Result: A subgradient method with space dilatation for minimizing convex functions

The paper considers the optimization problem: minimize f(x), \(x\in {\mathbb{R}}^ n\), where f is a real-valued function which is continuous, convex, but not everywhere differentiable. The approach to generating solutions to this problem is a subgradient algorithm where the direction at each iteration is determined by a space-dilatation operator. The space-dilatation operator in the direction S is: \(R_{\alpha}(S)X=X+(\alpha -1)SS^ TX\) where \(\| S\| =1\), \(\alpha >0\). The global convergence of the algorithm is established and it is illustrated on a simple example.