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Assuming binary decision variable $x_{ijd}$ indicates whether teams $i$ and $j$ play each other on day $d$, introduce a decision variable $z$, and maximize $z$ subject to linear constraints $$z \le \f …

Your example shows that your objective function is instead $$\sum_x \min_{i: b_i = 1} f_i(x)$$ You can linearize the problem as follows: \begin{align} &\text{minimize} &\sum_x \sum_i c_{i,x} f_i(x) \\ …

You can formulate this as a set covering problem. For each circle $j$, define a binary variable $x_j$ that indicates whether circle $j$ is selected. Let $C_i$ be the set of circles that intersect li …

Yes, the linearization of a product of binary variables is well known: https://or.stackexchange.com/questions/37/how-to-linearize-the-product-of-two-binary-variables

For binary $P$, we have $\min\{P_{k,i},P_{l,j}\} = P_{k,i} P_{l,j}$. In your linearization, you have introduced $r_{i,k,l,j}$ to represent this product. Because of the linear constraints $$\sum_k P_ …

You can solve the problem via binary quadratic programming as follows. Let binary decision variable $x_{id}$ indicate whether row $i$ is rotated $d$ places. The problem is to maximize $$\sum_{j=0}^{ …

I will simplify the notation to illustrate the idea. You want to minimize $$\sigma \max\left(\sum_{i,j} d_{ij} x_i x_j - \alpha, 0\right).$$ Introduce binary decision variable $y_{ij}$ to represent t …

This is a variant of the integer equality knapsack problem and can be solved via dynamic programming. The complexity is described here. For a DP recursion, first define $$\Psi_k(x):=\min_{\begin{arra …

Let $L_i$ be a constant lower bound on $\mathbf{w}^T \mathbf{x}_i$. You can linearize the problem by introducing binary decision variable $y_i\in\{0,1\}$ to indicate whether $\mathbf{w}^T \mathbf{x}_i …

By introducing a dummy depot node, you can think of this as a special case of the time-dependent traveling salesman problem. The dummy node is adjacent to all other nodes, with zero-cost links. You mi …