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A Fast Space-decomposition Scheme for Nonconvex Eigenvalue Optimization

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Abstract

In this paper, a nonsmooth bundle algorithm to minimize the maximum eigenvalue function of a nonconvex smooth function is presented. The bundle method uses an oracle to compute separately the function and subgradient information for a convex function, and the function and derivative values for the smooth mapping. Using this information, in each iteration, we replace the smooth inner mapping by its Taylor-series linearization around the current serious step. To solve the convex approximate eigenvalue problem with affine mapping faster, we adopt the second-order bundle method based on 𝓥𝓤-decomposition theory. Through the backtracking test, we can make a better approximation for the objective function. Quadratic convergence of our special bundle method is given, under some additional assumptions. Then we apply our method to some particular instance of nonconvex eigenvalue optimization, specifically: bilinear matrix inequality problems.

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References

  1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, applied mathematical sciences. Vol 75. 3rd edn. Springer-Verlag, New York (2002)

  2. Alizadeh, F., Haeberly, J.-P.A., Overton, M.L.: Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM J. Optim. 8, 746–768 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apkarian, P., Noll, D., Prot, O.: A trust region spectral bundle method for nonconvex eigenvalue optimization. SIAM J. Optim. 19(1), 281–306 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Apkarian, P., Noll, D., Thevenet, J.-B., Tuan, H.D.: A spectral quadratic-SDP method with applications to fixed-order H 2 and H 8 synthesis. Eur. J. Control 10, 527–538 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellman, R., Fan, K.: On systems of linear inequalities in Hermitian matrix variables. In: Klee, V.L. (ed.) Convexity, Volume 7 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, pp. 1–11 (1963)

    Google Scholar 

  6. Cullum, J., Donath, W., Wolfe, P.: The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Nondifferentiable Optimization, Math. Program. Stud. 3, 35–55 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clarke, F.: Optimization and Nonsmooth Analysis, Canadian Math. society series. Wiley, New York (1983)

    Google Scholar 

  8. Díaz, A.R., Kikuchi, N.: Solution to shape and topology eigenvalue optimization problems using a homogenization method. Int. J. Numer. Methods Eng. 35, 1487–1502 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (1987)

  10. Fletcher, R.: Semi-definite matrix constraints in optimization, SIAM. J. Control Optim. 23, 493–513 (1985)

  11. Fukuda, M., Kojima, M.: Branch-and-cut algorithms for the bilinear matrix inequality eigenvalue problem. Comput. Optim. Appl. 19, 79–105 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goemans, M.X.: Semidefinite programming in combinatorial optimization. Math. Program. 79, 143–161 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. 2nd corrected edn, Springer-Verlag (1993)

  14. Helmberg, C., Kiwiel, K.C.: A spectral bundle method with bounds. Math. Program. 93, 173–194 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, M., Pang, L.P., Liang, X.J., Xia, Z.Q.: The space decomposition theory for a class of semi-infinite maximum eigenvalue optimizations. Abstr. Appl. Anal. 2014, 12p (2014). Article ID 845017. doi:10.1155/2014/845017

    MathSciNet  Google Scholar 

  16. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673–696 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Helmberg, C., Rendl, F., Weismantel, R.: A semidefinite programming approach to the quadratic knapsack problem. J. Comb. Optim. 4, 197–215 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Grund. der math. Wiss, vols. 305–306. Springer, Berlin (1993)

  19. Hiriart-Urruty, J.B., Ye, D.: Sensitivity analysis of all eigenvalues of a symmetric matrix. Numer. Math. 70, 45–72 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kojima, M., Shida, M., Shindoh, S.: Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs. Math. Program. 80, 129–160 (1998)

    MathSciNet  MATH  Google Scholar 

  21. Kiwiel, K.C.: Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization. Math. Program. 52, 285–302 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lemaréchal, C.: Lagrangian relaxation. In: Jünger, M., Naddef, D. (eds.) Computational Combinatorial Optimization, pp. 112–156. Lecture Notes in Computer Science 2241. Springer, New York (2001)

