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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 $$x\in\mathbb{R}^n$$, where $$f: \mathbb{R}^n\to\mathbb{R}$$ is a nondifferentiable convex function. The method is developed using the ideas of the Moreau-Yosida regularization [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 $$\mathbb{R}^n$$.
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.

MSC:
 65K05 Numerical mathematical programming methods 90C25 Convex programming 90C53 Methods of quasi-Newton type 52A41 Convex functions and convex programs in convex geometry
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