Globally maximizing the sum of squares of quadratic forms over the unit sphere

1. Aparicio, G., Casado, L.G., G-Tóth, B., Hendrix, E.M.T. and García, I.: Heuristics to reduce the number of simplices in longest edge bisection refinement of a regular \(n\)-Simplex. In: Computational science and its applications ICCSA. Lecture notes in computer science, vol. 8580, pp 115–125. Springer International Publishing, Cham (2014)

2. Aparicio, G., Casado, L.G., G-Tóth, B., Hendrix, E.M.T., García, I.: On the minimum number of simplex shapes in longest edge bisection refinement of a regular \(n\)-simplex. Informatica 26(1), 17–32 (2015)

   Article  MathSciNet  Google Scholar 

3. Aparicio, G., Salmerón, J.M., Casado, L.G., Asenjo, R., Hendrix, E.M.T.: Parallel algorithms for computing the smallest binary tree size in unit simplex refinement. J. Parallel Distrib. Comput. 112, 166–178 (2018)

   Article  Google Scholar 

4. Dickinson, P.J.: On the exhaustivity of simplicial partitioning. J. Global Optim. 58(1), 189–203 (2014)

   Article  MathSciNet  Google Scholar 

5. Goldfarb, D., Liu, S.: An \(O(n^3L)\) primal interior point algorithm for convex quadratic programming. Math. Program. 49(1–3), 325–340 (1990)

   Article  Google Scholar 

6. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1989)

   MATH  Google Scholar 

7. Hao, C.L., Cui, C.F., Dai, Y.H.: A sequential subspace projection method for extreme Z-eigenvalues of supersymmetric tensor. Numer. Linear Algebra Appl. 22(2), 283–298 (2015)

   Article  MathSciNet  Google Scholar 

8. Henrion, D., Lasserre, J.B., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

   Article  MathSciNet  Google Scholar 

9. Horst, R.: A New Branch and Bound Approach for Cconcave Minimization Problems. Lecture Notes in Computer Science, vol. 41, pp. 330–337. Springer, Berlin (1975)

   Google Scholar 

10. Horst, R.: An algorithm for nonconvex programming problems. Math. Program. 10(1), 312–321 (1976)

    Article  MathSciNet  Google Scholar 

11. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Springer, Berlin (2000)

    Book  Google Scholar 

12. Jiang, B., Ma, S.Q., Zhang, S.Z.: Alternating direction method of multipliers for real and complex polynomial optimization models. Optimization 63(6), 883–898 (2014)

    Article  MathSciNet  Google Scholar 

13. Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884 (2002)

    Article  MathSciNet  Google Scholar 

14. Kuczynski, J., Wozniakowski, H.: Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start. SIAM J. Matrix Anal. Appl. 13(4), 1094–1122 (1992)

    Article  MathSciNet  Google Scholar 

15. Li, Z.N., He, S.M., Zhang, S.Z.: Approximation methods for polynomial optimization: models, algorithms and applications. Springerbriefs in Optimization. Springer, New York (2012)

    Book  Google Scholar 

16. Mourrain, B., Pavone, J.P.: Subdivision methods for solving polynomial equations. J. Symbol. Comput. 44(3), 292–306 (2009)

    Article  MathSciNet  Google Scholar 

17. Mourrain, B., Trebuchet, P.: Generalized normal forms and polynomial system solving. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ACM, pp. 253–260 (2005)

18. Nesterov, Y.: Random walk in a simplex and quadratic optimization over convex polytopes. CORE Discussion Paper 2003/71, CORE-UCL, Louvain-La-Neuve (2003)

19. Nie, J., Wang, L.: Semidefinite relaxations for the best rank-1 tensor approximation. SIAM J. Matrix Anal. Appl. 35(3), 1155–1179 (2014)

    Article  MathSciNet  Google Scholar 

20. Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. Springer, New York (2014)

    Book  Google Scholar 

21. Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996)

    Article  MathSciNet  Google Scholar 

22. Salmerón, J.M.G., Aparicio, G., Casado, L.G., García, I., Hendrix, E.M.T., G-Tóth, B.: Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement. J. Comb. Optim. 33(2), 389–402 (2017)

    Article  MathSciNet  Google Scholar 

23. So, A.M.C.: Deterministic approximation algorithmsfor sphere constrained homogeneous polynomial optimization problems. Math. Program. 129(2), 357–382 (2011)

    Article  MathSciNet  Google Scholar 

24. Sturm, J.F.: SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

25. Wang, L., Xia, Y.: A linear-time algorithm for globally maximizing the sum of a generalized rayleigh quotient and a quadratic form on the unit sphere. SIAM J. Optim. 29(3), 1844–1869 (2019)

    Article  MathSciNet  Google Scholar 

26. Wang, Y.J., Zhou, G.L.: A hybrid second-order method for homogenous polynomial optimization over unit sphere. J. Oper. Res. Soc. China 5(1), 99–109 (2017)

    Article  MathSciNet  Google Scholar 

27. Xia, Y.: On local convexity of quadratic transformations. J. Oper. Res. Soc. China 2, 341–350 (2014)

    Article  MathSciNet  Google Scholar 

28. Zhou, G., Caccetta, L., Teo, K.L., Wu, S.Y.: Nonnegative polynomial optimization over unit spheres and convex programming relaxations. SIAM J. Optim. 22(3), 987–1008 (2012)

    Article  MathSciNet  Google Scholar 

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