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IPI

In the paper, we present an algorithm framework for the more general problem of minimizing
the sum $f(x)+\psi(x)$, where $f$ is smooth and $\psi$
is convex, but possible nonsmooth.
At each step, the search direction of the algorithm is obtained by solving an optimization
problem involving a quadratic term with diagonal Hessian and Barzilai-Borwein steplength plus $ \psi(x)$.
The nonmonotone strategy is combined with -Borwein steplength to accelerate the convergence process.
The method with the nomonotone line search techniques is showed to
be globally convergent.
In particular, if $f$ is convex, we show that the method shares a sublinear
global rate of convergence.
Moreover, if $f$ is strongly convex,
we prove that the method converges R-linearly.
Numerical experiments with compressive sense problems show that our approach is competitive with
several known methods for some standard $l_2-l_1$ problems.

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