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5

You can formulate this as an instance of the quadratic assignment problem by duplicating the workers and incurring the batching cost only for pairs of duplicate workers. Here's an alternative MIQP formulation that does not double the number of workers. Let binary decision variable $x_{i,j}$ indicate whether job $i$ is assigned to worker $j$. The problem ...

3

Here's one possible formulation, where $a_1,\dots, a_n$ are the values of the $n$ integers. Let binary decision variable $x_{i,j}$ indicate whether integer $i$ is assigned to subset $j\in\{1,\dots,k\}$. The problem is to maximize $z$ subject to \begin{align} \sum_j x_{i,j} &= 1 &&\text{for all $i$} \tag1\\ \sum_i a_i x_{i,j} &\ge z &&...

2

For a continuous function, all you need to do is prove that it's (i) non-convex, and (ii) monotonic. (i) can be shown using the eigenvalues of the hessian matrix, and (ii) using the gradient. However, in your case your domain is $\mathbb{Z}$, therefore derivatives are generally not defined, and neither is the concept of (pseudo)convexity. You can show ...

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