All convex functions are quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity.4.2 Basic properties A function is quasiconvex if and only dom fdom f is
show that f is obtained from simple convex functions by operations that preserve convexity ? ? ? ? ? ? nonnegative weighted sum composition with a?ne function pointwise maximum and supremum composition minimization perspective IOE 611: Nonlinear Programming, Winter 2011 3. Convex functions Page 3–...
(iii). Strict convexity: a function f is strictly convex if and only if ∇2f is positive definite in domf . (iv). Strict concavity: a function f is strictly concave if and only if ∇2f is negative definite in domf . Where ∇2f is the Hessian matrix of f which is defined at...
In addition, one of the main result of this section (Theorem 9.5) gives, under suitable assumptions, a characterization of convexity of a function in terms of its associated Hessian matrix. This boils down to check positive (semi)definiteness of matrices and thus, there will be separate ...
expression int,x(t), andx'(t) t - independent variable x(t) - unknown function (or list of functions) Description • TheConvex(f, t, x(t))command determines if the integrand is convex. • If the integrand is convex, the functionalJ=∫abft,x,x,'...
Reduction to a scalar function ? Showing that f is obtained through operations preserving convexity Convex Optimization 7 Lecture 3 Second-Order Conditions Let f be twice di?erentiable and let dom(f ) = Rn [in general, it is required that dom(f ) is open] The Hessian ?2f (x) is a ...
the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete ...
More generally, a continuous, twice differentiable function of several variables isconvexon a convex setif and only if its Hessian matrix is positive semidefiniteon the interior of the convex set. (Wikipedia) If we want to check if a function is convex, one easy way is to use our old frien...
Since it is the perspective (perspective preserves convexity and concavity) of log(1+x) , andlog(1+x) is concave. In this regard, the formulated problem is convex (since it is a maximization problem for concave function). △ ...
Further, if there exists α > 0 such that f(γx,y(t)) is an α-strongly convex function of t, for all x, y∈ A, then f is called α-strongly convex. For differentiable functions, it is possible to write down first-order and second-order characterizations of convexity (Udriste, ...