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For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex.
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.
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:
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 ...
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
The objective function in a linear-fractional problem is both quasiconcave and quasiconvex (hence quasilinear) with a monotone property, pseudoconvexity, which is a stronger property than quasiconvexity. A linear-fractional objective function is both pseudoconvex and pseudoconcave, hence pseudolinear.
The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand. The functional must be bounded from below to have a minimizer. This means