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Concept

Gradient

The vector of a function's partial derivatives, pointing in the direction of steepest increase. It vanishes at every minimum and at a great many other places, which is why a vanishing gradient certifies an optimum only under convexity.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A line under every point of x². The curve x² with 4 tangent lines drawn, each extended across the whole interval and each staying below the curve throughout.

A line under every point

The chord above the curve is one definition of convexity. There is a second — a line under the curve at every point, staying under everywhere — and it is the one that turns a statement about a derivative at a point into a statement about the whole function.

analysis · Convexity One minimum, or several. Two curves side by side with their local minima marked: a convex one with a single minimum, and a fourth-power well with 2.

Where the guarantee stops

Convexity converts every downhill method into a correct one, and its absence removes the guarantee entirely rather than degrading it. What is left is a collection of partial answers, and knowing which of them apply to a given problem is most of what non-convex optimisation is.

analysis · Convexity

Named alongside it

The objects these essays reach for when they reach for this one.

ConvexityLocal minimumOptimisationComplexityCounterexampleDerivativeInequalityRelaxationSaddle pointSupporting lineTangent

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