Global minimization of a difference of two convex functions

Abstract

We study the problem of finding the global minimum of the difference between two convex functions f(x)−g(y), under linear constraints of the form: x∈X, y∈Y, Ax+By+c≤0, where $X \subset \mathbb{R}^{n_1 } , Y \subset \mathbb{R}^{n_2 }$, are convex polyhedral sets. The proposed solution method consists in converting the problem into a concave minimization problem in $\mathbb{R}^{n_2 + 1}$ and applying the outer approximation method to the latter problem. Using a special type of separating hyperplanes the same result could also be obtained by applying the generalized Benders’ decomposition method with a proper change of variable in the master problem. As specialized to the indefinite quadratic programming problem, the algorithm is convergent, provided only the set Y ₀={y∈Y:(∃x∈X) Ax+By+c≤0} is bounded.