Abstract
In this paper we present an inexact scalarization proximal point algorithm to solve unconstrained multiobjective minimization problems where the objective functions are quasiconvex and locally Lipschitz. Under some natural assumptions on the problem, we prove that the sequence generated by the algorithm is well defined and converges. Then providing two error criteria we obtain two versions of the algorithm and it is proved that each sequence converges to a Pareto–Clarke critical point of the problem; furthermore, it is also proved that assuming an extra condition, the convergence rate of one of these versions is linear when the regularization parameters are bounded and superlinear when these parameters converge to zero.
Similar content being viewed by others
References
Attouch, H., Bolte, J., & Svaiter, B. (2013). Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods. Mathematical Programing Series A, 137, 91–129.
Apolinario, H., Papa Quiroz, E. A., & Oliveira, P. R. (2016). A scalarization proximal point method for quasiconvex multiobjective minimization. Journal of Global Optimization, 64, 79–96.
Auslender, A., & Teboulle, M. (2006). Interior gradient and proximal methods for convex and conic optimization. SIAM Journal of Optimization, 16(3), 697–725.
Auslender, A., Teboulle, M., & Ben-Tiba, S. (1999). Interior proximal and multiplier methods based on second order homogeneous functional. Mathematics of Operations Research, 24(3), 645–668.
Auslender, A., Teboulle, M., & Ben-Tiba, S. (1999b). A logarithmic–quadratic proximal method for variational inequalities. Computational Optimization and Applications, 12, 31–40.
Aussel, D. (1998). Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach. Journal of Optimization Theory and Applications, 97(1), 29–45.
Bento, G., Ferreira, O., & Oliveira, P. R. (2010). Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds. Nonlinear Analysis: Theory, Methods and Applications, 72(3), 564–572.
Baygorrea, N., Papa Quiroz, E. A., & Maculan, N. (2017). On the convergence rate of an inexact proximal point algorithm for quasiconvex minimization on Hadamard manifolds. Journal of the Operations Research Society of China, 5(4), 457–467.
Baygorrea, N., Papa Quiroz, E. A., & Maculan, N. (2016). Inexact proximal point methods for quasiconvex minimization on Hadamard manifolds. Journal of the Operations Research Society of China, 4(4), 397–424.
Berkovitz, L. (2003). Convexity and optimization in\( {\mathbb{R}}^n\). New York: Wiley.
Bolte, J., Daniilidis, A., Lewis, A., & Shiota, M. (2007). Clarke subgradients of stratifiable functions. SIAM Journal on Optimization, 18, 556–572.
Bonnel, H., Iusem, A. N., & Svaiter, B. F. (2005). Proximal methods in vector optimization. SIAM Journal on Optimization, 15, 953–970.
Brito, A. S., da Cruz Neto, J. X., Lopez, J. O., & Oliveira, P. R. (2012). Interior proximal algorithm for quasiconvex programming and variational inequalities with linear constraints. Journal of Optimization Theory and Applications, 154, 217–234.
Burachik, R. S., & Iusem, A. (1998). Generalized proximal point algorithm for the variational inequality problem in Hilbert space. SIAM Journal of Optimization, 8, 197–216.
Burachik, R. S., & Scheimberg, S. (2000). A proximal point method for the variational inequality problem in Banach spaces. SIAM Journal on Control and Optimization, 39(5), 1633–1649.
Ceng, L., & Yao, J. (2007). Approximate proximal methods in vector optimization. European Journal of Operational Research, 183(1), 1–19.
Chen, J. S., & Pan, S. (2008). A proximal-like algorithm for a class of nonconvex programming. Pacific Journal of Optimization, 4, 319–333.
Chen, G., & Teboulle, M. (1993). Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM Journal on Optimization, 3, 538–543.
Clarke, F. (1990). Optimization and nonsmooth analysis. Philadelphia: SIAM.
Clarke, F. (2013). Functional analysis, calculus of variations and optimal control. New York: Springer.
Cunha, F. G. M., da Cruz Neto, J. X., & Oliveira, P. R. (2010). A proximal point algorithm with \(\phi -\)divergence to quasiconvex programming. Optimization, 59, 777–792.
Custodio, A. L., Madeira, J. F. A., Vaz, A. I. F., & Vicente, L. N. (2011). Direct Multisearch for multiobjective optimization. SIAM Journal on Optimization, 21, 1109–1140.
Eckstein, J. (1998). Approximate iterations in Bregman-function-based proximal algorithms. Mathematical Programming, 83, 113–123.
Ehrgott, M. (2005). Multicriteria optimization. New York: Springer.
Güler, O. (1992). New proximal point proximal algorithms for convex minimization. SIAM Journal Control and Optimization., 2, 649–664.
Goudou, X., & Munier, J. (2009). The gradient and heavy ball with friction dynamical system: The quasiconvex case. Mathematical Programming, Serie B, 116, 173–191.
