Abstract
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.
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Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not SOS-convex. Math. Program. 135(1–2), 275–292 (2012)
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)
Belousov, E.G., Klatte, D.: A Frank–Wolfe type theorem for convex polynomial programs. Comput. Optim. Appl. 22(1), 37–48 (2002)
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)
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)
Currie, J.: A free Matlab toolbox for optimization, OPTI toolbox, version 2.15. (2013). http://www.i2c2.aut.ac.nz/Wiki/OPTI/index.php/Main/HomePage
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)
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)
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)
Fujiwara, Y., Kuroiwa, D.: Lagrange duality in canonical DC programming. J. Math. Anal. Appl. 408(2), 476–483 (2013)
Goberna, M.A., Jeyakumar, V., Dinh, N.: Dual characterizations of set containments with strict convex inequalities. J. Glob. Optim. 34(1), 33–54 (2006)
Harada, R., Kuroiwa, D.: Lagrange-type duality in DC programming. J. Math. Anal. Appl. 418(1), 415–424 (2014)
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)
Helton, J.W., Nie, J.W.: Semidefinite representation of convex sets. Math. Program. 122(1), 21–64 (2010)
Jeyakumar, V.: Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim. 13(4), 947–959 (2003)
Jeyakumar, V., Li, G.: Characterizing robust set containments and solutions of uncertain linear programs without qualifications. Oper. Res. Lett. 38(3), 188–194 (2010)
Jeyakumar, V., Li, G.: Exact SDP relaxations for classes of nonlinear semidefinite programming problems. Oper. Res. Lett. 40(6), 529–536 (2012)
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)
Jeyakumar, V., Li, G.: A new class of alternative theorems for SOS-convex inequalities and robust optimization. Appl. Anal. 94(1), 56–74 (2015)
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)
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)
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)
Jeyakumar, V., Li, G., Vicente-Pérez, J.: Robust SOS-convex polynomial optimization problems: exact SDP relaxations. Optim. Lett. 9(1), 1–18 (2015)
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)
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)
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)
Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103(1), 45–62 (2005)
Lasserre, J.B.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4), 1995–2014 (2008)
Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2010)
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)
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)
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)
Lemaire, B.: Duality in reverse convex optimization. SIAM J. Optim. 8(4), 1029–1037 (1998)
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)
Löfberg J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)
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)
Nie, J.: Polynomial matrix inequality and semidefinite representation. Math. Oper. Res. 36(3), 398–415 (2011)
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)
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)
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)
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)
Reznick, B.: Extremal PSD forms with few terms. Duke Math. J. 45(2), 363–374 (1978)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
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)
Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66(2), 399–415 (1978)
Volle, M.: Concave duality: application to problems dealing with difference of functions. Math. Program. 41(2), 261–278 (1988)
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)
Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)
Acknowledgements
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.
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Lee, J.H., Lee, G.M. On minimizing difference of a SOS-convex polynomial and a support function over a SOS-concave matrix polynomial constraint. Math. Program. 169, 177–198 (2018). https://doi.org/10.1007/s10107-017-1210-z
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DOI: https://doi.org/10.1007/s10107-017-1210-z
Keywords
- DC programming
- Set containment
- SOS-convex polynomials
- SOS-concave matrix
- Sums of squares polynomials
- Strong duality