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 −

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 −

From the graph of the above function, it is clear that the global minima occurs at $x=-1$.