed to find a global solution for almost all GCDT instances (and slightly perturbed ones in some exceptionally rare cases), in exact arithmetic (including eigenvalue computation). Our algorithm is based on the approach proposed by Iwata, Nakatsukasa and Takeda (2015) for computing the signed distance between overlapping ellipsoids. Our algorithm computes all the Lagrange multipliers of GCDT by solving a two-parameter linear eigenvalue problem, obtains the corresponding KKT points, and finds a global solution as the KKT point with the smallest objective value. In practice, in finite precision arithmetic, our algorithm requires O(n 6 log log u −1) computational time where n is the number of variables and u is the unit roundoff. Although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed, numerical experiments illustrate that our algorithm performs well in finite precision arithmetic.

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

“…This paper is largely an outgrowth of [1], extending the scope from TRS to QCQP, relaxing the convexity assumption in the constraint, and fully analyzing the degenerate cases. We also note [33], which solves QCQP with an additional ball constraint (generalized CDT problem; GCDT) via a two-parameter eigenvalue problem. It is in principle possible to impose a ball constraint with sufficiently large radius to convert QCQP (2) to GCDT, then use the algorithm in [33].…”

“…We also note [33], which solves QCQP with an additional ball constraint (generalized CDT problem; GCDT) via a two-parameter eigenvalue problem. It is in principle possible to impose a ball constraint with sufficiently large radius to convert QCQP (2) to GCDT, then use the algorithm in [33]. However, this would be very inefficient, requiring O(n 6 ) operations: our algorithm here needs at most O(n 3 ) operations, and can be faster when sparsity structure is present.…”

A Survey of Hidden Convex Optimization

“…where Q 1 ≻ 0. Recently, the polynomial-time solvability of (CDT) has been proved in [16,24,63]. Hidden convex reformulation even for the diagonal (CDT) (i.e., Q 0 , Q 1 and Q 2 are all diagonal matrices) remains unknown.…”

Quadratic optimization with two ball constraints

“…Then, there exist two nonnegative multipliers γ and µ such that KKT conditions (7) hold. We have [25] Thus,…”

“…Numerical results for several classes of test problems are presented in Section 3 to show the efficiency of the new approach compared to Sakaue et. al algorithm [25], the recent Alternating Direction Method of Multipliers (ADMM) algorithm suggested in [2] and CVX software [12] for small, medium and large-scale problems.…”