1 Convexity, concavity and quasi-concavity. (SB )

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1 UNIVERSITY OF MARYLAND ECON 600 Summer 2010 Lecture Two: Unconstrained Optimization 1 Convexity, concavity and quasi-concavity. (SB ) For any two points, x, y R n, we can trace out the line of points joining them by varying t [0, 1] as tx + (1 t)y. This is a convex combination of x and y. Definition 1. A subset C R n is convex if for every x, y C and every t [0, 1], tx+(1 t)y C. Definition 2. A function f : R n R is concave if for all x, y R n, f(tx + (1 t)y) tf(x) + (1 t)f(y). A function is strictly concave if the inequality is strict for all x, y. Note that what this says is that if we think of the graph of f in R n+1,, then the line that joins any points (x, f(x)) and (y, f(y)) lies below the graph of the function. See Figure 1. 1

2 Definition 3. A function g : R n R is convex if for all x, y R n, g(tx + (1 t)y) tg(x) + (1 t)f(y). A function g is convex if g is concave. The definitions imply that the graph of a concave function outlines a convex set in R n+1. PS2 asks you to show explicitly that for f concave, {(x, y) y f(x)} is convex. (What is the corresponding result for f convex) Fact 1. If f : R n R and f is C2, then f is concave if and only if f (x) 0 for all x. Using the idea of convex combinations, for any concave function f : R n R and any two points x, y we can create a concave function g : R R in the following manner, g(t) = f(tx + (1 t)y). That is, along any slice in R n a concave function is concave. Suppose that f : R n R is C1. Then for any two points, x, y, f(y) f(x) + f(x)(y x). This can be verbalized by the statement that any concave function lies below its sub-gradient (subgradient is used technically to deal with the possibility that f is not C1). Geometrically, look at the graph of f and pick any point x. Now take the tangent to the graph of f at f(x). The function lies below that tangent line. See Figure 2. Definition 4. Let A be an n n symmetric matrix. We say that A is negative semi-definite if x Ax 0 for all x. If in addition, x Ax = 0 implies that x = 0, then we say A is negative definite. Definition 5. Let A be an n n symmetric matrix. We say that A is positive semi-definite if x Ax 0 for all x. If in addition, x Ax = 0 implies that x = 0, then we say A is positive definite. 2

3 Definition 6. Let A be an n n matrix. The k th order leading principal minor of A is the determinant of the k k matrix obtained by deleting the last n k rows and columns. (This is distinguished from principal minors in that the matrix for a principal minor is obtained by deleting any identical n k rows and columns.) Fact 2. An n n symmetric matrix is positive definite if all its n leading principal minors are (strictly) positive. A is negative definite if its n leading principal minors alternate in sign with a 11 < 0. See Blume and Simon, p. 383 for the relationship between negative semi-definiteness and principal minors. Note that there are 2 n 1 principal minors for a given matrix. Negative semidefiniteness and positive semi-definiteness require checking all of them! Theorem 1. Suppose that f : R n R is C2. Then f is strictly concave if and only if D 2 f(x) which is an n n symmetric matrix is negative definite for all x. f is concave if and only if D 2 f(x) which is an n n symmetric matrix is negative semidefinite for all x. (There are parallel results relating convexity and positive definiteness.) Suppose a function f : R n R is concave. Show that the set of points in R n given by {x f(x) b 0} is a convex set. (See PS2.) Definition 7. Recall the definition of a hyperplane: H a,b = {x a x = b}. A hyperplane defines two half-spaces given by Ha,b u = {x a x b} and Ha,b l = {x a x b}. Note that half-spaces are convex sets. Definition 8. A function f : R n R is quasi-concave if for all c R, the set C c = {x R n f(x) c} is a convex set, i.e., the upper contour sets are convex sets. An alternative definition is f(tx + (1 t)y) min{f(x), f(y)} for all x, y R n and all t 0[0, 1]. f is strictly quasi-concave if f(tx + (1 t)y) = min{f(x), f(y)} for 0 < t < 1 implies x = y. Observe that in R, strictly quasi-concave functions have single MODES or maximums. This makes strictly quasi-concave functions very useful in optimization problems. Definition 9. Suppose f : R n R is C2. Let H (the Bordered Hessian of f) be the n + 1 n + 1 3

