The authors consider the optimization problem \[ g(x)- h(x)\to \inf_{x\in\mathbb{R}^n}, \] where the functions \(g\) and \(h\) are convex. For this d.c. optimization problem, the authors give duality, local and global optimization conditions and a numerical algorithm. The given algorithm is applied for the solution of a trust-region problem of the form \[ {1\over 2}\cdot x^T\cdot A\cdot x+ b^T\cdot x\to \inf_{\| x\|\leq r}. \] Numerical experiments are given and relations to the Goldstein-Levitin-Polyak gradient projection algorithm in the convex case are discussed.