Semistrictly quasiconcave approximation and an application to. A function is concave if the chord joining any two points on any plane profile of its graph lies everywhere on or. Mathematics 2 for economics institute for statistics. The level curves isoquants, indifference curves are convex to the origin diminishing marginal rate of substitution. While boosting system throughput, ca has also put forward challenges to the resource allocation problems. Monotone linlcage, quasiconcavity, greedy optimization. Global rates of convergence of the mle for multivariate interval censoring gao, fuchang and wellner, jon a. How to prove a function is a quasiconcave function. Good news is, though, that typically there will not be too many of those. Quasi concave and convex best online classes for eco.
Quasiconcave functions can have local maxima which are not global maxima, as easy examples show. Another result provides conditions under which a point of constrained global. If f is quasiconcave, then its upper level sets are convex sets. Like the result for functions of a single variable, it is used to show that stationary points are global maximizers of concave functions and global minimizers of convex functions. Pi r, the minimum split function of its maximum join linkage coincides with f assertion 3. In production theory it is commonly assumed that the production function is increasing and quasiconcave. You can download a program called r and run a demo file, where you can. We limit ourselves to giving just some chronological references in order to emphasize the. Quasiconcave is a mathematical concept that has several applications in economics. One result provides conditions under which a point satisfying the kuhntucker conditions is a point of constrained global maximum. The third picture shows that the opposite isnt true. Global rates of convergence in logconcave density estimation kim, arlene k.
But such critical points need not exist and even if they do, they are not necessar. The class of multiextremal problems in which the minimum of a quasiconcave function in a convex compact set is sought, is considered. Limit distribution theory for maximum likelihood estimation of a logconcave density balabdaoui, fadoua, rufibach, kaspar, and wellner, jon a. Inference for the mode of a logconcave density doss, charles r. Econ 205 slides from lecture joel sobel september 10, 2010. In this paper, we firstly construct the energy efficiency optimization problem and prove that the function is strictly quasi concave. I if f is a monotonic transformation of a concave function, it is quasiconcave. A theorem on the additivity of the quasiconcave closure of an additive convex function uzi segal nufleld college, oxford ox1 inf, uk received september 1982, accepted january 1983 in this paper a necessary and sufficient condition for the additivity of the quasiconcave closure. Global properties theorem if f is concave and x is a local maximizer of f, then x is a global maximizer.
If a function is increasing, then is it both quasiconcave. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. A sufficient condition for global concavity is that the hessian of f is everywhere negative definite, and this requires det. A note on the maximum of quasiconcave functions springerlink. Methods are given for finding the global extremum, consisting of the minimization of linear functions of a specially chosen family. Fromnow on we will assume thatx is aconvex subset of rn. Quasiconcavity and therefore also concavity of functions gj implies that. Can gradient descent find the maximum of a concave function. A necessary and sufficient condition for a unique maximum. Sufficient condition for global maximum of strictly quasi. In what follows, we cover the elements of this theory by focusing on its two main results.
Graduate student, professor, faculty of industrial and management engineering. On the global minimum of a quasiconcave functional. This also means that if a monotonic transformation of. To understand the significance of the terms applications in economics, it is useful to begin with a brief consideration of the origins and meaning of the term in mathematics. However, quasi concave and quasi convex functions posses enough structure that is important for. Specifically, it is almost but definitely not exactly true that if f. Quasi convex and quasi concave functions are used extensively in economics and form a major portion of the the subject mathematical methods in economics. In mathematics, a concave function is the negative of a convex function. This structures allow building efficient algorithms that find minimal sets on which values of quasiconcave functions are maximum. If f is quasiconcave, then the hessian matrix is negative semi. If f is a function of one variable and is singlepeaked, then f is quasiconcave.
Methods of finding the global minimum of a quasiconcave. If f is globally strictly concave, then a critical point x. Near a local maximum in the interior of the domain of a function, the function must be concave. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. Functions which are increasing, coradiant and quasiconcave have found many applications in microeconomic analysis. Then any local minimum maximum is akzo a global minimum. Regularization for density estimation maximum likelihood estimation of densities max f2f xn i1 log fx i over any reasonably large class f yields. Then by a previous result, for every point x, no point on the graph of f lies above the tangent to f at x. A local maximum of a quasiconcave function need not also be a global maximum of the function. The set argmaxffx jx2dgof maximizers of fon dis either empty or convex. On global minima of semistrictly quasiconcave functions.
Were not going to say much more about quasiconcave, but it is the feature which guarantees a unique maximum. Maximum likelihood estimation of a logconcave probability density is formulated as a convex optimization problem and shown to. On global extrema for a class of symmetric functions1. The origins of the notion of quasiconcave function are considered, with special interest in some. A note on the quasiconvex jensen divergences and the quasiconvex. This paper studies the global behaviour of semistrictly quasiconcave functions with possibly nonconvex domain in the presence of global minima. Energy efficient power allocation strategy for 5g carrier.
Likewise in consumer theory one often assumes that the utility function has these properties. A much weaker property holds for the linkage functions. In mathematics, a quasiconvex function is a realvalued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form. More significantly, first order conditions are not in general sufficient to identify global of quasi concave or convex optimization problems. To show this, we make use of the fact that any monotone increasing transformation of a concave function is quasiconcave. A strictly concave function will have at most one global maximum. Jensens inequality logarithmically concave function quasiconcave function concavification.
Quasiconcave functions defined on semilattices, antimatroids and convex. Monotone linkage clustering and quasiconvex set functions. Strictly quasiconvex concave functions and duality in. The negative of a quasiconvex function is said to be quasiconcave. Let 0 be strictly quasiconvex strictly quasiconcave on a convex set c c en. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. For a function of a single variable, along any stretch of the curve the highest. Lecture 14 outline 1 convexity 2 concave and convex functions 3 cardinal and ordinal properties. Any local maximum of a concave function is also a global maximum. Yes, place it as a convex function by flipping the function and then, since there is one major local minimum the global minimum, gradient descent will definitely find it.
Center of mathematical statistics of the academy, bucharest, romania. Carrier aggregation ca is considered to be a potential technology in next generation wireless communications. A function f is quasiconcave if hx gfx is a concave function for some strictly increasing function gfrom and 00 df x dx is it true that that x0 is a maximizer of. Global minimization of the difference of increasing co. Ktl are sufficient conditions for a constrained maximum.