Global optimization of a convex-concave fraction plus a convex function using hidden sawtooth-curve bounds via a two-layer dual approach

Abstract

We address a class of optimization problems, denoted by (SFC), of minimizing the sum of a convex-concave fraction and a convex function over a convex set. It is shown that problem (SFC) can be reformulated into an equivalent one-dimensional optimization problem, where each subproblem is evaluated by solving an associated convex programming. The optimal Lagrangian multipliers of the convex subproblems are utilized to construct sawtooth-curve and wave-curve lower bounds, which play a crucial role in devising the branch-and-bound algorithm for globally solving (SFC). In this paper, we propose a two-layer dual approach to get hidden sawtooth-curve lower bounds, which leads to a new efficient branch-and-bound algorithm for solving (SFC). Moreover, we improve the iterative complexity $O\left(\frac{1}{ϵ}\right)$ with wave-curve bounds to $O\left(\frac{1}{\sqrt{ϵ}}\right)$ for finding an ϵ-approximate optimal solution. Numerical results demonstrate that it is more efficient than the recent branch-and-bound algorithm based on wave-curve bounds.