Strictly concave global maximum download

With concave functions, solving maximization problems is so much easier. It is shown that the class of weakly nonconstant functions possesses the property that every local maximum is global. A note on the maximum of quasiconcave functions springerlink. To download right click on to be added take a look at sheet1. Convex preferences get that name because they make upper contour sets convex. For example, x4 is concave upward but its second derivative equals to 0 when x 0. Sufficient condition for global maximum of strictly quasiconcave functions unconstrained. Thus a local maximum of profit function is global in this case see bellow. Therefore, and the objective function value in of is so, is also the objective function value of in problem.

Biodiversitys contribution to human welfare has become a key argument for maintaining and enhancing biodiversity in managed ecosystems. You should give the formula for your function and then sketch it. Show that the sum of strictly concave function is strictly concave. Global properties theorem if f is concave and x is a local maximizer of f, then x is a global maximizer. Strictly quasiconvex concave functions and duality in. Theorem 1 is a constant function on ucb mathematics. I if f is a monotonic transformation of a concave function, it is quasi concave. It is called strictly concave, if its domain df is an interval and f1. John riley minor corrections 25 july 2016 concave functions in economics 1. Im going to give the argument in two dimensions because its simpler to explain there and the idea is the same in higher dimensions, but you have to d. Why is a maximum of a strictly convex function over a convex. More over, distinction between convexity and quasiconvexity arises from the fact that a strictly convex function cannot be even weakly concave and a strictly concave function cannot be even weakly convex. Explicitly quasiconvex function strictly convex space least squares problem. A sufficient condition for global concavity is that the hessian of f is everywhere negative definite, and this requires det.

Notice this does not guarantee that a solution exists. Then, f is non strictly concave upwarddownward if and. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. Local maximum points of explicitly quasiconvex functions core. Bee1020basicmathematicaleconomics week7,lecturetuesday17.

Prove that a strictly concave function of a strictly concave function maybe strictly convex. If there is a local max then it is the unique global max on the interval where the function. The proofs already given are good, but heres another one that i think will offer some insight. Strictly concave function has a unique maximizer stack exchange. The previous two statements imply that is the unique optimal solution to. And here, if you look at the dotted line, what it tells me is that a concave function, and the property were going to be using is that if a strictly concave function has a maximum, which is not always the case, but if it has a maximum, then it actually must besorry, a local maximum, it must be a global maximum. Karamardian, s strictly quasiconvex concave functions and duality in. But if i think about something like logx, it is strictly concave, but clearly does not have a maximizer.

Sufficient condition for global maximum of strictly quasi concave functions unconstrained. John riley minor corrections 25 july 2016 ucla econ. Quasi concave functions have nice properties for maximization. The functional relationship between biodiversity b and economic value v is, however, insufficiently understood, despite the premise of a positiveconcave bv relationship that dominates scientific and political arenas. Conditions under which stationary point is global optimum. Whenever it exists, the stationary point furnishes the global minimum. Kuhn tucker condition is sufficient for a global optimum. More explicitly, a convex problem is of the form min f x s. This also means that if a monotonic transformation of f is concave, then f is concave. Grenander 1956 establish that the nonparametric maximum likelihood esti mator of f is. I if f is concave, then it is quasi concave, so you might start by checking for concavity.

Convex functions basic properties and examples operations that preserve convexity the conjugate function. Econ 205 slides from lecture joel sobel september 10, 2010. Then any local minimum maximum is akzo a global minimum maximum. This allows us to take first order conditions to obtain that maximum.

On global minima of semistrictly quasiconcave functions. If f is globally strictly concave, then a critical point x. Therefore, optimization methods for ibm model 1 specically, the em algorithm are typically only guaranteed to reach a global maximum of the objective function see the appendix for a simple example contrasting convex and strictly convex functions. Suppose production function fx is concave and the cost function cx is convex. Let 0 be strictly quasiconvex strictly quasi concave on a convex set c c e. A strictly concave function will have at most one global maximum. Then, f is concave upwarddownward if and only if f0is increasingdecreasing. Concave function the function f is concave on if for any and any definition 2. Figure 5 illustrates concave and strictly concave functions. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an.

