strictsteep

Strictsteep is a term used in numerical optimization to denote a conservative variant of the steepest descent method that imposes a strict decrease of the objective function at every iteration. In this framework, the search direction is the steepest descent direction p_k = -∇f(x_k). A backtracking line search is applied along this direction to locate a step size α_k that yields a strict decrease: f(x_k + α_k p_k) < f(x_k). If the line search fails to find any α_k within the allowed range, the algorithm may terminate with stagnation or switch to a different update rule.

The line search typically uses a reduction factor β in (0,1) and a decrease condition similar to, but strictly enforcing, an Armijo-type condition: f(x_k + α p_k) ≤ f(x_k) + c α ∇f(x_k)ᵀ p_k with c ∈ (0,1). The strict version requires a strict inequality and may include a minimum step size or a maximum number of backtracking steps.

Convergence properties of strictsteep follow those of gradient-based methods under standard assumptions: if f is continuously differentiable with Lipschitz continuous gradient and is convex, the sequence {x_k} typically converges to a minimizer; in nonconvex settings it converges to a stationary point under suitable conditions. The practical performance is influenced by the choice of line-search parameters and the landscape of f, with potential slow convergence in ill-conditioned or high-curvature regions.

Strictsteep is related to other gradient methods that employ line searches, such as gradient descent with Armijo or Wolfe conditions, but distinguishes itself by the explicit requirement of strict objective decrease at each iteration.