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CONVEX WEB PROGRAMMING
Convex optimization is a subfield of mathematical optimization.
A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity.
If its second derivative is positive then it is strictly convex, but the converse does not hold, as shown by f(x) = x4. More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set. Any local minimum of a convex function is also a global minimum.
A strictly convex function will have at most one global minimum. For a convex function f, the sublevel sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function; such a function is called a quasiconvex function.
If its second derivative is positive then it is strictly convex, but the converse does not hold, as shown by f(x) = x4. More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set. Any local minimum of a convex function is also a global minimum.
A strictly convex function will have at most one global minimum. For a convex function f, the sublevel sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function; such a function is called a quasiconvex function.
