Projection onto the exponential cone: a univariate root-finding problem

The exponential function and its logarithmic counterpart are essential corner stones of nonlinear mathematical modelling. In this paper, we treat their conic extensions, the exponential cone and the relative entropy cone, in primal, dual and polar form, and show that finding the nearest mapping of a point onto these convex sets all reduce to a single univariate root-finding problem. This leads to a fast projection algorithm shown numerically robust over a wide range of inputs.