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A hybrid algorithm for the two-trust-region subproblem

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Abstract

Two-trust-region subproblem (TTRS), which is the minimization of a general quadratic function over the intersection of two full-dimensional ellipsoids, has been the subject of several recent research. In this paper, to solve TTRS, a hybrid of efficient algorithms for finding global and local-nonglobal minimizers of trust-region subproblem and the alternating direction method of multipliers (ADMM) is proposed. The convergence of the ADMM steps to the first-order stationary condition is proved under certain conditions. On several test problems, we compare the new algorithm against three competitors: the Snopt software, the algorithm proposed by Sakaue et al. (SIAM J Optim 26:1669–1694, 2016) and the CADMM algorithm proposed by Huang and Sidiropoulos (IEEE Trans Signal Process 64:5297–5310, 2016). The numerical results show that the new algorithm is competitive.

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References

  • Adachi S, Iwata S, Nakatsukasa Y, Takeda A (2017) Solving the trust region subproblem by a generalized eigenvalue problem. SIAM J Optim 27(1):269–291

    Article  MathSciNet  MATH  Google Scholar 

  • Ai W, Zhang S (2008) Strong duality for the CDT subproblem: a necessary and sufficient condition. SIAM J Optim 19(4):1735–1756

    Article  MathSciNet  MATH  Google Scholar 

  • Bai X, Scheinberg K (2015) Alternating direction methods for non convex optimization with applications to second-order least-squares and risk parity portfolio selection. Optim Online

  • Beck A, Eldar YC (2006) Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J Optim 17(3):844–860

    Article  MathSciNet  MATH  Google Scholar 

  • Bertsekas D (2014) Constrained optimization and Lagrange multiplier methods. Academic press, New York

    MATH  Google Scholar 

  • Bienstock D (2016) A note on polynomial solvability of the CDT problem. SIAM J Optim 26(1):488–498

    Article  MathSciNet  MATH  Google Scholar 

  • Bomze Im, Overton Ml (2015) Narrowing the difficulty gap for the Celis-Dennis-Tapia problem. Math Program 151(2):459–476

    Article  MathSciNet  MATH  Google Scholar 

  • Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3(1):1–122

    Article  MATH  Google Scholar 

  • Burer S, Anstreicher KM (2013) Second-order-cone constraints for extended trust-region subproblems. SIAM J Optim 23(1):432–451

    Article  MathSciNet  MATH  Google Scholar 

  • Burer S, Yang B (2015) The trust region subproblem with non-intersecting linear constraints. Math Program 149(1–2):253–264

    Article  MathSciNet  MATH  Google Scholar 

  • Celis MR, Dennis JE, Tapia RA (1984) A trust region algorithm for nonlinear equality constrained optimization. Numer Optim 1984:71–82

    Google Scholar 

  • Chen X, Yuan Y (2001) On maxima of dual function of the CDT subproblem. J Comput Math 19(1):113–124

    MathSciNet  MATH  Google Scholar 

  • Conn AR, Gould NI, Toint PL (2000) Trust region methods. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Grant M, Boyd S (2013) CVX: Matlab software for disciplined convex programming, version 2.0 beta. http://cvxr.com/cvx

  • Hajinezhad D, Shi Q (2018) Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications. J Glob Optim 70(1):261–288

    Article  MathSciNet  MATH  Google Scholar 

  • Heinkenschloss M (1994) On the solution of a two ball trust region subproblem. Math Program 64(1–3):249–276

    Article  MathSciNet  MATH  Google Scholar 

  • Hong M, Luo ZQ, Razaviyan M (2016) Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM J Optim 26(1):337–364

    Article  MathSciNet  MATH  Google Scholar 

  • Huang K, Sidiropoulos ND (2016) Consensus-ADMM for general quadratically constrained quadratic programming. IEEE Trans Signal Process 64(20):5297–5310

