Abstract

In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. We define a linesearch-based solution method, and we show that it converges to a set of Pareto stationary points. To this aim, we carry out a theoretical analysis of the problem by only assuming Lipschitz continuity of the functions; more specifically, we give new optimality conditions that take explicitly into account the bound constraints, and prove that the original problem is equivalent to a bound constrained problem obtained by penalizing the nonlinear constraints with an exact merit function. Finally, we present the results of some numerical experiments on bound constrained and nonlinearly constrained problems, showing that our approach is promising when compared to a state-of-the-art method from the literature.

Keywords

  1. derivative-free multiobjective optimization
  2. Lipschitz optimization
  3. inequality constraints
  4. exact penalty functions

MSC codes

  1. 90C30
  2. 90C56
  3. 65K05
  4. 49J52

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
C. Audet, G. Savard, and W. Zghal, Multiobjective optimization through a series of single-objective formulations, SIAM J. Optim., 19 (2008), pp. 188--210.
2.
S. Bandyopadhyay, S. K. Pal, and B. Aruna, Multiobjective GAs, quantitative indices, and pattern classification, IEEE Trans. Syst. Man Cybernet. Part B Cybernet. 34 (2004), pp. 2088--2099.
3.
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
4.
A. Conn, K. Scheinberg, and L. N. Vicente, Introduction to Derivative-Free Optimization, MPS-SIAM Ser. Optim. 8, SIAM, Philadelphia, 2009.
5.
A. L. Custódio, M. Emmerich, and J. F. A. Madeira, Recent developments in derivative-free multiobjective optimization, Comput. Technol. Rev., 5 (2012), pp. 1--30.
6.
A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente, Errata to Direct Multisearch for Multiobjective Optimization, http://www.mat.uc.pt/~lnv/papers/errata-dms.pdf.
7.
A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente, Direct multisearch for multiobjective optimization, SIAM J. Optim., 21 (2011), pp. 1109--1140.
8.
K. Deb, A. Pratap, S. Agarwal, and T. A. M. T. Meyarivan, A fast and elitist multiobjective genetic algorithm: IEEE Trans. Evol. Comput., 6 (2002), pp. 182--197.
9.
E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), pp. 201--213.
10.
G. Fasano, G. Liuzzi, S. Lucidi, and F. Rinaldi, A linesearch-based derivative-free approach for nonsmooth constrained optimization, SIAM J. Optim., 24 (2014), pp. 959--992.
11.
E. H. Fukuda, L. M. Gran͂a Drummond, and F. M. P. Raupp, An external penalty-type method for multicriteria, TOP, 24 (2016), pp. 493--513.
12.
T. Glad and E. Polak, A multiplier method with automatic limitation of penalty growth, Math. Program., 17 (1979), pp. 140--155.
13.
J. B. Hiriart-Urruty, On optimality conditions in nondifferentiable programming, Math. Program., 14 (1978), pp. 73--86.
14.
X. X. Huang and X. Q. Yang, Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization, SIAM J. Optim., 13 (2002), pp. 675--692.
15.
Y. Ishizuka and K. Shimizu, Necessary and sufficient conditions for the efficient solutions of nondifferentiable multiobjective problems, IEEE Trans. Syst. Man Cybernet., SMC-14 (1984), pp. 624--629.
16.
J. Jahn, Introduction to the Theory of Nonlinear Optimization, Springer, Berlin, 1996.
17.
J. Jahn, Vector Optimization, Springer, Berlin, 2009.
18.
N. Karmitsa, Test Problems for Large-Scale Nonsmooth Minimization, Technical report No., B. 4/2007, Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland, 2007.
19.
C.-J. Lin, S. Lucidi, L. Palagi, A. Risi, and M. Sciandrone, Decomposition algorithm model for singly linearly-constrained problems subject to lower and upper bounds, J. Optim. Theory Appl., 141 (2009), pp. 107--126.
20.
G. Liuzzi, S. Lucidi, F. Parasiliti, and M. Villani, Multiobjective optimization techniques for the design of induction motors, IEEE Trans. Magnet., 39 (2003), pp. 1261--1264.
21.
S. Lucidi, New results on a continuously differentiable exact penalty function, SIAM J. Optim., 2 (1992), pp. 558--574.
22.
O. L. Mangasarian, Nonlinear Programming, Classics Appl. Math., SIAM, Philadelphia, 1994.
23.
K. Shimizu, Y. Ishizuka, and J. F. Bard, Nondifferentiable and Two-level Mathematical Programming, Kluwer, Norwell, MA, 1997.
24.
A. Suppapitnarm, K. A. Seffen, G. T. Parks, and P. J. Clarkson, A simulated annealing algorithm for multiobjective optimization, Eng. Optim., 33 (2000), pp. 59--85.
25.
E. Zitzler and L. Thiele, Multiobjective Optimization using Evolutionary Algorithms --- A Comparative Case Study, in Parallel Problem Solving from Nature --- PPSN V: 5th International Conference Amsterdam, The Netherlands, A. E. Eiben, T. Bäck, M. Schoenauer, and H.-P. Schwefel, eds., Springer, Berlin 1998, pp. 292--301.

Information & Authors

Information

Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 2744 - 2774
ISSN (online): 1095-7189

History

Submitted: 1 September 2015
Accepted: 29 August 2016
Published online: 8 December 2016

Keywords

  1. derivative-free multiobjective optimization
  2. Lipschitz optimization
  3. inequality constraints
  4. exact penalty functions

MSC codes

  1. 90C30
  2. 90C56
  3. 65K05
  4. 49J52

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media