Gradient ascent ($x_{n+1}=x_n+{\eta }_n\nabla J(x_n)$)

Gradient ascent seeks a maximum by stepping along the gradient of a differentiable objective. It is gradient descent applied to $-J$. Taylor expansion gives $J(x+\eta \nabla J)=J(x)+\eta \Vert \nabla J{\Vert }^2+O({\eta }^2)$, so sufficiently small positive steps improve the objective when the gradient is nonzero. Line search, projection onto admissible controls, and noise-aware gradient estimates are common practical modifications; a local stationary result need not be globally optimal.