When trying to find the minimum of a function f using the steepest descent method, which of the following is a plausible termination criteria?

Question

BlackTom AI · Accepted Answer

In optimization, especially with the steepest descent (gradient descent) method, the termination criterion signals when we should stop iterating because progress is effectively complete or numerical limits are reached.

- The block of text shows the question and one provided answer: "The norm of the gradient of f is nearly zero". This is a widely used terminating criterion: when the gradient norm ||∇f(x_k)|| is close to zero, we are near a stationary point where no short descent direction exists, indicating a potential minimum (or saddle/maximum, depending on the function's curvature).

- Since there are no other answer options listed, we cannot analyze alternatives in this case. However, the idea behind the gradient-norm criterion is rooted in the first-order optimality condition: at an unconstrained minimum, the gradient is zero. In practice, we use a tolerance ε > 0 and stop when ||∇f(x_k)|| ≤ ε, balancing precision and computational effort.

- Exploring why this criterion is plausible: when the gradient is nearly zero, any further update by the steepest descent method will yield only tiny changes in f, implying diminishing returns from continuing iterations. This aligns with the goal of finding a stationary point efficiently.

- Why other, non-listed criteria might be less appropriate in this context: if there were criteria based on function value changes, step size, or iteration count alone, they might fail to detect convergence in flat regions or could terminate prematurely before reaching a stationary point. The gradient-based stop condition directly ties to optimality conditions rather than solely monitoring objective improvement.

- Summary of the key idea: using the gradient norm as a termination signal leverages the fundamental property of stationary points in unconstrained optimization, making it a standard and robust choice for algorithms like steepest descent.

When trying to find the minimum of a function f using the steepest descent method, which of the following is a plausible termination criteria?单项选择题

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