Trust Region Methods
Given a twice differentiable function $f\colon\mathbf{R}^n\to\mathbf{R}$ and $x\in\mathbf{R}^n$, this application determines $\Delta > 0$ and $s\in\mathbf{R}^n$ from solving the trust-region subproblem: \[ \min_{\|s\|\leqslant \Delta}\; \frac{1}{2}s^THs+g^Ts,\;\mathrm{where}\;H=\nabla^2f(x)\;\mathrm{and}\;g=\nabla f(x), \] and then $x_\mathrm{new}$ from the formula \[ x_\mathrm{new} = x + s \] such that $f(x_\mathrm{new}) < f(x)$.
Determining $\Delta$, $s$, and $x_\mathrm{new}$ is in one iteration of iterative methods for solving unconstrained optimization problems. The results $\Delta$ and $x_\mathrm{new}$ can also be used in the next iteration. But this application does just one iteration. If you are looking for a complete solver, unconstrained optimization is recommendeded.
Input Data
This application requires for $f(x)$ and $x$ the same input format as Line Search Calculator. The initial value of trust-region radius $(\Delta)$ must be greater than zero, default is 1.