    Article  MathSciNet  MATH  Google Scholar 

  • Li GD, Yuan Y (2005) Compute a Celis–Dennis–Tapia step. J Comput Math 23(1):463–478

    MathSciNet  MATH  Google Scholar 

  • Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second-order cone programming. Linear Algebra Appl 284(1–3):193–228

    Article  MathSciNet  MATH  Google Scholar 

  • Luo H, Sun X, Wu H (2008) Convergence properties of augmented Lagrangian methods for constrained global optimization. Optim Methods Softw 23(5):763–778

    Article  MathSciNet  MATH  Google Scholar 

  • Luo HZ, Sun XL, Li D (2007) On the convergence of augmented Lagrangian methods for constrained global optimization. SIAM J Optim 18(4):1209–1230

    Article  MathSciNet  MATH  Google Scholar 

  • Martínez JM (1994) Local minimizers of quadratic functions on Euclidean balls and spheres. SIAM J Optim 4(1):159–176

    Article  MathSciNet  MATH  Google Scholar 

  • Nesterov Y, Wolkowicz H, Ye Y (2000) Semidefinite programming relaxations of nonconvex quadratic optimization, handbook of semidefinite programming. Springer, Boston, pp 361–419

    MATH  Google Scholar 

  • Peng J-M, Yuan Y (1997) Optimality conditions for the minimization of a quadratic with two quadratic constraints. SIAM J Optim 7(3):579–594

    Article  MathSciNet  MATH  Google Scholar 

  • Sakaue S, Nakatsukasa Y, Takeda A, Iwata S (2016) Solving generalized CDT problems via two-parameter eigenvalues. SIAM J Optim 26(3):1669–1694

    Article  MathSciNet  MATH  Google Scholar 

  • Salahi M, Taati A (2017) Alternating direction method of multipliers for the extended trust region subproblem. Iran J Numer Anal Optim 7(1):107–117

    MATH  Google Scholar 

  • Salahi M, Taati A, Wolkowicz H (2016) Local nonglobal minima for solving large scale extended trust region subproblems. Comput Optim Appl 66(2):223–244

    Article  MathSciNet  MATH  Google Scholar 

  • Shen Y, Wen Z, Zhang Y (2014) Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization. Optim Methods Softw 29(2):239–263

    Article  MathSciNet  MATH  Google Scholar 

  • Xu L, Yu B, Zhang Y (2017) An alternating direction and projection algorithm for structure-enforced matrix factorization. Comput Optim Appl 68(2):33–362

    Article  MathSciNet  MATH  Google Scholar 

  • Yang B, Burer S (2016) A two-variable approach to the two-trust-region subproblem. SIAM J Optim 26(1):661–680

    Article  MathSciNet  MATH  Google Scholar 

  • Yuan Y (1991) A dual algorithm for minimizing a quadratic function with two quadratic constraints. J Comput Math 9(4):348–359

    MathSciNet  MATH  Google Scholar 

  • Yuan Y (1990) On a subproblem of trust region algorithms for constrained optimization. Math Program 47(1–3):53–63

    Article  MathSciNet  MATH  Google Scholar 

  • Ye Y, Zhang S (2003) New results on quadratic minimization. SIAM J Optim 14(1):245–267

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Y (1992) Computing a Celis–Dennis–Tapia trust-region step for equality constrained optimization. Math Program 55(1–3):109–124

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank both reviewers for their useful comments and questions which improved the paper and University of Guilan for supporting this research.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    Saeid Ansary Karbasy & Maziar Salahi

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  1. Saeid Ansary Karbasy
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  2. Maziar Salahi
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Correspondence to Maziar Salahi.

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Communicated by Orizon Pereira Ferreira.

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Ansary Karbasy, S., Salahi, M. A hybrid algorithm for the two-trust-region subproblem. Comp. Appl. Math. 38, 115 (2019). https://doi.org/10.1007/s40314-019-0864-y

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  • DOI: https://doi.org/10.1007/s40314-019-0864-y

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