Local Minimum
Calculus & Advanced MathA local minimum is a point on a function that is lower than all nearby points, like the bottom of a valley.
Formula
f(c) \le f(x) \text{ for all } x \text{ near } c
Definition
A local minimum is the lowest point in a neighborhood, like the bottom of a valley. All nearby points are higher.
Example
The lowest point of a bowl is a local minimum. Everything around it is higher up.
Key Insight
Just like a local maximum is a hilltop, a local minimum is a valley floor, and the word "local" means you only compare nearby points.
Definition
$f$ has a local minimum at $c$ if $f(c) \le f(x)$ for all $x$ in some open interval around $c$. At a differentiable local min, $f'(c) = 0$ and $f'$ changes from negative to positive at $c$.
Example
$f(x) = x^2 - 6x + 9 = (x-3)^2$: $f'(x) = 2x - 6 = 0$ at $x = 3$. $f''(3) = 2 > 0$, confirming a local minimum at $f(3) = 0$.
Key Insight
For a differentiable function, the derivative transitions from negative (decreasing) to positive (increasing) at a local minimum.
Definition
$c$ is a local minimum if $f(c) \le f(x)$ for all $x$ in a deleted neighborhood of $c$. Second-order sufficient condition: $f'(c) = 0$ and $f''(c) > 0$. In multivariable optimization, the Hessian must be positive definite at the critical point.
Example
In machine learning, gradient descent seeks local minima of the loss function $L(\theta)$. For convex $L$, any local minimum is the global minimum, which is why convexity is so valued in optimization.
Key Insight
The KKT conditions generalize local minimum conditions to constrained optimization problems, forming the foundation of modern convex programming.