What is a concave function in the context of optimization?

In the context of optimization, a concave function is defined as a curve that is bowl-shaped downward. This characteristic means that for any two points on the curve, the line segment connecting those points lies above the curve itself. Consequently, this implies that if you take any combination (or weighted average) of two points on a concave function, that combination will also lie above the curve, showing diminishing returns.

This property of concavity is important in optimization because it guarantees that any local maximum is also a global maximum within the feasible region. This trait simplifies analysis and ensures that optimization techniques will yield consistent results.

The other options depict forms that do not align with the definition of a concave function. For example, a curve that is bowl-shaped upward represents a convex function, exhibiting increasing returns. A straight line indicates linearity, and a flat line represents constant values with no variability, both of which do not possess the concave structure that is crucial for this definition in optimization contexts.