In this talk, the question of finding optimal success probabilities of linear optics quantum gates is linked to the theory of convex optimization. This question of optimalsuccess probability is important in the framework of quantum computation with linear optics and selective photon number measurements only, in order to assess the scalability of a specific scheme. Based on earlier work by other authors, it is shown that by exploiting this link, upper bounds for the success probability of gates involving single modes and arbitrary photon numbers can be derived that hold in all generality, and restrictions do not have to be imposed such as the requirement of a certain finite number of modes, of optical elements in the network or of photon numbers. The concept of Lagrange duality provides then rigorous proofs for bounds on success probabilities, without the need to resort to numerical means. As a corollary, the previously formulated conjecture is proven that the optimal success probability of a non-linear sign shift gate is exactly one quarter. In an extended outlook, other applications of the tools of semi-definite programming in quantum information theory will be sketched, and complete hierarchies of efficient criteria for multi-particle entanglement will be presented.