Hidden conic quadratic representation of some nonconvex quadratic optimization problems

Ben‐Tal, Aharon; Hertog, Dick den

doi:10.1007/s10107-013-0710-8

Cited by 77 publications

(123 citation statements)

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“…So the total complexity is then O( L x0−x * 2 ǫ ) (this may be roughly considered as O( φ 4 ǫξ 3 ) due to L ≤ φ and x 0 − x * 2 ≤ (2R) 2 ≤ O(φ 3 /ξ 3 ), which is worse than our complexity that is proportional to N φ 3 √ ǫξ 5 as shown in Theorem 4.1. Next let us give comparisons with the results in [2,15,21]. The method in [2] involves a process in simultaneously diagonalizing two matrices, whose computation is not given there.…”

Section: Main Algorithmmentioning

confidence: 99%

“…When the constraint in (GTRS) is a unit ball, the problem reduces to the classical trust region subproblem (TRS). The TRS first arose in trust region methods for nonlinear optimization [6] and also finds applications in the least square problems [31] and robust optimization [2]. Various approaches have been derived to solve the TRS and its variant with additional linear constraints, see [17,19,29,22,25,30,4,5,28].…”

mentioning

confidence: 99%

“…Numerous methods have been developed for solving the GTRS under various assumptions; see, for example, [18,24,3,25,7]. 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.…”

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confidence: 99%

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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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Section: Main Algorithmmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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A Linear-Time Algorithm for Generalized Trust Region Subproblems

Jiang¹,

Li²

2020

SIAM J. Optim.

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“…One important application of Lemma 2.6 will be presented in section 5. In what follows we list various extensions of Lemma 2.5, which will be very useful for our analysis.…”

Section: Introductionmentioning

confidence: 98%

Connectivity of Quadratic Hypersurfaces and Its Applications in Optimization, Part I: General Theory

Yan¹,

Zhang²,

Sun³

2015

SIAM J. Optim.

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In this paper we establish a basic theory on the connectivity of quadratic hypersurfaces. In addition, we examine how both the connectivity and the hidden convexity of quadratic functions are related to each other and prove several connectivity theorems on quadratic functions that combine and generalize Dines's and Brickman's results on the hidden convexity of quadratic mappings in real field. Many classical results in quadratic optimization are reproved by the unified approach, and more properties involving the matrix pencils are given.

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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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Hidden conic quadratic representation of some nonconvex quadratic optimization problems

Cited by 77 publications

References 20 publications

A Linear-Time Algorithm for Generalized Trust Region Subproblems

A Linear-Time Algorithm for Generalized Trust Region Subproblems

Connectivity of Quadratic Hypersurfaces and Its Applications in Optimization, Part I: General Theory

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

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