Let's break down the formulation into 2 clear steps: --- **Step 1: Define the Breakpoints and Variables** Let \( m \) be the number of intervals (with \( m \geq 3 \)), and let the breakpoints be \[ x_i = \frac{2\pi i}{m}, \quad \text{for } i = 0, 1, ..., m \] with corresponding function values \[ y_i = \sin(x_i), \quad \text{for } i = 0, 1, ..., m \] For any \( x \) in \([0, 2\pi]\), we want to express \( y \approx \sin(x) \) as a convex combination between two neighboring breakpoints. Introduce binary variables \( z_i \) to indicate which interval \( x \) lies in: \[ z_i = \begin{cases} 1 & \text{if } x \in [x_i, x_{i+1}] \\ 0 & \text{otherwise} \end{cases} \] for \( i = 0, 1, ..., m-1 \), with the constraint: \[ \sum_{i=0}^{m-1} z_i = 1 \] **(Explanation: This step divides the interval into \( m \) pieces and introduces binary variables to model which segment is active for the given \( x \).)** --- **Step 2: Piecewise Linear Constraints** For \( x \) in interval \([x_i, x_{i+1}]\), define a local variable \( \lambda \in [0, 1] \) so that: \[ x = \sum_{i=0}^{m-1} z_i \left[ (1-\lambda) x_i + \lambda x_{i+1} \right] \] \[ y = \sum_{i=0}^{m-1} z_i \left[ (1-\lambda) y_i + \lambda y_{i+1} \right] \] But so that only one interval is active (where \( z_i = 1 \)), we can write: \[ x = \sum_{i=0}^{m-1} z_i \left[ (1-\lambda_i) x_i + \lambda_i x_{i+1} \right] \] \[ y = \sum_{i=0}^{m-1} z_i \left[ (1-\lambda_i) y_i + \lambda_i y_{i+1} \right] \] where for each \( i \), \( 0 \leq \lambda_i \leq 1 \) and \( \sum_{i=0}^{m-1} z_i = 1 \). Add the constraints: - \( 0 \leq \lambda_i \leq 1 \) for all \( i \) - \( \lambda_i = 0 \) if \( z_i = 0 \) - \( \lambda_i \) is free in \([0,1]\) if \( z_i = 1 \) This can be enforced using "big-M" constraints, e.g. \( \lambda_i \leq z_i \). **(Explanation: This step encodes the piecewise linear interpolation using binary selection variables and ensures that \( x \) and \( y \) are on the appropriate segment.)** --- ### **Final Mixed Integer Program Constraints** For arbitrary \( m \), the formulation is: \[ \begin{align*} & x = \sum_{i=0}^{m-1} z_i \left[(1-\lambda_i)x_i + \lambda_i x_{i+1}\right] \\ & y = \sum_{i=0}^{m-1} z_i \left[(1-\lambda_i)y_i + \lambda_i y_{i+1}\right] \\ & \sum_{i=0}^{m-1} z_i = 1 \\ & 0 \leq \lambda_i \leq 1 \quad \forall i \\ & 0 \leq z_i \leq 1 \quad \forall i, \quad z_i \in \{0,1\} & \lambda_i \leq z_i \quad \forall i \end{align*} \] where \( x_i = \frac{2\pi i}{m} \), \( y_i = \sin(x_i) \). --- **This piecewise linear mixed-integer program ensures that \( y \) approximates \(\sin(x)\) using \( m \) line segments over \([0, 2\pi]\).**