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Quasi-Newton bundle-type methods for nondifferentiable convex optimization. (English) Zbl 0927.65074
The authors present a method for solving the nondifferentiable convex programming problem: $\min f(x)$ subject to $x\in\bbfR^n$, where $f: \bbfR^n\to\bbfR$ is a nondifferentiable convex function. The method is developed using the ideas of the Moreau-Yosida regularization [{\it K. Yosida}, Functional analysis (1965; Zbl 0126.11504)], the bundle method, and the quasi-Newton method. It is shown that in this method the minimizing solution of the given programming problem is exactly the solution of a subproblem of minimizing the Moreau-Yosida regularization of $f$ over $\bbfR^n$. The entire discussion in the paper is devoted to the solution of the subproblem, and a globally and superlinearly convergent quasi-Newton bundle-type algorithm is described for solving the subproblem. No numerical results are given.

65K05Mathematical programming (numerical methods)
90C25Convex programming
90C53Methods of quasi-Newton type
52A41Convex functions and convex programs (convex geometry)
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