FW4PDE: Frank--Wolfe algorithms for PDE-constrained optimization

The package implements conditional gradient methods for the solution of the PDE-constrained optimization problems

$$ \min_{u \in U_{\text{ad}}} J(S(u)) + \beta \|u\|_{L^1(D)}, $$

where $\beta \geq 0$, $S(u)$ is the solution to a potentially nonlinear PDE, and $U_{\text{ad}} = \{ u \in L^2(D) : a \leq u \leq b \}$. Here $a$, $b \in L^2(D)$ with $a \leq b$.

Examples

Examples of convex PDE-constrained problems can be found in convex and of potentially nonconvex ones in nonconvex.

The implementation can be used to optimally design renewable tidal-stream energy farms.

Installation

conda env create -f environment.yml
conda activate FW4PDE

Dependencies

The following packages are required:

• FEniCS
• dolfin-adjoint
• moola

See environment.yml for a complete list of dependencies.

References

• M. Besançon, A. Carderera, S. Pokutta (2022) FrankWolfe.jl: A High-Performance and Flexible Toolbox for Frank–Wolfe Algorithms and Conditional Gradients. INFORMS Journal on Computing 0(0).

• Dunn, J.C.: Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals. SIAM J. Control Optim. 17(2), 187–211 (1979)

• Dunn, J.C.: Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J. Control Optim. 18(5), 473–487 (1980)

• J.C. Dunn, S. Harshbarger, Conditional gradient algorithms with open loop step size rules, Journal of Mathematical Analysis and Applications, Volume 62, Issue 2, February 1978, Pages 432-444

• Harchaoui, Z., Juditsky, A. and Nemirovski, A. Conditional gradient algorithms for norm-regularized smooth convex optimization. Math. Program. 152, 75–112 (2015).

• K. Kunisch, D. Walter, On fast convergence rates for generalized conditional gradient methods with backtracking stepsize, preprint, https://arxiv.org/abs/2109.15217, 2021

A complete list of references is provided in lib.md.