17 . Convex Function A function f ( x ) is said to be strictly convex at x , if for any two other distinct points x1 and x2 .
f { λx1 + ( 1 — λ ) x2 } < λf ( x1 ) + ( 1 — λ ) f ( x2 ), where 0 < λ < 1 .
And a function f ( x ) is strictly concave , if — f ( x ) is strictly convex .
18 . Convex Polyhedron The set of all convex combinations of finite number of points is called the convex polyhedron generated by these points .
Important Points to be Remembered ( i ) A hyperplane is a convex set .
( ii ) The closed half spaces H1 = { x : cx ≥ z } and H2 = { x : cx ≤ z } are convex sets .
( iii ) The open half spaces : { x : cx > z } and { x : cx < z } are convex sets .
( iv ) Intersection of two convex sets is also a convex sets .
( v ) Intersection of any finite number of convex sets is also a convex set .
( vi ) Arbitrary intersection of convex sets is also a convex set .
( vii ) The set of all convex combinations of a finite number of points X1 , X2 ,….,
Xn is convex set .
( viii ) A set C is convex , if and only if every convex linear combination of points in C , also belongs to C .
( ix ) The set of all feasible solutions ( if not empty ) of a LPP is a convex set .
( x ) Every basic feasible solution of the system Ax = b , x ≥ 0 is an extreme point of the convex set of feasible solutions and conversely .
( xi ) If the convex set of the feasible solutions of Ax = b , x ≥ 0 is a convex polyhedron , then atleast one of the extreme points gives an optimal solution . ( xii ) If the objective function of a LPP assumes its optimal value at more than one extreme point , then every convex combination of these extreme points gives the optimal value of the objective function .