Abstract
In this paper, we propose a bounded degree hierarchy of both primal and dual conic programming relaxations involving both semi-definite and second-order cone constraints for solving a nonconvex polynomial optimization problem with a bounded feasible set. This hierarchy makes use of some key aspects of the convergent linear programming relaxations of polynomial optimization problems (Lasserre in Moments, positive polynomials and their applications, World Scientific, Singapore, 2010) associated with Krivine–Stengle’s certificate of positivity in real algebraic geometry and some advantages of the scaled diagonally dominant sum of squares (SDSOS) polynomials (Ahmadi and Hall in Math Oper Res, 2019. https://doi.org/10.1287/moor.2018.0962; Ahmadi and Majumdar in SIAM J Appl Algebra Geom 3:193–230, 2019). We show that the values of both primal and dual relaxations converge to the global optimal value of the original polynomial optimization problem under some technical assumptions. Our hierarchy, which extends the so-called bounded degree Lasserre hierarchy (Lasserre et al. in Eur J Comput Optim 5:87–117, 2017), has a useful feature that the size and the number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. As a special case, we provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for classes of polynomial optimization problems. This includes one-step convergence for a new class of first-order SDSOS-convex polynomial programs. In this case, we also show how a global solution is recovered from the level one SOCP relaxation. We finally derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by concave polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification.
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Notes
This was kindly pointed out to us by one of the referees
It is worth noting that [2, Proposition 3.15] provides a further class of examples for convex quadratic functions which are not SDSOS.
References
Ahmadi, A.A., Hall, G.: On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity. Math. Oper. Res. (2019). https://doi.org/10.1287/moor.2018.0962
Ahmadi, A.A., Majumdar, A.: DSOS and SDSOS optimization: more tractable alternatives to sum of squares and semidefinite optimization. SIAM J. Appl. Algebra Geom. 3, 193–230 (2019)
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)
Bertsimas, D., Freund, R.M., Sun, X.A.: An accelerated first-order method for solving SOS relaxations of unconstrained polynomial optimization problems. Optim. Methods Softw. 28, 424–441 (2013)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Chuong, T.D., Jeyakumar, V.: Convergent conic linear programming relaxations for cone convex polynomial programs. Oper. Res. Lett. 45(3), 220–226 (2017)
Chuong, T.D., Jeyakumar, V.: Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers. J. Global Optim. 72(4), 655–678 (2018)
D’Angelo, P., Putinar, M.: Polynomial Optimization on Odd-Dimensional Spheres, in Emerging Applications of Algebraic Geometry. Springer, New York (2008)
Fidalgo, C., Kovacec, A.: Positive semidefinite diagonal minus tail forms are sums of squares. Math. Z. 269, 629–645 (2011)
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht (1999)
Ghaddar, B., Vera, J.C., Anjos, M.F.: A dynamic inequality generation scheme for polynomial programming. Math. Program. 156, 21–57 (2016)
Ghasemi, M., Marshall, M.: Lower bounds for polynomials using geometric programming. SIAM J. Optim. 22(2), 460–473 (2012)
Henrion, D., Lasserre, J.B., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24, 761–779 (2009)
Horn, R., Johnson, C.R.: Matrix Analysis, 2nd edn, p. xviii+643. Cambridge University Press, Cambridge (2013)
Hu, S., Li, G., Qi, L.: A tensor analogy of Yuan’s theorem of the alternative and polynomial optimization with sign structure. J. Optim. Theory Appl. 168(2), 446–474 (2016)
Helton, J.W., Nie, J.W.: Semidefinite representation of convex sets. Math. Program. 122, 21–64 (2010)
Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136(1), 31–41 (2008)
