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If I read your question right, then the following steps of expanding and simplifying should work: $\sum_{i=1}^{K}y_{i}\left(\frac{x_{i}}{y_{i}}-\frac{X}{Y}\right)^{2}$ $\sum_{i=1}^{K}y_{i}\left(\frac{ …

What is the complexity of and how to go about solving the following task? Given $a, b \in \mathbb{R}_+^n$ and $n \ge k\in\mathbb{N}$, find $$ x_{\min} := \arg \min_{x \in \lbrace 0,1 \rbrace^n, x^T …

Question: given: $$\begin{align}&\phantom{=}\lbrace \left(x_i,y_i\right),\ x_i\in\mathbb{X}, y_i\in \mathbb{R}\rbrace_{i=1}^n\\ &\phantom{=}f: (x\in\mathbb{X},\,p_1,\dots,\, p_k\in\mathbb{R})\mapsto …

defining as the initial set of grid points $\lbrace (x_i,y_j,z_k)\rbrace$, where the indices resemble initally given point numbers, I would suggest to do the following: repeatedly calculate the ( …

In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ …

Question: what upper bounds are known on the number of non-metric entries of finite dimensional square matrices $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with strictly positive off-diagonal elements $ …

I had posted the problem because I couldn't see how to solve it for some time but for some strange reason I found a simple answer not long after I put it on MO, so it is rather to be seen as a comment …

The distribution has a stunning similarity to the density function of random Delaunay angles mentioned on page 4 of The Expected Extremes in a Delaunay Triangulation An explanation might be, that op …

Given a combinatorial optimization problem, say the Traveling Salesman Problem, the optimal solution is a set of elements (edges in this case) that satisfy certain constraints (constituting to a conne …

If the "unit distance" corresponds to exactly two elements having exchanged positions, then the answer to this question may be in the line of what you are looking for. It contains code for determining …

Instances with long shortest tours can be obtained from the optimal solutions to packing $n$ circles of largest equal radius into a square or, can be derived from such solutions via appropriate scalin …

Regarding the 3rd question as to whether the minimum weight matching is the best choice of edges to be traversed twice, I meanwhile found a counterexample based on the following observation: Every c …

Just found the answer in the 1969 paper THE TRAVELING-SALESMAN PROBLEM AND MINIMUM SPANNING TREES by Held and Karp; in which it is shown that it is not always possible to find the otimal solution to t …

I just realized that there is a trivial reduction to the TSP problem by deleting all diagonals of every cycle, i.e. all edges of set $D$ in the question. The optimal tour in the thus modified graph is …

Question: given a complete symmetric graph $G(V,E)$ with $n$ vertices and edges $e_{ij}$ having weight $\omega_{ij}$, does there always exists a vector of vertex potentials $(\pi_1,\,\dots,\,\pi_n)$ t …