An accelerated rearrangement method for two-phase composite optimization

Abstract

We propose and analyze an Accelerated Rearrangement Method (ARM) for solving a class of nonconvex optimization problems involving two-phase composites. These problems include maximizing the (work) energy of a membrane governed by the Poisson equation and minimizing the principal eigenvalue of a weighted Dirichlet-Laplacian, both subject to material distribution constraints. Building on the classical rearrangement method, we introduce momentum-like acceleration by extrapolating the Fréchet derivative, leading to a provably convergent algorithm. We also introduce a restarted variant that guarantees monotonic improvement of the objective. In one dimension, we derive asymptotic convergence rates for ARM and prove that they improve upon the classical rearrangement method. Numerical experiments in both two and three dimensions confirm the accelerated convergence and demonstrate practical efficiency.

Figures (7)

• Figure 1: We consider the extremal Poisson problem \ref{['e:PoissonOpt']} on the cuboid $\Omega = [0,2]\times[0,1]\times [0,1]$, with $f_- = 1$, $f_+ = 10$, and $\delta = 0.28$. We plot the iteration $k$ vs. $J^* - J(f_k)$(top left) and $\|f_{k+1} - f_{k}\|_{L^2(\Omega)}$ (top right) for the rearrangement method (RM) \ref{['e:RM']}, accelerated RM (ARM)\ref{['e:ARM']}, and restarted ARM (RARM) \ref{['e:RARM']}. In the middle panels, we plot the initial density $f_0$(middle left) and initial solution $u_0$(middle right). The density and solution are plotted on a "diagonal slice" though the cuboid mesh. In the lower panels, we plot the final density $f^*$(bottom left) and final solution $u^*$(bottom right). See \ref{['s:CompCuboid']} for details.
• Figure 2: Convergence of $y_k$(left) and $J^* - J(y_k)$(right) for two choices of parameters and initial conditions. (top)$f_- = 1, f_+ = 100, \delta = 0.02$, and $f_0 = f_- + (f_+ -f_-) \chi_{(0.8-\delta,0.8+\delta)}$. (bottom)$f_- = 1, f_+ = 10, \delta = 0.5$, and $f_0 = f_- + (f_+ -f_-) \chi_{(0.5-\delta,0.5+\delta)}$. See \ref{['s:PropARMc']} for details.
• Figure 3: We consider \ref{['e:PoissonOpt']} on the rectangle $\Omega = [0,2]\times[0,1]$ with $f_- = 1$, $f_+ = 10$, and $\delta = 0.2$. We plot the iteration $k$ vs. $J^* - J(f_k)$(top left) and $\|f_{k+1} - f_{k}\|_{L^2(\Omega)}$ (top right) for the rearrangement method (RM) \ref{['e:RM']}, accelerated RM (ARM) \ref{['e:ARM']}, and restarted ARM (RARM) \ref{['e:RARM']}. In the middle panels, we plot the initial density $f_0$(middle left) and initial solution $u_0$(middle right). In the lower panels, we plot the final density $f^*$(bottom left) and final solution $u^*$(bottom right). See \ref{['s:CompRes2DPoisson']} for details.
• Figure 4: We consider the extremal eigenvalue problem \ref{['e:EigOpt']} on the rectangle $\Omega = [0,2]\times[0,1]$ with $f_- = 1$, $f_+ = 10$, and $\delta = 0.2$. We plot the iteration $k$ vs. $\lambda_1^* - \lambda_1(f_k)$(top left) and $\|f_{k+1} - f_{k}\|_{L^2(\Omega)}$ (top right) for the rearrangement method (RM) \ref{['e:RM']}, accelerated RM (ARM) \ref{['e:ARM']}, and restarted ARM (RARM) \ref{['e:RARM']}. In the middle panels, we plot the initial density $f_0$(middle left) and initial principal eigenfunction $u_0$(middle right). In the lower panels, we plot the final density $f^*$(bottom left) and final principal eigenfunction $u^*$(bottom right). See \ref{['s:CompRes2Deig']} for details.
• Figure 5: We consider \ref{['e:PoissonOpt']} on a punctured starfish domain with $f_- = 1$, $f_+ = 10$, and $\delta = 0.25$. We plot the iteration $k$ vs. $J^* - J(f_k)$(top left) and $\|f_{k+1} - f_{k}\|_{L^2(\Omega)}$ (top right) for the rearrangement method (RM) \ref{['e:RM']}, accelerated RM (ARM) \ref{['e:ARM']}, and restarted ARM (RARM) \ref{['e:RARM']}. In the middle panels, we plot the initial density $f_0$(middle left) and initial solution $u_0$(middle right). In the lower panels, we plot the final density $f^*$(bottom left) and final solution $u^*$(bottom right). See \ref{['s:CompStarfish']} for details.

Theorems & Definitions (10)

• Theorem 2.1: Stationarity of accumulation points for RARM
• proof
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 2.7: Geometric asymptotic convergence rate for optimal ARM \ref{['e:modifiedARM']}
• proof