A dimension reduction procedure for the selection of Two-spring lattice-spring topologies with minimal fabrication cost and required weighted force-resistance performance

Starting from a problem in elastoplasticity, we consider an optimization problem $C(c_1,c_2)=c_1+c_2\to \min$ under constraints $F_R^k(c_1,c_2)=a\cdot F^k(c_1,c_2)+b\cdot R^k(c_1,c_2)\ge 1$ and $F^k(c_1,c_2)\ge 1$, where both $F^k$ and $R^k$ non-linear, $a,b$ are constants, and $i\in \{1,2\}$ is an index. For each $(a,b)$ we determine which of the two values of $i\in \{1,2\}$ leads to the smaller minimum of the optimization problem. This way we obtain an interesting curve bounding the region where $k=1$ outperforms $k=2$.