Jeyakumar, V., Lee, G.M., Li, G.: Alternative theorems for quadratic inequality systems and global quadratic optimization. SIAM J. Optim. 2, 667–690 (2009)
Jeyakumar, V., Li, G.: Exact conic programming relaxations for a class of convex polynomial cone programs. J. Optim. Theory Appl. 172(1), 156–178 (2017)
Jeyakumar, V., Kim, S., Lee, G.M., Li, G.: Solving global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets. J. Global Optim. 65, 175–190 (2016)
Josa, C., Molzahn, D.: Lasserre hierarchy for large scale polynomial optimization in real and complex variables. SIAM J. Optim. 28, 1017–1048 (2018)
Krivine, J.L.: Anneaux préordonnés. J. Anal. Math. 12, 307–326 (1964)
Kim, S., Kojima, M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26(2), 143–154 (2003)
Kuang, X., Ghaddar, B., Naoum-Sawaya, J., Zuluaga, L.F.: Alternative SDP and SOCP approximations for polynomial optimization. Eur. J. Comp. Optim. 7, 153–175 (2019)
Lasserre, J.B.: A Lagrangian relaxation view of linear and semidefinite hierarchies. SIAM J. Optim 23(3), 1742–1756 (2013)
Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. World Scientific, Singapore (2010)
Lasserre, J.B.: Representation of nonnegative convex polynomial. Arch. Math. 91, 126–130 (2008)
Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, Vol. 149 of IMA Volumes in Mathematics and its Applications, vol. 149, pp. 157–270. Springer, Berlin (2009)
Lasserre, J.B., Toh, K.C., Yang, S.: A bounded degree SOS hierarchy for polynomial optimization. Eur. J. Comput. Optim. 5, 87–117 (2017)
Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications, Synthesis Lectures on Mathematics and Statistics, 14. Morgan & Claypool Publishers, Williston (2014)
Megretski, A.: SPOT (Systems polynomial optimization tools) Manual, 2010, http://web.mit.edu/ameg/www/images/spot_manual.pdf
Nie, J.W.: Polynomial matrix inequality and semidefinite representation. Math. Oper. Res. 36, 398–415 (2011)
Nie, J.W., Wang, L.: Regularization methods for SDP relaxations in large-scale polynomial optimization. SIAM J. Optim. 22, 408–428 (2012)
Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320 (2003)
Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Program. 77, 301–320 (1997)
Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006)
Weisser, T., Lasserre, J., Toh, K.: Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity. Math. Program. Comput. 5, 1–32 (2017)
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The authors would like to thank the referees for their valuable comments and suggestions which greatly improved the original version of the paper.
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Appendix: Test problems
Appendix: Test problems
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TP1 ([11, Test problem 3, page 24]): Consider the following nonconvex polynomial problem:
$$\begin{aligned} \begin{array}{ll} \displaystyle \min _{x \in {\mathbb {R}}^6} &{} -25(x_1 - 2)^2 - (x_2 - 2)^2 - (x_3 - 1)^2-(x_4 - 4)^2 - (x_5 - 1)^2 - (x_6 - 4)^2 \\ \text{ s.t. } &{} (x_3-3)^2+x_4\ge 4, (x_5-3)^2+x_6\ge 4, x_1-3x_2\le 2, -x_1\\ &{}+x_2\le 2, x_1+x_2\le 6,\\ &{} x_1+x_2\ge 2, x_1\ge 0, x_2\ge 0, 1\le x_3\le 5, 0\le x_4\le 6, 1\le x_5\le 5, 0\le x_6\le 10. \end{array} \end{aligned}$$In [11, Test problem 3, page 24], it is shown that the optimal value is \(-310\) and is attained at \(x^*=(5,1,5,0,5,10)^T\).
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TP2 ([11, Test problem 3, page 7]): Consider the following nonconvex polynomial problem:
$$\begin{aligned} \begin{array}{ll} \displaystyle \min _{x \in {\mathbb {R}}^{13}} &{} 50x_1+50x_2+50x_3+50x_4-\sum _{i=1}^450x_i^2-\sum _{i=5}^{13} x_i \\ \text{ s.t. } &{} 2x_1+ 2x_2+ x_{10}+x_{11}\le 10, 2x_1\\ &{} + 2x_3+ x_{10}+x_{12}\le 10, 2x_2+ 2x_3+ x_{11}+x_{12}\le 10,\\ &{} -8x_1+x_{10}\le 0, -8x_2+x_{11}\le 0, -8x_3+x_{12}\le 0, -2x_4-x_5+x_{10}\le 0, \\ &{} -2x_6-x_7+x_{11}\le 0, -2x_8-x_9+x_{12}\le 0, 0\le x_i\le 3, i=10,11,12,\\ &{} 0\le x_i\le 1, i=1,\ldots ,9,13. \end{array} \end{aligned}$$In [11, Test problem 3, page 7], it is shown that the optimal value is \(-15\) and is attained at \(x^*=(1,1,1, 1,1,1,1,1,1,3,3,3,1)^T\).
