Let 0 be strictly quasiconvex strictly quasiconcave on a convex set c c en. The level curves isoquants, indifference curves are convex to the origin diminishing marginal rate of substitution. Near a local maximum in the interior of the domain of a function, the function must be concave. Regularization for density estimation maximum likelihood estimation of densities max f2f xn i1 log fx i over any reasonably large class f yields. The third picture shows that the opposite isnt true. In this paper, we firstly construct the energy efficiency optimization problem and prove that the function is strictly quasi concave. 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.
Fromnow on we will assume thatx is aconvex subset of rn. Maximum likelihood estimation of a logconcave probability density is formulated as a convex optimization problem and shown to have an equivalent dual. This also means that if a monotonic transformation of. Quasiconcavity and therefore also concavity of functions gj implies that. In mathematics, a concave function is the negative of a convex function. In production theory it is commonly assumed that the production function is increasing and quasiconcave. On global extrema for a class of symmetric functions1. Were not going to say much more about quasiconcave, but it is the feature which guarantees a unique maximum. Ktl are sufficient conditions for a constrained maximum. A note on the maximum of quasiconcave functions springerlink. To show this, we make use of the fact that any monotone increasing transformation of a concave function is quasiconcave. Specifically, it is almost but definitely not exactly true that if f. Lecture 14 outline 1 convexity 2 concave and convex functions 3 cardinal and ordinal properties. Sufficient condition for global maximum of strictly quasi.
Monotone linlcage, quasiconcavity, greedy optimization. Strictly quasiconvex concave functions and duality in. Limit distribution theory for maximum likelihood estimation of a logconcave density balabdaoui, fadoua, rufibach, kaspar, and wellner, jon a. You can download a program called r and run a demo file, where you can. In what follows, we cover the elements of this theory by focusing on its two main results. The negative of a quasiconvex function is said to be quasiconcave. A sufficient condition for global concavity is that the hessian of f is everywhere negative definite, and this requires det. But such critical points need not exist and even if they do, they are not necessar.
Then any local minimum maximum is akzo a global minimum. We can not show analytically this function is quasiconcave. Global rates of convergence of the mle for multivariate interval censoring gao, fuchang and wellner, jon a. If f is quasiconcave, then its upper level sets are convex sets. On the global minimum of a quasiconcave functional. This paper studies the global behaviour of semistrictly quasiconcave functions with possibly nonconvex domain in the presence of global minima.
A note on the quasiconvex jensen divergences and the quasiconvex. Energy efficient power allocation strategy for 5g carrier. Methods of finding the global minimum of a quasiconcave. Graduate student, professor, faculty of industrial and management engineering.
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. Center of mathematical statistics of the academy, bucharest, romania. If a function is increasing, then is it both quasiconcave. Semistrictly quasiconcave approximation and an application to. If f is globally strictly concave, then a critical point x.
A strictly concave function will have at most one global maximum. 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. Quasiconcave functions can have local maxima which are not global maxima, as easy examples show. Likewise in consumer theory one often assumes that the utility function has these properties. On global minima of semistrictly quasiconcave functions. Functions which are increasing, coradiant and quasiconcave have found many applications in microeconomic analysis. Jensens inequality logarithmically concave function quasiconcave function concavification. Can gradient descent find the maximum of a concave function. Inference for the mode of a logconcave density doss, charles r.
A function is concave if the chord joining any two points on any plane profile of its graph lies everywhere on or. The set argmaxffx jx2dgof maximizers of fon dis either empty or convex. Good news is, though, that typically there will not be too many of those. Monotone linkage clustering and quasiconvex set functions. Any local maximum of a concave function is also a global maximum. Quasi concave and convex best online classes for eco. The class of multiextremal problems in which the minimum of a quasiconcave function in a convex compact set is sought, is considered. We limit ourselves to giving just some chronological references in order to emphasize the. While boosting system throughput, ca has also put forward challenges to the resource allocation problems.
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. Carrier aggregation ca is considered to be a potential technology in next generation wireless communications. Quasiconcave is a mathematical concept that has several applications in economics. Quasiconcave functions defined on semilattices, antimatroids and convex. Proving that a cobbdouglas function is concave if the sum. More significantly, first order conditions are not in general sufficient to identify global of quasi concave or convex optimization problems. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. Another result provides conditions under which a point of constrained global. Pi r, the minimum split function of its maximum join linkage coincides with f assertion 3. The origins of the notion of quasiconcave function are considered, with special interest in some. Global rates of convergence in logconcave density estimation kim, arlene k.
How to prove a function is a quasiconcave function. If f is quasiconcave, then the hessian matrix is negative semi. Global properties theorem if f is concave and x is a local maximizer of f, then x is a global maximizer. A local maximum of a quasiconcave function need not also be a global maximum of the function. Global minimization of the difference of increasing co. One result provides conditions under which a point satisfying the kuhntucker conditions is a point of constrained global maximum. If f is a function of one variable and is singlepeaked, then f is quasiconcave. Mathematics 2 for economics institute for statistics. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. Methods are given for finding the global extremum, consisting of the minimization of linear functions of a specially chosen family. It is shown that the class of weakly nonconstant functions possesses the property that every local maximum is global. A necessary and sufficient condition for a unique maximum.
I if f is a monotonic transformation of a concave function, it is quasiconcave. 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. Maximum likelihood estimation of a logconcave probability density is formulated as a convex optimization problem and shown to. This structures allow building efficient algorithms that find minimal sets on which values of quasiconcave functions are maximum.
Quasi convex and quasi concave functions are used extensively in economics and form a major portion of the the subject mathematical methods in economics. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. 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 much weaker property holds for the linkage functions. For a function of a single variable, along any stretch of the curve the highest. However, quasi concave and quasi convex functions posses enough structure that is important for.
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