Continuous optimization with a Euclidean TSP objective

I am trying to solve a problem of the form $$\min_{x_1,\dots,x_n} f(x_1,\dots,x_n)$$ subject to a constraint that $$\mathrm{length}(\mathrm{TSP}(x_1,\dots,x_n))\leq c$$, where $$x_1,\dots,x_n$$ are all points in $$\mathbb{R}^2$$, $$f$$ is a convex function, and $$\mathrm{TSP}()$$ denotes the shortest TSP tour through the points. There are obvious ways to formulate this as an IP or a QIP by introducing a binary variable $$y_{ij}$$ representing the tour and doing the usual tricks (big $$M$$ for IP, and taking products $$y_{ij}\|x_i-x_j\|$$.

I am wondering if anyone has had any experience with problems of this sort, and if there are additional tricks that could improve performance? A reference would help too as I don't know what to google to find similar results.

• You may try to solve the problem as a black-box problem, using Concord or even a TSP heuristic to solve the TSP subproblem. It wouldn't be exact since the TSP constraint is certainly non-convex, but might be enough to find good solutions in reasonable time Commented Dec 10, 2023 at 21:49
• That's a great idea! Any recommendations for a decent black-box solver? Commented Dec 11, 2023 at 2:11
• Either a derivative free solver such as Nomad, see doi.org/10.1007/s10898-021-01085-0 for other ones, or a derivative based solver such as Ipopt (open source) or Knitro (commercial) Commented Dec 11, 2023 at 8:41
• Out of curiosity, what is the objective to optimize? Commented Dec 11, 2023 at 11:34
• It's just $\sum_i \|x_i - p_i \|$, for given points $p_i$. And in fact I'm not even married to this specific version, I'd be happy to reverse the objective and constraint, or even just take a weighted combintation of the two. Commented Dec 11, 2023 at 17:42