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TP3 ([30, Example P4-4, page 109]): Consider the following nonconvex polynomial problem:
$$\begin{aligned} \begin{array}{ll} \displaystyle \min _{x \in {\mathbb {R}}^{4}} &{} x_1^4-x_2^4+x_3^4-x_4^4 \\ \text{ s.t. } &{} 0\le 2x_1^4+3x_2^2+2x_1x_2+2x_3^4+3x_4^2+2x_3x_4\le 1,\\ &{} 0\le 3x_1^2+2x_2^2-4x_1x_2+3x_3^2+2x_4^2-4x_3x_4\le 1, \\ &{} 0\le x_1^2+6x_2^2-4x_1x_2+x_3^2+6x_4^2-4x_3x_4\le 1,\\ &{} 0\le x_1^2+4x_2^4-3x_1x_2+x_3^2+4x_4^4-3x_3x_4\le 1,\\ &{} 0\le 2x_1^2+5x_2^2+3x_1x_2+2x_3^2+5x_4^2+3x_3x_4\le 1, x\in [0,1]^4. \end{array} \end{aligned}$$In [30, Example P4-4, page 109], it is shown that the optimal value is \(-0.033539\).
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TP4 ([30, Example P8-4, page 111]): Consider the following nonconvex polynomial problem:
$$\begin{aligned} \begin{array}{ll} \displaystyle \min _{x \in {\mathbb {R}}^{8}} &{} x_1^4-x_2^4+x_3^4-x_4^4+x_5^4-x_6^4+x_7^4-x_8^4+x_1-x_2 \\ \text{ s.t. } &{} 0\le 2x_1^4+3x_2^2+2x_1x_2+2x_3^4+3x_4^2+2x_3x_4+2x_5^4+3x_6^2+2x_5x_6+2x_7^4+3x_8^2\\ &{} +2x_7x_8\le 1,\\ &{} 0\le 3x_1^2+2x_2^2-4x_1x_2+3x_3^2+2x_4^2-4x_3x_4+3x_5^2+2x_6^2-4x_5x_6\\ &{} +3x_7^2+2x_8^2-4x_7x_8\le 1, \\ &{} 0\le x_1^2+6x_2^2-4x_1x_2+x_3^2+6x_4^2-4x_3x_4+x_5^2+6x_6^2-4x_5x_6+x_7^2\\ &{} +6x_8^2-4x_7x_8\le 1,\\ &{} 0\le x_1^2+4x_2^4-3x_1x_2+x_3^2+4x_4^4-3x_3x_4+x_5^2+4x_6^4-3x_5x_6+x_7^2\\ &{} +4x_8^4-3x_7x_8\le 1,\\ &{} 0\le 2x_1^2+5x_2^2+3x_1x_2+2x_3^2+5x_4^2+3x_3x_4+2x_5^2+5x_6^2+3x_5x_6+2x_7^2\\ &{} +5x_8^2+3x_7x_8\le 1,\\ &{} x\in [0,1]^8. \end{array} \end{aligned}$$In [30, Example P8-4, page 111], it is shown that the optimal value is \(-0.43603\).
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TP5 ([30, Example P20-4, page 113]): Consider the following nonconvex polynomial problem:
$$\begin{aligned} \begin{array}{ll} \displaystyle \min _{x \in {\mathbb {R}}^{20}} &{} x_1^4-x_2^4+x_3^2-x_4^2+x_5^2-x_6^2+x_7^2-x_8^2+x_9^2-x_{10}^2+x_{11}^2-x_{12}^2\\ &{} +x_1-x_2+x_{13}^2-x_{14}^2+x_{15}^2\\ &{} -x_{16}^2+x_{17}^2-x_{18}^2+x_{19}^2-x_{20}^2 \\ \text{ s.t. } &{} 0\le 2x_1^2+3x_2^2+2x_1x_2+2x_3^2+3x_4^2+2x_3x_4+2x_5^2+3x_6^2\\ &{} +2x_5x_6+2x_7^2+3x_8^2+2x_7x_8\\ &{} +2x_9^2+3x_{10}^2+2x_9x_{10}+2x_{11}^2+3x_{12}^2+2x_{11}x_{12}\\ &{} +2x_{13}^2+ 3x_{14}^2+2x_{13}x_{14}+2x_{15}^2 +3x_{16}^2\\ &{} +2x_{15}x_{16}+2x_{17}^2+3x_{18}^2+2x_{17}x_{18}+2x_{19}^2\\ &{} +3x_{20}^2+2x_{19}x_{20}\le 1,\\ &{} 0\le 3x_1^2+2x_2^2-4x_1x_2+3x_3^2+2x_4^2-4x_3x_4+3x_5^2+2x_6^2-4x_5x_6\\ &{} +3x_7^2+2x_8^2-4x_7x_8\\ &{} +3x_9^2+2x_{10}^2-4x_9x_{10}+3x_{11}^2+2x_{12}^2-4x_{11}x_{12}+3x_{13}^2\\ &{} + 2x_{14}^2-4x_{13}x_{14}+3x_{15}^2 +2x_{16}^2\\ &{} -4x_{15}x_{16}+3x_{17}^2+2x_{18}^2-4x_{17}x_{18}+3x_{19}^2\\ &{} +2x_{20}^2-4x_{19}x_{20}\le 1, \\ &{} 0\le x_1^2+6x_2^2-4x_1x_2+x_3^2+6x_4^2-4x_3x_4+x_5^2+6x_6^2-4x_5x_6\\ &{} +x_7^2+6x_8^2-4x_7x_8\\ &{} +x_9^2+6x_{10}^2-4x_9x_{10}+x_{11}^2+6x_{12}^2-4x_{11}x_{12}+x_{13}^2\\ &{} + 6x_{14}^2-4x_{13}x_{14}+x_{15}^2 +6x_{16}^2\\ &{} -4x_{15}x_{16}+x_{17}^2+6x_{18}^2-4x_{17}x_{18}+x_{19}^2+6x_{20}^2-4x_{19}x_{20}\le 1,\\ &{} 0\le x_1^2+4x_2^2-3x_1x_2+x_3^2+4x_4^2-3x_3x_4+x_5^2+4x_6^2-3x_5x_6\\ &{} +x_7^2+4x_8^2-3x_7x_8\\ &{} +x_9^2+4x_{10}^2-3x_9x_{10}+x_{11}^2+4x_{12}^2-3x_{11}x_{12}+x_{13}^2+ 4x_{14}^2\\ &{} -3x_{13}x_{14}+x_{15}^2 +4x_{16}^2\\ &{} -3x_{15}x_{16}+x_{17}^2+4x_{18}^2-3x_{17}x_{18}+x_{19}^2+4x_{20}^2\\ &{} -3x_{19}x_{20}\le 1,\\ &{} 0\le 2x_1^2+5x_2^2+3x_1x_2+2x_3^2+5x_4^2+3x_3x_4+2x_5^2+5x_6^2\\ &{} +3x_5x_6+2x_7^2+5x_8^2+3x_7x_8\\ &{} +2x_9^2+5x_{10}^2+3x_9x_{10}+2x_{11}^2+5x_{12}^2+3x_{11}x_{12}+2x_{13}^2\\ &{} + 5x_{14}^2+3x_{13}x_{14}+2x_{15}^2 +5x_{16}^2\\ &{} +3x_{15}x_{16}+2x_{17}^2+5x_{18}^2+3x_{17}x_{18}+2x_{19}^2\\ &{} +5x_{20}^2+3x_{19}x_{20}\le 1,\\ &{} x\in [0,1]^{20}. \end{array} \end{aligned}$$In [30, Example P20-4, page 113], it is shown that the optimal value is \(-0.43603\).
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TP6 ([30, Example P6-6, page 110]): Consider the following nonconvex polynomial problem:
$$\begin{aligned} \begin{array}{ll} \displaystyle \min _{x \in {\mathbb {R}}^{6}} &{} x_1^6-x_2^6+x_3^6-x_4^6+x_5^6-x_6^6+x_1-x_2 \\ \text{ s.t. } &{} 0\le 2x_1^6+3x_2^2+2x_1x_2+2x_3^6+3x_4^2+2x_3x_4+2x_5^6+3x_6^2+2x_5x_6\le 1,\\ &{} 0\le 3x_1^2+2x_2^2-4x_1x_2+3x_3^2+2x_4^2-4x_3x_4+3x_5^2+2x_6^2-4x_5x_6\le 1, \\ &{} 0\le x_1^2+6x_2^2-4x_1x_2+x_3^2+6x_4^2-4x_3x_4+x_5^2+6x_6^2-4x_5x_6\le 1,\\ &{} 0\le x_1^2+4x_2^6-3x_1x_2+x_3^2+4x_4^6-3x_3x_4+x_5^2+4x_6^6-3x_5x_6\le 1,\\ &{} 0\le 2x_1^2+5x_2^2\!+\!3x_1x_2\!+\!2x_3^2\!+\!5x_4^2\!+\!3x_3x_4\!+\!2x_5^2\!+\!5x_6^2+3x_5x_6\le 1, x\in [0,1]^6. \end{array} \end{aligned}$$In [30, Example P6-6, page 110], it is shown that the optimal value is \(-0.41288\).
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Chuong, T.D., Jeyakumar, V. & Li, G. A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs. J Glob Optim 75, 885–919 (2019). https://doi.org/10.1007/s10898-019-00831-9
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DOI: https://doi.org/10.1007/s10898-019-00831-9