Abstract
In this paper, we propose a projected subgradient method for solving constrained nondifferentiable quasiconvex multiobjective optimization problems. The algorithm is based on the Plastria subdifferential to overcome potential shortcomings known from algorithms based on the classical gradient. Under suitable, yet rather general assumptions, we establish the convergence of the full sequence generated by the algorithm to a Pareto efficient solution of the problem. Numerical results are presented to illustrate our findings.
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Notes
Motivating the algorithm and its parameters using this example was prompted to us by one of the anonymous referees.
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Acknowledgements
The authors are grateful to the handling editor and the anonymous referees for their valuable remarks, comments and new references, which helped to improve the original presentation. The research of Xiaopeng Zhao was supported in part by the National Natural Science Foundation of China (Grant Number 11801411) and the Natural Science Foundation of Tianjin (Grant Number 18JCQNJC01100). The work of M. Köbis was carried out during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme of the author at the Norwegian University of Science and Technology. The research of Yonghong Yao was partially supported in part by University Innovation Team of Tianjin [Grant Number TD13-5033]. The research of Jen-Chih Yao was supported in part by the Grant MOST 108-2115-M-039-005-MY3.
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Appendix: Analytically Finding Pareto Optimal Solutions in the Small Test Cases
Appendix: Analytically Finding Pareto Optimal Solutions in the Small Test Cases
The (weak) Pareto optimal solutions in Examples 5.1–5.3 can be obtained by applying the techniques from [14, 15]. We partially adapt the notation of these two references to outline the analysis below:
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Example 5.1 Here, we have \(f_1 = x^2\), \(f_2 = \exp (-x)\) and no constraints (i.e., \(C = \mathbb {R}\)). In this example, we obtain \(\mathop {{\text {argmin}}}_{x \in C} f_1 = \lbrace 0 \rbrace = [0,0]\mathbin {=:}[\alpha ,\beta ]\) (bounded) and \(\mathop {{\text {argmin}}}_{x \in C} f_2 = \emptyset \) (empty), corresponding to ‘Case 2’ in [14, p. 95]. Then, the inclusion \({[}0,\infty {)} \subseteq E_{\mathrm {w}}\) follows from [14, Theorem 4.3(b)]. On the other hand, note that
$$\begin{aligned} B_- {:=} \lbrace x \in C: x \ge \beta ,\, f_2(x) = f_2(\beta )\rbrace = [ 0, 0] \mathrel {=:}[\beta , \beta ^-] \end{aligned}$$and that \(\lbrace x \in C: x < 0,\, \exp (-x) = \exp (0) = 1\rbrace = \emptyset \). So, we obtain from the second case in [14, Theorem 4.3(b)] that \(E_{\mathrm {w}} \cap \lbrace x \in C: x < 0 \rbrace = \emptyset \). Consequently, \(E_{\mathrm {w}} = {[} 0, \infty {)}\).
Furthermore, note that \(f_2\) is strictly decreasing. From [15, Corollary 5.5(b)], it can be obtained immediately that \(E = {[}0, \infty {)}\).
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Example 5.2 For \(f_1 = \tfrac{x^5}{5}\), \(f_2 = \tfrac{x^3}{3}\), \(C = [-1,1]\), we get
$$\begin{aligned} \mathop {{\text {argmin}}}_{x \in C} f_1 \cap \mathop {{\text {argmin}}}_{x \in C} f_2 = \lbrace -1 \rbrace \ne \emptyset \,. \end{aligned}$$From [14, Proposition 3.1] (resp. [15, Proposition 2.1]), we therefore conclude directly that \(E_{\mathrm {w}} = \lbrace -1 \rbrace \) (resp. \(E = \lbrace -1 \rbrace \)).
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Example 5.3 For a single objective, we always have the set of (classically) optimal solutions to this one dimension coinciding with E (if existent). From that (or formally also from [14, Proposition 3.1] and [15, Proposition 2.1]), we get \(E = E_{\mathrm {w}} = \lbrace 0 \rbrace \).
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Zhao, X., Köbis, M.A., Yao, Y. et al. A Projected Subgradient Method for Nondifferentiable Quasiconvex Multiobjective Optimization Problems. J Optim Theory Appl 190, 82–107 (2021). https://doi.org/10.1007/s10957-021-01872-5
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DOI: https://doi.org/10.1007/s10957-021-01872-5
Keywords
- Multiobjective optimization
- Pareto optimality
- Quasiconvex functions
- Projected subgradient method
- Quasi-Fejér convergence