能否在Cplex中建模分段非线性成本函数并保留二次项？

Absolutely, you can model this exact piecewise cost function (quadratic for x < x0, linear for x ≥ x0) directly in CPLEX without linearizing the quadratic term. Here's a practical, step-by-step approach that leverages CPLEX's native support for quadratic programming:

Step 1: Define Your Variables #

First, set up the variables you'll need:

• Continuous decision variable: x (the variable driving your cost function)
• Binary variable: y (acts as a switch: y=1 means x falls in the quadratic segment; y=0 means it's in the linear segment)

Step 2: Add Constraints to Enforce Segmentation #

Since CPLEX works with non-strict inequalities, we'll use big-M constraints to enforce the logical split between the two segments. You'll need to pick a reasonable M value—this should be slightly larger than the maximum possible value x can take (too large an M can hurt solver performance). Also, use a tiny positive ε (like 1e-6) to handle the strict x < x0 condition:

• When y=1 (quadratic segment): x ≤ x0 - ε + M*(1 - y)
• When y=0 (linear segment): x ≥ x0 - M*y

These constraints ensure that y correctly flags which segment x is in.

Step 3: Formulate the Piecewise Cost Function #

Now define the cost components and combine them using the binary switch variable:

• Quadratic cost (active when y=1): C_quad = a*x² + b*x + c (replace a, b, c with your actual quadratic coefficients)
• Linear cost (active when y=0): C_lin = d*x + e (replace d, e with your linear coefficients)
• Total cost: C_total = y*C_quad + (1 - y)*C_lin

Step 4: Set the Objective #

If you're minimizing the cost (the most common use case for such functions), your objective will be:

minimize C_total

Key Tips for Success #

• CPLEX handles this formulation as a Mixed-Integer Quadratic Programming (MIQP) problem, which it supports natively. Just ensure your quadratic term is convex (for minimization problems)—this is standard for most practical cost functions, and CPLEX will solve it efficiently.
• Keep ε small enough to not impact your solution's practicality but large enough to avoid numerical precision issues.
• Don't oversize M—stick to a value that's just above your upper bound for x to keep the problem tight.

内容的提问来源于stack exchange，提问作者Roozbeh Morsali