A Projected Subgradient Method for Nondifferentiable Quasiconvex Multiobjective Optimization Problems

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.

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:

• 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 $E = {[}0, \infty {)}$.

• 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. $E = \lbrace -1 \rbrace$).

• 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 $E = E_{\mathrm {w}} = \lbrace 0 \rbrace$.