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A function : defined on a convex subset of a real vector space is quasiconvex if for all , and [,] we have (+ ()) {(), ()}.In words, if is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then is quasiconvex.
Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let and ((,),) with (,) =.The Riesz-Markov-Kakutani representation theorem states that the dual space of () can be identified with the space of signed, finite Radon measures on it.
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, [1] whereas mathematical optimization is in general NP-hard. [2 ...
An important special case of concavification is where the original function is a quasiconcave function. It is known that: Every concave function is quasiconcave, but the opposite is not true. Every monotone transformation of a quasiconcave function is also quasiconcave.
Every convex function is pseudoconvex, but the converse is not true. For example, the function () = + is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function () = is quasiconvex but not pseudoconvex. This can be summarized schematically as:
Theorem: If the function f is differentiable, the gradient of f at a point is either zero, or perpendicular to the level set of f at that point. To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and decides to go in the direction where the slope is steepest.
The function () = has ″ = >, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points.
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1: