Treffer: A new super-memory gradient method with curve search rule

An unconstrained minimization problem of the form \[ \text{minimize }f(x),\quad x\in\mathbb{R}^n \] is considered. It is assumed that \(f\) is a continuously differentiable function satisfying the following conditions: (i) \(f\) has a lower bound on the level set \(L_0= \{x\in\mathbb{R}^n\mid f(x)\leq f(x_0)\}\), where \(x_0\) is given; (ii) the gradient of \(f\) is uniformly continuous in an open convex set \(B\) that contains \(L_0\); (iii) the gradient of \(f\) is Lipschitz continuous in \(B\). A new super-memory gradient method with curve search rule for solving the unconstraint minimization problem under the above assumptions is described. The method uses previous multi-step iterative information and curve search rule to generate new iterative points at each iteration. The suggested method has global convergence and linear convergence rate. Numerical experience with the method presented at the end of the paper shows a good computational effectiveness in practical computations.