In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix A is stable if and only if it can be written as A = (J - R)Q, where J = -J(T), R >= 0 and Q > 0 (that is, R is positive semidefinite and Q is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables (I, R, Q): (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require O(n(3)) operations per iteration, where n is the size of A. We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms. (C) 2017 Elsevier Ltd. All rights reserved.