Several recent studies demonstrated that higher order (non-linear) functionals can yield outstanding performances in the contexts of segmentation, co-segmentation and tracking. In general, higher order functionals result in difficult problems that are not amenable to standard optimizers, and most of the existing works investigated particular forms of such functionals. In this study, we derive general bounds for a broad class of higher order functionals. By introducing auxiliary variables and invoking the Jensenís inequality as well as some convexity arguments, we prove that these bounds are auxiliary functionals for various non-linear terms, which include but are not limited to several affinity measures on the distributions or moments of segment appearance and shape, as well as soft constraints on segment volume. From these general-form bounds, we state various non-linear problems as the optimization of auxiliary functionals by graph cuts. The proposed bound optimizers are derivative-free, and consistently yield very steep functional decreases, thereby converging within a few graph cuts. We report several experiments on color and medical data, along with quantitative comparisons to stateof-the-art methods. The results demonstrate competitive performances of the proposed algorithms in regard to accuracy and convergence speed, and confirm their potential in various vision and medical applications.