4 matrix given by H = 0 x 1 x D 2 f x n x n In general, we do not have a direct relationship between negative definiteness of matrices and quasi-concavity. However, there is a sufficient condition for quasi-concavity as follows: Theorem 2. (SB 21.19) Calculate the leading principal minors of H from k=3 onward. If they alternate in sign with the first leading principal minor 0, for all x, then f is quasi-concave. If they are all negative, the f is quasi-convex. 2 Necessary conditions For an Optimum (SB ) 2.1 Differentiable Functions The simplest form of a non-linear program is the unconstrained problem where the choice set is simply the Euclidean space, R n. This section addresses the following question: If x is a solution 4

5 to the problem, max f(x) x R n what can we say about characteristics of x or the behavior of f at x? Definition 10. A point x R n is a global maximum of a function f if for all x R n, f(x) f(x). Definition 11. Let ɛ > 0. A set B ɛ (x) = {y x y < ɛ} is an open ball around x. Definition 12. A point x R n is a local maximum of a function f if there exists an open ball around x, B ɛ (x) such that for all x B ɛ (x), f(x) f(x). A couple of perhaps obvious points, if x is a global maximum of f it is also a local maximum of f. If x is a local maximum of f it may or may not be a global maximum. Recall that the gradient of f at x can be used to approximate the amount f changes with a small move away from x in direction v. f(x + v) f(x) + v f(x) This observation leads to the next theorem. Theorem 3. Suppose f is C1. If x is a local maximum of f, then f(x) = 0. Proof. Suppose that f(x) is not identically zero. In particular, suppose that f(x) i < 0.(This is WLOG). Consider the vector v = t (0, 0,... 1, 0, 0, 0) where the nonzero term is in the i th component. Then f(x + v) f(x) is approximately t (x) x i > 0 and, of course, we can choose t less than 0 for any ball. This violates the claim that x is a local maximum. (This is not an exact proof because I have not addressed the exact way that the approximation occurs but this is the idea of the proof.) Notice that the theorem illustrates only that the gradient being zero is a NECESSARY and not a SUFFICIENT condition for a local maximum. It is not the only NECESSARY condition for a local maximum. Theorem 4. If x is a local maximum of f, then there is an open ball around x, B ɛ (x), such that f is concave on B ɛ (x). Corollary 1. Suppose f is C2. If x is a local maximum of f, then 2 f(x) (that is, the Hessian of f at x) is negative semi-definite. 5

6 Recalling an earlier remark, since these are necessary conditions for LOCAL maxes, we cannot conclude from them that x is a GLOBAL max. To ensure that it is a global max, we usually take one of two approaches: 1. Solve for all the local maximums, and since we know that at least one of them must be the global maximum, just compare them among each other, provided that a global maximum exists. 2. Check for additional features on f which ensure that if it is a local maximum, it is also a global maximum. (Usually these are conditions that ensure that there exist at most ONE local maximum, (or perhaps a connected set of local maxes) which, therefore implies that if a global max exists, the two coincide.) 2.2 Non-Differentiable Functions The idea of the gradient of a function f can be generalized to the case where f is not differentiable. Definition 13. The superdifferential of a concave function, f at a point x is the set of ALL supporting hyperplanes of the graph of f at the point, (x, f(x ). (Rockafeller, pp ) For convex functions, replace supe with sub. Definition 14. A supergradient of a function, f at a point x is an element of the superdifferential of f at x. (Rockafeller, pp ) Theorem 5. If x is an unconstrained local maximum of a function, f : R n R, then the vector of n zeroes, 0 must be an element of the super-differential of f at x. Note, if f is differentiable at x then the super-differential and the gradient are one and the same (the only element of the super-differential is the gradient of f). This then implies that for differentiable cases, a necessary condition for x to be an unconstrained maximum of f : R n R is that (x ) x i = 0, i. But the next example shows how to use the more general case. Suppose that, x if x 1 f(x) = 2 x if x 1 6

7 This is continuous and obviously reaches its maximum at x = 1. The gradient of f is not defined at x though. The super-differential at x is the set of all hyperplanes that touch the graph at (x, f(x )). It is the set of all lines through (1, 1) with slope between 1 and 1. See the Figure 4 for a few of them. Observe that the line with slope zero is in this set. 3 Sufficient conditions for an Optimum (SB ) Theorem 6. If f : R n R is concave, and C1 then, f(x) = 0 implies that x is a global maximum of f. That is, if f is concave, then f(x) = 0 is both a necessary and a sufficient condition for x to maximize f. 4 Minimizing versus maximizing Theorem 7. Suppose f : R n R. Then x = arg max f(x) iff. x = arg min f(x). x Rn x R n Further Work: Show that the first order necessary condition for a local minimum is the exactly the same. Find corresponding necessary second order conditions and a corresponding condition on f to ensure that f(x) = 0 is both a necessary and a sufficient condition for x to minimize f. 7