Continuations - Top3d

In this tutorial, you will learn how to apply continuations strategy to derive the final solution to global minima.

Convexity is a very preferable property since every local minima is also the global minima, and what the program is solving for is the global minima. Unfortunately, the use of SIMP method to achieve binary solution will destroy the convexity of the optimization problem. For such problems, it is possible that for different starting points the program converges to totally different local minima. In order to penalize intermediate densities and mitigate the premature convergence to one of the multiple local minima when solving the non-convex problem, one could perform a continuations step. The use of SIMP method will destroy convexity of the optimization problem. As previously presented by Groenwold (2010), the continuations step is given as

$$p^k = \begin{cases} 1 & k \leqslant 20, \\ \min\{p^{\max}, 1.02p^{k-1}\} & k > 20, \\ \end{cases}$$

where $k$ is the iteration number, and $p^{\text{max}}$ is the maximum penalization power.

Though this methodology is not proven to converge to the global optimum, it regularizes the algorithm and allows the comparison of different optimization strategies.

Implementing the continuations strategy is done by adding a single line after line 68:

if loop <= 20, penal = 1; else penal = min(pmax,1.02*penal); end