There is a known problem of touring a sequence of $n$ polygons: given a starting point $s$, an ending point $t$ and a sequence of polygons $P_1,\dots,P_k$ with a total of $n$ vertices, find points $q_1 \in P_1,\dots,q_k \in P_k$ such that the length of the path $sq_1\dots q_k t$ is minimal, that is, $|sq_1|+|q_1q_2|+\dots+|q_k t|$ is minimal, where $|\cdot|$ is the Euclidean ($L_2$) length of a segment.
If $P_i$ are convex (not necessarily disjoint), then there is an $O(kn^2 \log(n))$ algorithm, which finds the exact solution for this problem, see https://doi.org/10.1145/780542.780612.
Are there any known polynomial algorithms for minimising the sum of the squared lengths of the path segments (which can also be called the “energy” of the path), that is, the sum $|sq_1|^2+|q_1q_2|^2+\dots+|q_k t|^2$ with $q_i \in P_i$ for each $i$, in the case when $P_1,\dots,P_k$ are convex (not necessarily disjoint)?
