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.