Necessary and sufficient conditions for a maximum 10 5. Convex preferences get that name because they make upper contour sets convex. John riley minor corrections 25 july 2016 concave functions in economics 1. On the functional relationship between biodiversity and. A new technique for generating quadratic programming test. This also means that if a monotonic transformation of. 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. Increasing and decreasing functions, maximums and minimums of. In the theory of the firm it is almost always postulated that there are gains to input diversification. You dont need a formal proof, but you do need a convincing argument. Karamardian, s strictly quasiconvex concave functions and duality in. Therefore, and the objective function value in of is so, is also the objective function value of in problem. The set argmaxffx jx2dgof maximizers of fon dis either empty or convex. The proofs already given are good, but heres another one that i think will offer some insight.
I if f is concave, then it is quasi concave, so you might start by checking for concavity. Strictly quasiconvex concave functions and duality in. Is the function of two strictly concave functions also concave. If is strictly concave, then equality occurs in 1 if and only if there exists.
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 distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated. The previous two statements imply that is the unique optimal solution to. Technical note on constant returns to scale production. 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. Then, for all, since, is a strictly concave function, and the maximum of over is attained uniquely at. 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. F is a strict global maximum of nlp if fx fy for all y. Let 0 be strictly quasiconvex strictly quasi concave on a convex set c c e. 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. On global minima of semistrictly quasiconcave functions. 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. These problems have a number of properties that make them useful for test purposes.
Any strictly increasing function is quasiconcave and quasiconvex check this. Bee1020basicmathematicaleconomics week7,lecturetuesday17. 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. Illustration of concave and strictly concave functions. Econ 205 slides from lecture joel sobel september 10, 2010. Any local maximum of a concave function is also a global maximum. This allows us to take first order conditions to obtain that maximum. More explicitly, a convex problem is of the form min f x s. Conditions under which stationary point is global optimum. Sufficient condition for global maximum of strictly quasi. Convex functions basic properties and examples operations that preserve convexity the conjugate function.
Introduction to convex constrained optimization researchgate. Suppose production function fx is concave and the cost function cx is convex. Necessary and sufficient conditions for a maximum if is a differentiable concave function then the following. This paper describes a new technique for generating convex, strictly concave and indefinite bilinear or not quadratic programming problems. John riley minor corrections 25 july 2016 ucla econ. Then, f is nonstrictly concave upwarddownward if and. For example, x4 is concave upward but its second derivative equals to 0 when x 0. So the global maximizer of the function is a stationary point. Concave function the differentiable function f is concave on x if for any x x x01, and any. Convex optimization minima and maxima tutorialspoint. I if f is a monotonic transformation of a concave function, it is quasi concave. Near a local maximum in the interior of the domain of a function, the function must be concave. Lastly fc00 and 0 is neither a local maximum nor a local minimum.
Concave function the function f is concave on if for any and any definition 2. Quasi concave functions have nice properties for maximization. Its a well known result that strictly concave functions have a unique maximizer. On a converse of jensens discrete inequality journal of. If f is globally strictly concave, then a critical point x. Strictly concave function has a unique maximizer stack exchange. Then any local minimum maximum is akzo a global minimum maximum. 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. Let x0 be any local maximum of f, but not a global maximum. Global optimization for the sum of concaveconvex ratios problem. 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. A note on the maximum of quasiconcave functions springerlink.
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. 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. Local maximum points of explicitly quasiconvex functions core. Proof a let x0 be any local maximum of f, but not a global maximum.
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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Global properties theorem if f is concave and x is a local maximizer of f, then x is a global maximizer. Introduction to convex constrained optimization march 4, 2004. 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. For example, strictly concave quadratic problems with their global. Theorem 1 is a constant function on ucb mathematics. Theorem 18 the set of points at which a concave function f attains its maximum over c is a convex set. Prove that a concave function of a concave function is not necessarily concave. Then, f is non strictly concave upwarddownward if and. Sufficient condition for global maximum of strictly quasiconcave functions unconstrained.
This also means that if a monotonic transformation of f is concave, then f is concave. A strictly concave function will have at most one global maximum. To download right click on to be added take a look at sheet1. In mathematics, a concave function is the negative of a convex function. I if f is a monotonic transformation of a concave function, it is quasiconcave. If there is a local max then it is the unique global max on the interval where the function. I if you are at a maximum, then moving \forward in any direction must not increase the objective function. Prove that a strictly concave function of a strictly concave function maybe strictly convex. You should give the formula for your function and then sketch it.
Notice this does not guarantee that a solution exists. Whenever it exists, the stationary point furnishes the global minimum. Elsewhere module we have discussed necessary conditions for a maximum for the following. A note on the maximum of quasiconcave functions springer. 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. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an. Grenander 1956 establish that the nonparametric maximum likelihood esti mator of f is. Biodiversitys contribution to human welfare has become a key argument for maintaining and enhancing biodiversity in managed ecosystems. It is shown that the class of weakly nonconstant functions possesses the property that every local maximum is global. Convex set convex function strictly convex function concave. Concave linear fractional programs share some important properties with concave linear programs, due to the generalized concavityconvexity of the objective function qxfxgx 2. Global optimization for the sum of concaveconvex ratios. Concave andquasiconcave functions 1 concaveandconvexfunctions 1.
Figure 5 illustrates concave and strictly concave functions. But such critical points need not exist and even if they do, they are not necessarily maximizers of the function consider fxx3. Stationary point of a strictly concave function if the. A necessary and sufficient condition for a unique maximum. Sufficient condition for global maximum of strictly quasi concave functions unconstrained. But if i think about something like logx, it is strictly concave, but clearly does not have a maximizer. Kuhn tucker condition is sufficient for a global optimum. Then by a previous result, for every point x, no point on the graph of f lies above the tangent to f at x.
With concave functions, solving maximization problems is so much easier. I if f is concave, then it is quasiconcave, so you might start by checking for concavity. It is called strictly concave, if its domain df is an interval and f1. Let 0 be strictly quasiconvex strictly quasi concave on a convex set c c en. Then any local minimum maximum is akzo a global minimum. Thus a local maximum of profit function is global in this case see bellow.195 458 997 228 1518 8 463 50 1004 527 861 339 300 30 1342 591 604 460 1319 478 813 1470 863 189 549 508 1194 402 1000 892 1096 861 967 1094 866 601 212 1465 633 129 281 1245 146 1369 445 1024 985