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Convex Optimization - Minima and Maxima
Local Minima or Minimize
$ar{x}in :S$ is said to be local minima of a function $f$ if $fleft ( ar{x} ight )leq fleft ( x ight ),forall x in N_varepsilon left ( ar{x} ight )$ where $N_varepsilon left ( ar{x} ight )$ means neighbourhood of $ar{x}$, i.e., $N_varepsilon left ( ar{x} ight )$ means $left | x-ar{x} ight |< varepsilon$
Local Maxima or Maximizer
$ar{x}in :S$ is said to be local maxima of a function $f$ if $fleft ( ar{x} ight )geq fleft ( x ight ), forall x in N_varepsilon left ( ar{x} ight )$ where $N_varepsilon left ( ar{x} ight )$ means neighbourhood of $ar{x}$, i.e., $N_varepsilon left ( ar{x} ight )$ means $left | x-ar{x} ight |< varepsilon$
Global minima
$ar{x}in :S$ is said to be global minima of a function $f$ if $fleft ( ar{x} ight )leq fleft ( x ight ), forall x in S$
Global maxima
$ar{x}in :S$ is said to be global maxima of a function $f$ if $fleft ( ar{x} ight )geq fleft ( x ight ), forall x in S$
Examples
Step 1 − find the local minima and maxima of $fleft ( ar{x} ight )=left | x^2-4 ight |$
Solution −
![Min](/convex_optimization/images/min.jpg)
From the graph of the above function, it is clear that the local minima occurs at $x= pm 2$ and local maxima at $x = 0$
Step 2 − find the global minima af the function $fleft (x ight )=left | 4x^3-3x^2+7 ight |$
Solution −
![Min 2](/convex_optimization/images/min2.jpg)
From the graph of the above function, it is clear that the global minima occurs at $x=-1$.
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