This paper deals with the Robust Constraint Feasibility (RCF) problem which aims finding all θ such that there exists a vector x satisfying the constraint f(x, θ) < 0. An interval based algorithm will answer the question but with a crippling computational complexity which is exponential with respect to the dimension of the vector (x, θ). If the constraint function is assumed to be convex in x when θ is fixed, the paper shows that the complexity of the problem becomes polynomial with respect to the dimension of vector x and exponential with respect to the dimension of vector θ. Another contribution of the paper is to provide an algorithm which combines convex optimization and interval analysis to solve the problem with the reduced complexity. As an illustration, a simple numerical example is given.