A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces

Abstract

The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.

Figures (5)

• Figure 1: A Pareto optimal control computed with Algorithm \ref{['algo:nonsmooth_descent_method']} for mesh size $h_{\max} = 0.02$, initial control $u_0 \equiv 8$ and the constant obstacle $\psi \equiv 1$.
• Figure 2: Qualitative analysis of the solutions derived by Algorithm \ref{['algo:nonsmooth_descent_method']} for different discretizations for the constant obstacle. Subfigures (b) - (d) use the reference solution $u_{\mathrm{ref}}^*$ corresponding to mesh size $h_{\max} = 0.01$.
• Figure 3: A Pareto optimal control computed with Algorithm \ref{['algo:nonsmooth_descent_method']} for mesh size $h_{\max} = 0.02$, initial control $u_0 \equiv 8$ and the piecewise constant obstalce $\psi$ defined in \ref{['eq:piecewise_constant_obstacle']}.
• Figure 4: Qualitative analysis of the solutions derived by Algorithm \ref{['algo:nonsmooth_descent_method']} for different discretizations for the nonconstant obstacle. Subfigures (b)-(d) use the reference solution $u_{\mathrm{ref}}^*$ corresponding to mesh size $h_{\max} = 0.01$.
• Figure 5: Size of the approximated subdifferential for each iteration. Results obtained by Algorithm \ref{['algo:nonsmooth_descent_method']} for the piecewise constant obstacle with mesh size $h_{\max} = 0.02$ and initial control $u_0 \equiv 8$.

Theorems & Definitions (47)

• Definition 2.1: Miettinen1998
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9
• proof