An explicit algorithm for the minimization of an ℓ 1 penalized least squares functional, with non-separable ℓ 1 term, is proposed. Each step in the iterative algorithm requires four matrix vector multiplications and a single simple projection on a convex set (or equivalently thresholding). Convergence is proven and a 1/N convergence rate is derived for the functional. In the special case where the matrix in the ℓ 1 term is the identity (or orthogonal), the algorithm reduces to the traditional iterative soft-thresholding algorithm. In the special case where the matrix in the quadratic term is the identity (or orthogonal), the algorithm reduces to a gradient projection algorithm for the dual problem.By replacing the projection with a simple proximity operator, other convex nonseparable penalties than those based on an ℓ 1 -norm can be handled as well.

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We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painlevé sense (impossibility to add any term to the Hamiltonian without destroying the Painlevé property).

The quartic Hénon-Heiles Hamiltonian passes the Painlevé test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case (α, β, γ) = (0, 0, 0). We integrate them by building a birational transformation to two fourth-order firstdegree equations in the Cosgrove classification of polynomial equations that have the Painlevé property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Hénon-Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlevé integrability (the completeness property).