Hamiltonian descent for composite objectives

In optimization the duality gap between the primal and the dual problems is a measure of the suboptimality of any primal-dual point. In classical mechanics the equations of motion of a system can be derived from the Hamiltonian function, which is a quantity that describes the total energy of the system. In this paper we consider a convex optimization problem consisting of the sum of two convex functions, sometimes referred to as a composite objective, and we identify the duality gap to be the `energy' of the system. In the Hamiltonian formalism the energy is conserved, so we add a contractive term to the standard equations of motion so that this energy decreases linearly (ie, geometrically) with time. This yields a continuous-time ordinary differential equation (ODE) in the primal and dual variables which converges to zero duality gap, ie, optimality. This ODE has several useful properties: it induces a natural operator splitting; at convergence it yields both the primal and dual solutions; and it is invariant to affine transformation despite only using first order information. We provide several discretizations of this ODE, some of which are new algorithms and others correspond to known techniques, such as the alternating direction method of multipliers (ADMM). We conclude with some numerical examples that show the promise of our approach. We give an example where our technique can solve a convex quadratic minimization problem orders of magnitude faster than several commonly-used gradient methods, including conjugate gradient, when the conditioning of the problem is poor. Our framework provides new insights into previously known algorithms in the literature as well as providing a technique to generate new primal-dual algorithms.