Abstract
In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.
Similar content being viewed by others
References
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)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, Philadelphia (2001)
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, 220–226 (2017)
Flores-Bazan, F., Opazo, F.: Characterizing the convexity of joint-range for a pair of inhomogeneous quadratic functions and strong duality. Minimax Theory Appl. 1(2), 257–290 (2016)
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Nashua (2004)
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1, Mar 2014. http://cvxr.com/cvx
Helton, J.W., Nie, J.: Semidefinite representation of convex sets. Math. Program. Ser. A 122(1), 21–64 (2010)
Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals. Springer, Berlin (1993)
Jeyakumar, V., Lasserre, J.-B., Li, G.: On polynomial optimization over non-compact semi-algebraic sets. J. Optim. Theory Appl. 163, 707–718 (2014)
Jeyakumar, V., Li, G.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. Ser. A 147(1–2), 171–206 (2014)
Jeyakumar, V., Vicente-Perez, J.: Dual semidefinite programs without duality gaps for a class of convex minimax programs. J. Optim. Theory Appl. 162(3), 735–753 (2014)
Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)
Lasserre, J.B.: An Introduction to Polynomial and Semi-algebraic Optimization, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2015)
Luenberger, D.G.: A double look at duality. IEEE Trans. Autom. Control 73(10), 1474–1482 (1992)
Mordukhovich, B.S., Nam, N.M.: An easy path to convex analysis and applications. Synthesis Lectures on Mathematics and Statistics, vol. 14. Morgan & Claypool Publishers, Williston (2014)
Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. Program. Ser. A 146(1–2), 97–121 (2014)
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 203–206 (1993)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3—a MATLAB software for semidefinite–quadratic–linear programming. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html
Tuy, H., Tuan, H.D.: Generalized S-lemma and strong duality in nonconvex quadratic programming. J. Glob. Optim. 56(3), 1045–1072 (2013)
Acknowledgements
The authors are grateful to the referees for their valuable comments and suggestions, which have contributed to improving the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
T. D. Chuong’s research was supported by the UNSW Vice-Chancellor’s Postdoctoral Research Fellowship. V. Jeyakumar’s research was partially supported by a grant from the Australian Research Council.
Rights and permissions
About this article
Cite this article
Chuong, T.D., Jeyakumar, V. Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers. J Glob Optim 72, 655–678 (2018). https://doi.org/10.1007/s10898-018-0665-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-018-0665-7
Keywords
- Nonconvex polynomial programs
- Generalized Lagrangian duality
- Global optimality
- Sum of squares polynomials
- Quadratic programs
- Separable programs