Taiwanese Journal of Mathematics

Multi-objective Optimization Problems with SOS-convex Polynomials over an LMI Constraint

Liguo Jiao, Jae Hyoung Lee, Yuto Ogata, and Tamaki Tanaka

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

In this paper, we aim to find efficient solutions of a multi-objective optimization problem over a linear matrix inequality (LMI in short), in which the objective functions are SOS-convex polynomials. We do this by using two scalarization approaches, that is, the $\epsilon$-constraint method and the hybrid method. More precisely, we first transform the considered multi-objective optimization problem into their scalar forms by the $\epsilon$-constraint method and the hybrid method, respectively. Then, strong duality results, between each formulated scalar problem and its associated semidefinite programming dual problem, are given, respectively. Moreover, for each proposed scalar problem, we show that its optimal solution can be found by solving an associated single semidefinite programming problem, under a suitable regularity condition. As a consequence, we prove that finding efficient solutions to the considered problem can be done by employing any of the two scalarization approaches. Besides, we illustrate our methods through some nontrivial numerical examples.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 23 pages.

Dates
First available in Project Euclid: 30 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1572422530

Digital Object Identifier
doi:10.11650/tjm/191002

Subjects
Primary: 90C29: Multi-objective and goal programming 65K05: Mathematical programming methods [See also 90Cxx] 52A41: Convex functions and convex programs [See also 26B25, 90C25]

Keywords
multi-objective optimization semidefinite programming SOS-convex polynomials linear matrix inequality

Citation

Jiao, Liguo; Lee, Jae Hyoung; Ogata, Yuto; Tanaka, Tamaki. Multi-objective Optimization Problems with SOS-convex Polynomials over an LMI Constraint. Taiwanese J. Math., advance publication, 30 October 2019. doi:10.11650/tjm/191002. https://projecteuclid.org/euclid.twjm/1572422530


Export citation

References

  • A. A. Ahmadi and P. A. Parrilo, A convex polynomial that is not sos-convex, Math. Program. 135 (2012), no. 1-2, 275–292.
  • V. Chankong and Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology, North-Holland Series in System Science and Engineering 8, North-Holland, Amsterdam, 1983.
  • M. Ehrgott, Multicriteria Optimization, Second edition, Springer-Verlag, Berlin, 2005.
  • F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities: Image space analysis and separation, in: Vector Variational Inequalities and Vector Equilibria, 153–215, Nonconvex Optim. Appl. 38, Kluwer, Dordrecht, 2000.
  • M. C. Grant and S. P. Boyd, The CVX user's guide, release 2.0., User manual (2013). http://cvxr.com/cvx/doc/CVX. Accessed 7 April 2019.
  • J. W. Helton and J. Nie, Semidefinite representation of convex sets, Math. Program. 122 (2010), no. 1, 21–64.
  • J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Second edition, Springer-Verlag, Berlin, 2011.
  • V. Jeyakumar, A. M. Rubinov, B. M. Glover and Y. Ishizuka, Inequality systems and global optimization, J. Math. Anal. Appl. 202 (1996), no. 3, 900–919.
  • V. Jeyakumar, G. M. Lee and N. Dinh, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM J. Optim. 14 (2003), no. 2, 534–547.
  • ––––, Characterizations of solution sets of convex vector minimization problems, European J. Oper. Res. 174 (2006), no. 3, 1380–1395.
  • V. Jeyakumar and G. Li, Exact SDP relaxations for classes of nonlinear semidefinite programming problems, Oper. Res. Lett. 40 (2012), no. 6, 529–536.
  • ––––, A new class of alternative theorems for SOS-convex inequalities and robust optimization, Appl. Anal. 94 (2015), no. 1, 56–74.
  • V. Jeyakumar and M. J. Nealon, Complete dual characterizations of optimality for convex semidefinite programming, in: Constructive, Experimental, and Nonlinear Analysis, (Limoges, 1999), 165–173, CRC Math. Model. Ser. 27, CRC, Boca Raton, FL, 2000.
  • V. Jeyakumar, T. S. Ph\dam and G. Li, Convergence of the Lasserre hierarchy of SDP relaxations for convex polynomial programs without compactness, Oper. Res. Lett. 42 (2014), no. 1, 34–40.
  • L. Jiao and J. H. Lee, Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data, accepted in Annals of Operations Research, (2019).
  • L. Jiao, J. H. Lee and N. Sisarat, Multi-objective convex polynomial optimization and semidefinite programming relaxations, arXiv:1903.10137.
  • J. B. Lasserre, Convexity in semialgebraic geometry and polynomial optimization, SIAM J. Optim. 19 (2009), no. 4, 1995–2014.
  • ––––, An Introduction to Polynomial and Semi-algebraic Optimization, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2015.
  • J. H. Lee and L. Jiao, Solving fractional multicriteria optimization problems with sum of squares convex polynomial data, J. Optim. Theory Appl. 176 (2018), no. 2, 428–455.
  • ––––, Finding efficient solutions for multicriteria optimization problems with sos-convex polynomials, accepted in Taiwanese Journal of Mathematics, (2019).
  • C. Li, K. F. Ng and T. K. Pong, Constraint qualifications for convex inequality systems with applications in constrained optimization, SIAM. J. Optim. 19 (2008), no. 1, 163–187.
  • B. Reznick, Extremal PSD forms with few terms, Duke Math. J. 45 (1978), no. 2, 363–374.
  • L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38 (1996), no. 1, 49–95.