A univariate Heaviside function is the (discontinuous) indicator function of an (open or closed) interval. By its name, a Heaviside composite function is the composition of a Heaviside function with a multivariate function that may be nonconvex and nondifferentiable. Complementarity constraints are well known for their modeling breadth in many optimization problems, ranging from an indefinite quadratic program to bilevel optimization to structured sparsity selection. In principle, these two features: Heaviside functions and complementarity constraints, can be formulated by integer variables; yet the resulting optimization problems present great challenges for state-of-the-art integer programming methods beyond small-to-medium sizes, per solution by the well-known Gurobi solver. This talk presents an elementary yet very effective way, termed the progressive integer programming (PIP) method, to exploit the solver's capability and push it beyond what it can normally handle. Extensive computational results demonstrate PIP's great promise for solving many classes of nonconvex optimization problems that involve Heaviside composite functions and complementarity constraints, and beyond.