This paper presents new graph-cut based optimization algorithms for image processing problems. Popular graph-cut based algorithms give approximate solutions and are based on the concept of partition move. The main contribution of this work consists in proposing novel partition moves called multilabel moves to minimize Markov random field (MRF) energies with convex prior and any likelihood energy functions. These moves improve the optimum quality of the state-of-the-art approximate minimization algorithms while controlling the memory need of the algorithm at the same time. Thus, the two challenging problems, improving local optimum quality and reducing required memory for graph construction are handled with our approach. These new performances are illustrated on some image processing experiments, such as image restoration and InSAR phase unwrapping.