Mathematical Programming

, Volume 169, Issue 1, pp 177–198 | Cite as

On minimizing difference of a SOS-convex polynomial and a support function over a SOS-concave matrix polynomial constraint

Full Length Paper Series B


In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater’s condition. Using our set containment characterization, we derive a zero duality gap result for a DC optimization problem with a SOS-convex polynomial and a support function, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programs. Also, we present the relations of their solutions. Finally, through a simple numerical example, we illustrate our results. Particularly, in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem.


DC programming Set containment SOS-convex polynomials SOS-concave matrix Sums of squares polynomials Strong duality 

Mathematics Subject Classification

90C30 90C22 26A51 



This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIT) (NRF-2016R1A2B1006430). The authors would like to express their sincere thanks to anonymous referees for valuable suggestions and comments for the paper.


  1. 1.
    Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not SOS-convex. Math. Program. 135(1–2), 275–292 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ahmadi, A.A., Parrilo, P.A.: A complete characterization of the gap between convexity and SOS-convexity. SIAM J. Optim. 23(2), 811–833 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Belousov, E.G., Klatte, D.: A Frank–Wolfe type theorem for convex polynomial programs. Comput. Optim. Appl. 22(1), 37–48 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boţ, R.I., Wanka, G.: Duality for multiobjective optimization problems with convex objective functions and DC constraints. J. Math. Anal. Appl. 315(2), 526–543 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boţ, R.I., Hodrea, I.B., Wanka, G.: Some new Farkas-type results for inequality systems with DC functions. J. Glob. Optim. 39(4), 595–608 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Currie, J.: A free Matlab toolbox for optimization, OPTI toolbox, version 2.15. (2013).
  7. 7.
    Dinh, N., Jeyakumar, V., Lee, G.M.: Sequential Lagrangian conditions for convex programs with applications to semidefinite programming. J. Optim. Theory Appl. 125(1), 85–112 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dinh, N., Mordukhovich, B., Nghia, T.T.A.: Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs. Math. Program. 123(1), 101–138 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fang, D.H., Li, C., Yang, X.Q.: Stable and total Fenchel duality for DC optimization problems in locally convex spaces. SIAM J. Optim. 21(3), 730–760 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fujiwara, Y., Kuroiwa, D.: Lagrange duality in canonical DC programming. J. Math. Anal. Appl. 408(2), 476–483 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goberna, M.A., Jeyakumar, V., Dinh, N.: Dual characterizations of set containments with strict convex inequalities. J. Glob. Optim. 34(1), 33–54 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Harada, R., Kuroiwa, D.: Lagrange-type duality in DC programming. J. Math. Anal. Appl. 418(1), 415–424 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Härter, V., Jansson, C., Lange, M.: VSDP: a matlab toolbox for verified semidefinite-quadratic-linear programming. Technical report, Institute for Reliable Computing, Hamburg University of Technology (2012)Google Scholar
  14. 14.
    Helton, J.W., Nie, J.W.: Semidefinite representation of convex sets. Math. Program. 122(1), 21–64 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jeyakumar, V.: Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim. 13(4), 947–959 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jeyakumar, V., Li, G.: Characterizing robust set containments and solutions of uncertain linear programs without qualifications. Oper. Res. Lett. 38(3), 188–194 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jeyakumar, V., Li, G.: Exact SDP relaxations for classes of nonlinear semidefinite programming problems. Oper. Res. Lett. 40(6), 529–536 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jeyakumar, V., Li, G.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. 147(1–2), 171–206 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jeyakumar, V., Li, G.: A new class of alternative theorems for SOS-convex inequalities and robust optimization. Appl. Anal. 94(1), 56–74 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Jeyakumar, V., Vicente-Pérez, J.: Dual semidefinite programs without duality gaps for a class of convex minimax program1s. J. Optim. Theory Appl. 162(3), 735–753 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14(2), 534–547 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jeyakumar, V., Lee, G.M., Lee, J.H.: Generalized SOS-convexity and strong duality with SDP dual programs in polynomial optimization. J. Convex Anal. 22(4), 999–1023 (2015)MathSciNetMATHGoogle Scholar
