Convex Quartic Problems: Homogenized Gradient Method and Preconditioning
Résumé
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 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 O(κ 2 /K 2 ) and O(κ 2 /K 4 ) in relative accuracy, where K is the iteration counter. The constant κ 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 ℓ p norm composed with a quadratic map. Our construction involves a generalization of the so-called Lewis weights.
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