A Discussion on Non-Convex Optimization Problems Arising in Supply Chain Design and Finance

Abstract

Non-convex optimization problems belong to a class of classical nonlinear optimization problems, which are often difficult to solve. An optimization problem becomes non-convex due to the presence of non-convex functions in the objective function or constraints. A function is a convex function if its Hessian matrix is positive and semi-definite for all values; otherwise, it is a non-convex function. A Hessian matrix is called positive semi-definite when the eigenvalues of the matrix are non-negative. A non-convex function can be either a concave function or a function that is neither a concave nor a convex function. A concave function is always negative semi-definite, indicating that the eigenvalues of the matrix are non-positive. This chapter starts with a short introduction to non-convex problems, followed by a discussion on different non-convex problems arising in supply chain and finance. Thereafter, the authors discuss different algorithms used for solving non-convex problems. Finally, the chapter conclude with the limitations of different algorithms.