Article Dans Une Revue SIAM Journal on Optimization Année : 2025

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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Dates et versions

hal-05100908 , version 1 (06-06-2025)

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Radu-Alexandru Dragomir, Yurii Nesterov. Convex Quartic Problems: Homogenized Gradient Method and Preconditioning. SIAM Journal on Optimization, 2025, 35 (2), pp.651-677. ⟨10.1137/23M1583363⟩. ⟨hal-05100908⟩
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