An efficient algorithm for solving the generalized trust region subproblem

Salahi, Maziar; Taati, Akram

doi:10.1007/s40314-016-0349-1

Cited by 11 publications

(19 citation statements)

References 22 publications

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“…So we do not run further numerical experiments with MOSEK. We also tested the SOCP reformulation [7] under the simultaneous digonalization condition of the quadratic forms of the GTRS and the DB algorithm in [43] based on the simultaneous digonalization condition of the quadratic forms. The simultaneous digonalization condition naturally holds for problem (IP) when A is positive definite.…”

Section: Numerical Testsmentioning

confidence: 99%

“…Jiang et al [31] derived an SOCP reformulation for the GTRS when the problem has a finite optimal value and further derived a closed form solution when the SD condition fails. On the other hand, there is rich literature on iterative algorithms to solve the GTRS directly under mild conditions, for example, [37,44,41,43]. Pong and Wolkowicz proposed an efficient algorithm based on minimum generalized eigenvalue of a parameterized matrix pencil for the GTRS, which extended the results in [18] and [42] for the TRS.…”

Section: Introductionmentioning

confidence: 99%

“…Pong and Wolkowicz proposed an efficient algorithm based on minimum generalized eigenvalue of a parameterized matrix pencil for the GTRS, which extended the results in [18] and [42] for the TRS. Salahi and Taati [43] also derived a diagonalization-based algorithm under the SD condition of the quadratic forms. Recently, Adachi and Nakatsukasa [5] also developed a novel eigenvalue-based algorithm to solve the GTRS.…”

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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Section: Numerical Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…Recently, Ben-Tal and den Hertog [2] showed that if the two matrices in the quadratic forms are simultaneously diagonalizable (SD) (see [13] for more details about SD conditions), the GTRS can be then transformed into an equivalent second order cone programming (SOCP) formulation and thus can be solved efficiently. Salahi and Taati [23] also derived an efficient algorithm for solving (GTRS) under the SD condition. Jiang et al [15] derived an SOCP reformulation for the GTRS when the problem has a finite optimal value and further derived a closed form solution when the SD condition fails.…”

mentioning

confidence: 99%

A Linear-Time Algorithm for Generalized Trust Region Subproblems

Jiang¹,

Li²

2020

SIAM J. Optim.

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In this paper, we provide the first provable linear-time (in term of the number of non-zero entries of the input) algorithm for approximately solving the generalized trust region subproblem (GTRS) of minimizing a quadratic function over a quadratic constraint under some regularity condition. Our algorithm is motivated by and extends a recent linear-time algorithm for the trust region subproblem by Hazan and Koren [Math. Program., 2016, 158(1-2): 363-381]. However, due to the non-convexity and non-compactness of the feasible region, such an extension is nontrivial. Our main contribution is to demonstrate that under some regularity condition, the optimal solution is in a compact and convex set and lower and upper bounds of the optimal value can be computed in linear time. Using these properties, we develop a linear-time algorithm for the GTRS.

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“…It also has applications in double well potential problems [7] and compressed sensing for geological data [11]. In recent years, QCQP has received much attention in the literature and various methods have been developed to solve it [3,13,16,17,21,23]. In [16], Moré gives a characterization of the global minimizer of QCQP and describes an algorithm for the solution of QCQP which extends the one for TRS [15].…”

mentioning

confidence: 99%

A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint

Taati

Salahi

2019

Comput Optim Appl

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In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the sparsity of the involved matrices and solves the problem via solving a sequence of positive definite system of linear equations after identifying suitable generalized eigenvalues. Some numerical experiments are given to show the effectiveness of the proposed method and to compare it with some recent algorithms in the literature.We consider the following quadratically constrained quadratic programming (QCQP)where A, B ∈ R n×n are symmetric matrices with no definiteness assumed, a, b ∈ R n and β ∈ R. When B = I, b = 0 and β < 0, QCQP reduces to the classical trustregion subproblem (TRS), which arises in regularization or trust-region methods for unconstrained optimization [5,24]. Despite being nonconvex, numerous efficient algorithms have been developed to solve TRS [1,6,8,9,18]. The existing algorithms for TRS can be classified into two categories; approximate methods and accurate methods. The Steihaug-Toint algorithm is a well-known approximate method that

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An efficient algorithm for solving the generalized trust region subproblem

Cited by 11 publications

References 22 publications

Novel Reformulations and Efficient Algorithms for the Generalized Trust Region Subproblem

Novel Reformulations and Efficient Algorithms for the Generalized Trust Region Subproblem

A Linear-Time Algorithm for Generalized Trust Region Subproblems

A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint

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