Convex minimization

Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.

Given a real vector space ${\displaystyle X}$ together with a convex, real-valued function

defined on a convex subset ${\displaystyle {\mathcal {X}}}$ of ${\displaystyle X}$, the problem is to find any point ${\displaystyle x^{\ast }}$ in ${\displaystyle {\mathcal {X}}}$ for which the number ${\displaystyle f(x)}$ is smallest, i.e., a point ${\displaystyle x^{\ast }}$ such that

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