  23. Lewis, A.S., Overton, M.L.: Eigenvalue optimization. Acta Numer. 5, 149–190 (1996)

  24. Lemaréchal, C., Oustry, F., Sagastizabál, C.: The \(\mathcal {U}\)-Lagrangian of a convex function. Trans. AMS 352, 711–729 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lewis, A., Wright, S.: A proximal method for composite minimization. Math. Program. series A, 158(1), 501–546 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mangasarian, O.L.: Sufficiency of exact penalty minimization. SIAM J. Control Optim. 23(1), 30–37 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Noll, D., Apkarian, P.: Spectral bundle methods for nonconvex maximum eigenvalue functions: first-order methods. Math. Program. Ser. B 104, 701–727 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Noll, D., Apkarian, P.: Spectral bundle methods for nonconvex maximum eigenvalue functions: second-order methods. Math. Program. Ser. B 104, 729–747 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nesterov, Y.: Interior-point methods: an old and new approach to nonlinear programming. Math. Program. 79, 285–297 (1997)

    MathSciNet  MATH  Google Scholar 

  30. Nesterov, Y., Nemirovsky, A.: A general approach to polynomial-time algorithms design for convex programming, Technical report, Centr. Econ.& Math. Inst., USSR Academy of Sciences, Moscow USSR (1988)

  31. Noll, D., Torki, M., Apkarian, P.: Partially augmented Lagrangian method for matrix inequality constraints. SIAM J. Optim. 15, 161–184 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Oustry, F.: The U-Lagrangian of the maximum eigenvalue function. SIAM J. Optim. 9, 526–549 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Oustry, F.: A second-order bundle method to minimize the maximum eigenvalue function. Math. Program., Ser. A 89, 1–33 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Overton, M.L.: On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9, 256–268 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  35. Overton, M.L.: Large-scale optimization of eigenvalues. SIAM J. Optim. 2, 88–120 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. Overton, M.L., Womersley, R.S.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Program. 62, 321–357 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Overton, M.L., Womersley, R.S.: Second derivatives for optimizing eigenvalues of symmetric matrices. SIAM J. Matrix Anal. Appl. 16(3), 697–718 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  38. Potra, F.A., Sheng, R.: A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8, 1007–1028 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Polak, E., Wardi, Y.: Nondifferentiable optimization algorithm for designing of control systems having singular value inequalities. Automatica 18, 267–283 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rockafellar, R.T.: Convex Analysis. Princeton University Press, New Jersey (1970)

    Book  MATH  Google Scholar 

  41. Sagastizábal, C.: Composite proximal bundle method. Math. Program. Ser. B 140, 189–233 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Shapiro, A., Fan, M.K.H.: On eigenvalue optimization. SIAM J. Optim. 5(3), 552–569 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  43. Thevenet, J.B., Noll, D., Apkarian, P.: Nonlinear spectral SDP method for BMI-constrained problems: applications to control design. In: Informatics in Control, Automation and Robotics, pp. 61–72. Kluwer Academic Publishers (2006)

  44. Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable functions. In: Balinski, M. L., Wolfe, P. (eds.) Nondifferentiable Optimization, Math. Program. Stud. 3, pp. 145–173. North-Holland, Amsterdam (1975)

    Chapter  Google Scholar 

  45. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–55 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Dalian Maritime University, 116026, Dalian, China

    Ming Huang

  2. School of Control Science and Engineering, Dalian University of Technology, 116024, Dalian, China

    Ming Huang

  3. CORA, School of Mathematical Sciences, Dalian University of Technology, 116024, Dalian, China

    Li-Ping Pang & Zun-Quan Xia

  4. College of Science, Shenyang University, Shenyang, 110044, China

    Yuan Lu

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  1. Ming Huang
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  2. Li-Ping Pang
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  3. Yuan Lu
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  4. Zun-Quan Xia
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Correspondence to Ming Huang.

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Huang, M., Pang, LP., Lu, Y. et al. A Fast Space-decomposition Scheme for Nonconvex Eigenvalue Optimization. Set-Valued Var. Anal 25, 43–67 (2017). https://doi.org/10.1007/s11228-016-0365-8

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