Let's break the solution into **2 steps** with explanations and then provide the final answer. --- ### **Step 1: Define the Breakpoints and Linear Segments** Let’s define the breakpoints for the piecewise-linear approximation: - Divide \( [0, 2\pi] \) into \( m \) equal intervals. - The breakpoints are: \( x_k = \frac{2\pi k}{m} \) for \( k = 0, 1, ..., m \). - The corresponding function values are: \( y_k = \sin(x_k) \). **Explanation:** We sample the sine function at \( m+1 \) equally spaced points. Each interval \( [x_k, x_{k+1}] \) will be approximated by a straight line connecting \( (x_k, y_k) \) and \( (x_{k+1}, y_{k+1}) \). --- ### **Step 2: Formulate the Piecewise-Linear Function Using Mixed Integer Constraints** Introduce binary variables \( z_k \) for \( k = 0, 1, ..., m-1 \) to indicate which interval \( x \) belongs to. Only one \( z_k \) is active (equals 1) at any time: - \( \sum_{k=0}^{m-1} z_k = 1 \), \( z_k \in \{0, 1\} \) Let \( x \) be in \( [x_k, x_{k+1}] \) if \( z_k = 1 \). The linear approximation in this interval is: \[ y = y_k + \frac{y_{k+1} - y_k}{x_{k+1} - x_k} (x - x_k) \] To enforce this, use the following constraints: \[ x = \sum_{k=0}^{m-1} z_k \left( (x_k + (x_{k+1} - x_k) \lambda) \right) \] \[ y = \sum_{k=0}^{m-1} z_k \left( y_k + (y_{k+1} - y_k) \lambda \right) \] where \( \lambda \in [0, 1] \) for the active interval (so \( x = x_k + \lambda(x_{k+1} - x_k) \) when \( z_k = 1 \)), and \( \lambda \) is forced to be 0 otherwise. Alternatively, **enforce bounds** using big-M constraints: \[ x_k z_k \leq x \leq x_{k+1} z_k + (1-z_k)M \quad \forall k \] where \( M \) is a large constant. **Explanation:** The binary variables select which linear segment to use, and the constraints ensure \( x \) and \( y \) are linked according to the segment equation for the selected interval. --- ## **Final Answer** **The piecewise-linear approximation of \( y = \sin(x) \) over \( [0, 2\pi] \) with \( m \) intervals can be formulated in a mixed integer program as:** 1. **Breakpoints:** \( x_k = \frac{2\pi k}{m} \), \( y_k = \sin(x_k) \) for \( k = 0, ..., m \). 2. **Binary variables:** \( z_k \in \{0, 1\} \) for \( k = 0, ..., m-1 \), with \( \sum_{k=0}^{m-1} z_k = 1 \). 3. **Piecewise-linear constraints:** \[ x = \sum_{k=0}^{m-1} z_k (x_k + (x_{k+1} - x_k) \lambda_k) \] \[ y = \sum_{k=0}^{m-1} z_k (y_k + (y_{k+1} - y_k) \lambda_k) \] \[ 0 \leq \lambda_k \leq 1 \quad \forall k \] (where only one \( \lambda_k \) is active; others are zero). **This set of constraints models the piecewise-linear approximation of \( \sin(x) \) using \( m \) linear segments in a mixed integer program.**