CONVEX QUARTIC PROBLEMS: HOMOGENIZED GRADIENT METHOD AND PRECONDITIONING

We consider a convex minimization problem for which the objective is the sum of a homogeneous polynomial of degree four and a linear term. Such a task arises as a subproblem in algorithms for quadratic inverse problems with a difference-of-convex structure. We design a first-order method called homogenized gradient, along with an accelerated version, which enjoy fast convergence rates of, respectively, ο(k²/K²) and ο(k²/K⁴) in relative accuracy, where K is the iteration counter. The constant k is the quartic condition number of the problem. Then, we show that for a certain class of problems, it is possible to compute a preconditioner for which this condition number is √n, where n is the problem dimension. To establish this, we study the more general problem of finding the best quadratic approximation of an lp norm composed with a quadratic map. Our construction involves a generalization of the so-called Lewis weights.