A Gentle Introduction to Nonlinear Constrained Optimization with Piecewise Linear Approximations

Nonlinear constrained optimization problems can be addressed using piecewise linear (PWL) approximations. This technique involves transforming a problem into a separable form where decision variables appear in separate functions. Each nonlinear function is then approximated by a series of linear segments defined by breakpoints, allowing the problem to be modeled for an LP/MIP solver. The article uses a portfolio selection problem as a detailed example, walking through the mathematical formulation and referencing a Python implementation with Gurobi.