Throughout this paper represents a finite sequence of real numbers belonging to a fixed closed interval, and is a positive weight sequence associated with if is a convex function on, then the wellknown jensens inequality 1, 2 asserts that. Introduction to convex constrained optimization march 4, 2004. Then any heal minimum maximum is also a global minimum maximum. Stationary point of a strictly concave function if the. Is the function of two strictly concave functions also concave. But such critical points need not exist and even if they do, they are not necessarily maximizers of the function consider fxx3. Convex set convex function strictly convex function concave. Introduction to convex constrained optimization researchgate. Theorem 18 the set of points at which a concave function f attains its maximum over c is a convex set. Sufficient condition for global maximum of strictly quasi. Necessary and sufficient conditions for a maximum 10 5. Then by a previous result, for every point x, no point on the graph of f lies above the tangent to f at x. Since 0 is upper semicontinuous, if and only 0 is lower semicontinuous, it also follows that every strictly quasi concave and upper semicontinuous function is quasi concave.

I if f is a monotonic transformation of a concave function, it is quasiconcave. If is strictly concave, then equality occurs in 1 if and only if there exists. For example, strictly concave quadratic problems with their global. Global optimization for the sum of concaveconvex ratios problem. In the theory of the firm it is almost always postulated that there are gains to input diversification.

Increasing and decreasing functions, maximums and minimums of a function increasing and decreasing functions the functions can be increasing or decreasing along its domain or in a certain interval. Illustration of concave and strictly concave functions. Convex optimization minima and maxima tutorialspoint. I if you are at a maximum, then moving \forward in any direction must not increase the objective function. Then, for all, since, is a strictly concave function, and the maximum of over is attained uniquely at. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Near a local maximum in the interior of the domain of a function, the function must be concave. F is a strict global maximum of nlp if fx fy for all y. Then any local minimum maximum is akzo a global minimum.

A new technique for generating quadratic programming test. On a converse of jensens discrete inequality journal of. Concave function the differentiable function f is concave on x if for any x x x01, and any. Concave functions of two variables while we will not provide a proof here, the following three definitions are equivalent if the function f is differentiable. Let x0 be any local maximum of f, but not a global maximum. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. Necessary and sufficient conditions for a maximum if is a differentiable concave function then the following. Any strictly increasing function is quasiconcave and quasiconvex check this.

On the functional relationship between biodiversity and. Proof a let x0 be any local maximum of f, but not a global maximum. Any local maximum of a concave function is also a global maximum. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.

Concave linear fractional programs share some important properties with concave linear programs, due to the generalized concavityconvexity of the objective function qxfxgx 2. In mathematics, a concave function is the negative of a convex function. You dont need a formal proof, but you do need a convincing argument. Technical note on constant returns to scale production. Global optimization for the sum of concaveconvex ratios. Its a well known result that strictly concave functions have a unique maximizer. This also means that if a monotonic transformation of. Increasing and decreasing functions, maximums and minimums of.

This paper describes a new technique for generating convex, strictly concave and indefinite bilinear or not quadratic programming problems. A note on the maximum of quasiconcave functions springer. Then, f is nonstrictly concave upwarddownward if and. Just as in the one variable case we now show that if f is concave, the foc for a maximum are both necessary and sufficient. Argue graphically that all the required properties are satisfied. Prove that a concave function of a concave function is not necessarily concave. Elsewhere module we have discussed necessary conditions for a maximum for the following. So the global maximizer of the function is a stationary point. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with.

A necessary and sufficient condition for a unique maximum. Let 0 be strictly quasiconvex strictly quasi concave on a convex set c c en. Technical note on constant returns to scale production functions. The set argmaxffx jx2dgof maximizers of fon dis either empty or convex.