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Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being 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.
Extensions of convex optimization include the optimization of biconvex, pseudo-convex, and quasiconvex functions. Extensions of the theory of convex analysis and iterative methods for approximately solving non-convex minimization problems occur in the field of generalized convexity , also known as abstract convex analysis.
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:
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:
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.
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.