SUBGRADIENT Meaning and
Definition
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A subgradient refers to a concept in mathematical optimization that pertains to an extension of the notion of gradient to nonsmooth functions. More specifically, it is a generalization of the gradient, which is typically used for differentiable functions, to handle functions that are not strictly differentiable.
When applied to a function that is not differentiable at a particular point, a subgradient represents a set of vectors that characterize the possible slopes of tangents to the function at that point. It captures the possible rates of change of the function, allowing for optimization algorithms to make progress even when differentiability is not guaranteed.
In mathematical terms, given a nonsmooth function, a subgradient at a particular point x is a vector that satisfies the following condition: for any point y, the difference between the function values at y and x can be bounded from above by the inner product between the subgradient and the difference between y and x.
The concept of subgradient is particularly useful in convex optimization, as it enables the identification of optimal solutions and provides crucial information for convergence analysis. It allows for the development of subgradient methods, an important class of algorithms used to optimize nonsmooth and convex functions.
In summary, a subgradient is an extension of the gradient that allows optimization algorithms to handle nonsmooth functions by characterizing possible slopes of tangents at points where differentiability is not guaranteed.
Etymology of SUBGRADIENT
The word "subgradient" is a combination of the prefix "sub-" and the word "gradient".
The prefix "sub-" comes from the Latin word "sub", which means "under" or "below". It is commonly used to indicate that something is subordinate, less than, or smaller in relation to something else.
The word "gradient" is derived from the Latin word "gradiens", a present participle form of the verb "gradi", which means "to step" or "to walk". In mathematics, a gradient refers to the rate at which a quantity, such as temperature or pressure, changes in space.
When combined, "subgradient" refers to a gradient that is less than or below another gradient, or a subordinate or suboptimal gradient. In mathematics, it is often used to describe a set of weaker conditions that approximate a gradient.
