A descent method with relaxation type step

Abstract

A new bundle method for minimizing a convex nondifferentiable function f:ℜ → ℜ is presented. At each iteration a master problem is solved to get a search direction d. This master problem is a quadratic programming problem of the type

$$\begin{gathered}\min _d \tfrac{1}{2}d^T d \hfill \\s.t.\upsilon _c \geqslant g_i^T d - \varepsilon _i ,\forall i \in I \hfill \\\end{gathered}$$

where v _c is a parameter, which is an estimate the predicted decrease obtainable from the current iteration point and g: are ∈₁-subgradients at the current iteration point.

It is shown that each sequence of {x _k} generated by the algorithm minimizes f, i.e. f(x _k) ↓ inf{f(x) | x ∈ ℜ}, and that {x _k} converges to a minimum point whenever f attains its infimum.

Some numerical experiments on some nondifferentiable test problems found in the literature are performed with satisfactory and encouraging results.

Research supported by the Swedish Research Council for Engineering Sciences.