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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.
In LP, the objective and constraint functions are all linear. Quadratic programming are the next-simplest. In QP, the constraints are all linear, but the objective may be a convex quadratic function. Second order cone programming are more general. Semidefinite programming are more general. Conic optimization are even more general - see figure ...
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.
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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.