A Second-Order Cone Based Approach for Solving the Trust-Region Subproblem and Its Variants

Ho-Nguyen, Nam; Kılınç-Karzan, Fatma

doi:10.1137/16m1065197

Cited by 59 publications

(79 citation statements)

References 49 publications

(240 reference statements)

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“…We now give examples of problems where our assumptions hold. Corollary 1 recovers results associated with the epigraph of the TRS 2 and the GTRS (see [22,Theorem 13] and [37, Theorems 1 and 2]). inf x∈R 2 x 2 1 + x 2 2 + 10x 1 :…”

Section: Convex Hull Resultssupporting

confidence: 78%

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On Convex Hulls of Epigraphs of QCQPs

Wang

Kılınç-Karzan

2020

Integer Programming and Combinatorial Optimization

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Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

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Section: Convex Hull Resultssupporting

confidence: 78%

“…We provide several classes of problems that satisfy the assumptions of these theorems. In particular, we recover a number of results regarding the TRS [22], the GTRS [37], and the solvability of systems of quadratic equations [3].…”

Section: Introductionsupporting

confidence: 64%

On Convex Hulls of Epigraphs of QCQPs

Wang

Kılınç-Karzan

2020

Integer Programming and Combinatorial Optimization

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“…However, suffering from relatively large computational complexity, the SDP algorithm is not practical for large-scale applications. To overcome this difficulty, several recent papers [30,14,27] demonstrated that the TRS admits a second order cone programming (SOCP) reformulation. Ben-Tal and den Hertog [7] further showed an SOCP reformulation for the GTRS under a simultaneously diagonalizing (SD) procedure of the quadratic forms.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, this CQR was underappreciated in that time. Recently, people rediscovered this result; Wang and Xia [46] and Ho-Nguyen and Kilinc-Karzan [27] presented a linear time algorithm to solve the TRS by applying Nesterov's accelerated gradient descent algorithm to (1). We, instead, rewrite the epigraph reformulation for (1) as follows, min x,t {t :…”

Section: Introductionmentioning

confidence: 99%

“…To our best knowledge, our proposed CQRs are derived the first time for the GTRS. The reformulation (P 2 ) only occurs when the quadratic constraint is convex and thus can be solved by a slight modification of [46,27] in the accelerated gradient projection method by projecting, in each iteration, the current solution to the ellipsoid instead of the unit ball in the TRS case.…”

Section: Introductionmentioning

confidence: 99%

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Novel Reformulations and Efficient Algorithms for the Generalized Trust Region Subproblem

Jiang¹,

Li²

2019

SIAM J. Optim.

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We present a new solution framework to solve the generalized trust region subproblem (GTRS) of minimizing a quadratic objective over a quadratic constraint. More specifically, we derive a convex quadratic reformulation (CQR) via minimizing a linear objective over two convex quadratic constraints for the GTRS. We show that an optimal solution of the GTRS can be recovered from an optimal solution of the CQR. We further prove that this CQR is equivalent to minimizing the maximum of the two convex quadratic functions derived from the CQR for the case under our investigation. Although the latter minimax problem is nonsmooth, it is well-structured and convex. We thus develop two steepest descent algorithms corresponding to two different line search rules. We prove for both algorithms their global sublinear convergence rates. We also obtain a local linear convergence rate of the first algorithm by estimating the Kurdyka-Lojasiewicz exponent at any optimal solution under mild conditions. We finally demonstrate the efficiency of our algorithms in our numerical experiments. Problem (P) is known as the generalized trust region subproblem (GTRS) [44,41]. When Q 2 is an identity matrix I and b 2 = 0, c = −1/2, problem (P) reduces to the classical trust region subproblem (TRS). The TRS first arose in the trust region method for nonlinear optimization [15,49], and has found many applications including robust optimization [8] and the least square problems [50]. As a generalization, the GTRS also admits its own applications such as time of arrival problems [26] and subproblems of consensus ADMM in signal processing [29]. Over the past two decades, numerous solution methods have been developed for TRS (see [38,36,48,42,25,22,4] and references therein).Various methods have been developed for solving the GTRS under various assumptions (see [37,44,10,45,16,41,5] and references therein). Although it appears being nonconvex, the GTRS essentially enjoys Assumption 2.1. The set I P SD := {λ : Q 1 + λQ 2 0} ∩ R + is not empty, where R + is the nonnegative orthant.Assumption 2.2. The common null space of Q 1 and Q 2 is trivial, i.e., Null(Q 1 ) ∩ Null(Q 2 ) = {0}.Before introducing our CQR, let us first recall the celebrated S-lemma by definingf 1 (x) = f 1 (x) + γ with an arbitrary constant γ ∈ R.Lemma 2.3 (S-lemma [47,40]). The following two statements are equivalent: 1. The system off 1 (x) < 0 and f 2 (x) ≤ 0 is not solvable; 2. There exists µ ≥ 0 such thatf 1 (x) + µf 2 (x) ≥ 0 for all x ∈ R n .Using the S-lemma, the following lemma shows a necessary and sufficient condition under which problem (P) is bounded from below.

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On Conic Relaxations of Generalization of the Extended Trust Region Subproblem

Jiang

2019

Advances in Intelligent Systems and Computing

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A Second-Order Cone Based Approach for Solving the Trust-Region Subproblem and Its Variants

Cited by 59 publications

References 49 publications

On Convex Hulls of Epigraphs of QCQPs

On Convex Hulls of Epigraphs of QCQPs

Novel Reformulations and Efficient Algorithms for the Generalized Trust Region Subproblem

On Conic Relaxations of Generalization of the Extended Trust Region Subproblem

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