  23. 23.
    Jeyakumar, V., Li, G., Vicente-Pérez, J.: Robust SOS-convex polynomial optimization problems: exact SDP relaxations. Optim. Lett. 9(1), 1–18 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jeyakumar, V., Kim, S., Lee, G.M., Li, G.: Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets. J. Glob. Optim. 65(2), 175–190 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jeyakumar, V., Lee, G.M., Lee, J.H.: Sums of squares characterizations of containment of convex semialgebraic sets. Pac. J. Optim. 12(1), 29–42 (2016)MathSciNetMATHGoogle Scholar
  26. 26.
    Jeyakumar, V., Lee, G.M., Linh, N.T.H.: Generalized Farkas’ lemma and gap-free duality for minimax DC optimization with polynomials and robust quadratic optimization. J. Glob. Optim. 64(4), 679–702 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103(1), 45–62 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lasserre, J.B.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4), 1995–2014 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2010)MATHGoogle Scholar
  30. 30.
    Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by D.C. algorithms. J. Glob. Optim. 11(3), 253–285 (1997)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Le Thi, H.A., Van Ngai, H., Pham Dinh, T.: DC programming and DCA for general DC programs. In: Van Do, T., et al. (eds.) Advanced Computational Methods for Knowledge Engineering. Advances in Intelligent Systems and Computing, vol. 282, pp. 15–35. Springer, Berlin (2014)CrossRefGoogle Scholar
  33. 33.
    Lemaire, B.: Duality in reverse convex optimization. SIAM J. Optim. 8(4), 1029–1037 (1998)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Lemaire, B., Volle, M.: A general duality scheme for nonconvex minimization problems with a strict inequality constraint. J. Glob. Optim. 13(3), 317–327 (1998)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Löfberg J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)Google Scholar
  36. 36.
    Martinez-Legaz, J.E., Volle, M.: Duality in DC programming: the case of several DC constraints. J. Math. Anal. Appl. 237(2), 657–671 (1998)CrossRefMATHGoogle Scholar
  37. 37.
    Nie, J.: Polynomial matrix inequality and semidefinite representation. Math. Oper. Res. 36(3), 398–415 (2011)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Niu, Y.S., Pham Dinh, T.: DC programming approaches for BMI and QMI feasibility problems. In: Do Van, T., et al. (eds.) Advanced Computational Methods for Knowledge Engineering. Advances in Intelligent Systems and Computing, pp. 37–63. Springer, New York (2014)CrossRefGoogle Scholar
  39. 39.
    Niu, Y.S., Judice, J.J., Le Thi, H.A., Dinh, T.P.: Solving the quadratic eigenvalue complementarity problem by DC programming. In: Le Thi, H.A., et al. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences. Advances in Intelligent Systems and Computing, pp. 203–214. Springer, New York (2015)Google Scholar
  40. 40.
    Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P.A.: SOSTOOLS: sum of squares optimization toolbox for MATLAB, version 2.00. California Institute of Technology Pasadena (2004)Google Scholar
  41. 41.
    Reznick, B.: Some concrete aspects of Hilbert’s 17th problem. In: Delzell, C.N., Madden, J.J. (eds.) Real Algebraic Geometry and Ordered Structures. Contemporary Mathematics, vol. 253, pp. 251–272. American Mathematical Society, Providence, RI (2000)Google Scholar
  42. 42.
    Reznick, B.: Extremal PSD forms with few terms. Duke Math. J. 45(2), 363–374 (1978)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  44. 44.
    Sturm, J.F.: Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12(1–4), 625–653 (1999)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66(2), 399–415 (1978)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Volle, M.: Concave duality: application to problems dealing with difference of functions. Math. Program. 41(2), 261–278 (1988)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsPukyong National UniversityBusanRepublic of Korea

Personalised recommendations