Gromicho, J. (1998). Quasiconvex optimization and location theory. Dordrecht: Kluwer Academic Publishers.
Han, D., & He, B. (2001). A new accuracy criterion for approximate proximal point algorithms. Journal of Mathematical Analysis and Applications, 263, 343–354.
Hadjisavvas, N., Komlosi, S., & Shaible, S. (2005). Handbook of generalized convexity and generalized monotonicity (Vol. 76)., Nonconvex optimization and its applications New York: Springer.
Humes, C, Jr., & Silva, P. J. S. (2005). Inexact proximal algorithms and descent methods in optimization. Optimization and Engineering, 6, 257–271.
Iusem, A. N., & Sosa, W. (2010). A proximal point method for equilibrium problems in Hilbert spaces. Optimization, 59, 1259–1274.
Kaplan, A., & Tichatschke, R. (1998). Proximal point methods and nonconvex optimization. Journal of Global Optimization, 13, 389–406.
Kaplan, A., & Tichatschke, R. (2004). On inexact generalized proximal methods with a weakened error tolerance criterion. Optimization, 53, 3–17.
Kiwiel, K. C. (1997). Proximal minimization methods with generalized Bregman functions. SIAM Journal of Control and Optimization, 35, 1142–11268.
Langenberg, N. (2010). Pseudomonotone operators and the Bregman proximal point algorithm. Journal of Global Optimization, 47, 537–555.
Langenberg, N., & Tichatschke, R. (2012). Interior proximal methods for quasiconvex optimization. Journal of Global Optimization, 52, 641–661.
Martinet, B. (1970). Regularisation d’inéquations variationelles par approximations successives. Revue Frana̧ise Informatique Recherche Opérationnelle, 4, 154–158.
Mallma Ramirez, L., Papa Quiroz, E. A., & Oliveira, P. R. (2017). An inexact proximal method with proximal distances for quasimonotone equilibrium problems. Journal of the Operations Research Society of China, 5(4), 545–561.
Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic theory. New York, NY: Oxford University Press.
Miettinen, K. (2012). Nonlinear multiobjective optimization. New York: Springer.
Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation I: Basic theory (Vol. 330). New York: Springer Science and Business Media.
Pan, S., & Chen, J. S. (2007). Entropy-like proximal algorithms based on a second-order homogeneous distances function for quasiconvex programming. Journal of Global Optimization, 39, 555–575.
Papa Quiroz, E. A., & Oliveira, P. R. (2012). An extension of proximal methods for quasiconvex minimization on the nonnegative orthant. European Journal of Operational Research, 216, 26–32.
Papa Quiroz, E. A., Mallma Ramirez, L., & Oliveira, P. R. (2015). An inexact proximal method for quasiconvex minimization. European Journal of Operational Research, 246(3), 721–729.
Papa Quiroz, E. A., Mallma Ramirez, L., & Oliveira, P. R. (2018). An inexact algorithm with proximal distances for variational inequalities. RAIRO Operations Research, 52(1), 159–176.
Pennanen, T. (2002). Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Mathematics of Operations Research, 27, 170–191.
Polyak, B. T. (1987). Introduction to optimization. New York: Optimization Software.
Rockafellar, R. T. (1976). Monotone operations and the proximal point method. SIAM Journal on Control and Optimization, 14, 877–898.
Rockafellar, R. T., & Wets, R. (1998). Variational analysis. Grundlehren der Mathematischen, Wissenschaften (Vol. 317). New York: Springer.
Sawaragi, Y., Nakayama, H., & Tanino, T. (1985). Theory of multiobjetive optimization (Vol. 176)., Mathematics in science and engineering Orlando: Academic Press Inc.
Souza, S. S., Oliveira, P. R., da Cruz Neto, J. X., & Soubeyran, A. (2010). A proximal method with separable Bregman distance for quasiconvex minimization on the nonnegative orthant. European Journal of Operational Research, 201, 365–376.
Tang, G. J., & Huang, N. J. (2013). An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds. Operations Research Letters, 41(6), 586–591.
Van Tiel, J. (1984). Convex analysis: An introductory text, Chichester, UK. Hoboken: Wiley.
Villacorta, K. D., & Oliveira, P. R. (2011). An interior proximal method in vector optimization. European Journal of Operational Research, 214, 485–492.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Papa Quiroz, E.A., Cruzado, S. An inexact scalarization proximal point method for multiobjective quasiconvex minimization. Ann Oper Res 316, 1445–1470 (2022). https://doi.org/10.1007/s10479-020-03622-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-020-03622-8
Keywords
- Proximal point method
- Quasiconvex functions
- Multiobjective minimization
- Pareto optimality
- Clarke subdifferential
- Global convergence
- Convergence rate