Local Minimum
A 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. Formally, $f$ has a local minimum at $c$ if $f(c) \le f(x)$ for all $x$ in some open interval (or deleted neighborhood) around $c$; at a differentiable local min, $f'(c) = 0$ and $f'$ changes from negative to positive. The second-order sufficient condition is $f'(c) = 0$ and $f''(c) > 0$; in multivariable optimization, the Hessian must be positive definite at the critical point.
Example
The lowest point of a bowl is a local minimum, everything around it is higher up. For $f(x) = x^2 - 6x + 9 = (x-3)^2$: $f'(x) = 2x - 6 = 0$ at $x = 3$, and $f''(3) = 2 > 0$ confirms a local minimum at $f(3) = 0$. In machine learning, gradient descent seeks local minima of a 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
Just as a local maximum is a hilltop, a local minimum is a valley floor, and "local" means you only compare nearby points; for a differentiable function, the derivative transitions from negative (decreasing) to positive (increasing) at a local minimum. The KKT conditions generalize local minimum conditions to constrained optimization problems, forming the foundation of modern convex programming.