# What is difference between “Frank–Wolfe algorithm” and “Gradient steepest descent algorithm”?

In a minimization problem, i.e., $\min f(\mathbf{x})$, the procedure is

1. Start with an arbitrary $\mathbf{x}_1$.
2. At $i$-th iteration, calculate a gradient $\nabla f(\mathbf{x}_i)$.
3. Find the next point, $\mathbf{x}_{i+1}\in \mathbf{x}_i+\gamma \nabla f(\mathbf{x}_i)$, where $\gamma$ is a real value that minimizes $f(\mathbf{x}_i+\gamma \nabla f(\mathbf{x}_i))$.

I think both "Frank–Wolfe algorithm" and "Gradient steepest descent algorithm" are the same as the above. What is different?

• The Frank-Wolfe algorithm solves a constrained minimization problem, but your algorithm doesn't, so they're not the same. What description of the Frank-Wolfe algorithm are you basing your assumption on? – Brian Borchers Jun 14 '18 at 2:05