Key points. Assume (except for point 4) f : R n R is twice continuously differentiable. strictly above the graph of f except at x

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1 Key points Assume (except for point 4) f : R n R is twice continuously differentiable 1 If Hf is neg def at x, then f attains a strict local max at x iff f(x) = 0 In (1), replace Hf(x) negative definite by Hf( ) negative (semi) definite : replace local maximum with (weak) global maximum global negative semi-definiteness buys you a weak global max; local negative semi.definiteness buys you nothing 2 If f is concave and C 2 then Hf is globally negative semi-definite 3 If Hf(x) is negative definite then the tangent plane to f at x is locally strictly above the graph of f except at x In (3), replace Hf(x) neg def with Hf( ) neg (semi) def: replace is locally with is globally (weakly) 4 f is quasi-concave upper quasi-convex if all of its lower contour sets are convex sets for strict quasi-concavity, replace convex with strictly convex and add: no flat spots 5 A sufficient condition for f to be quasi-concave strictly quasi-concave is that at each x neg semi def Hf(x) is globally neg def on the subspace orthogonal to f(x). () September 23, / 17

2 First order necessary conditions for a max Theorem: If f : R n R is C 2 and Hf( x) is neg def, then f attains a strict local maximum at x R n iff f( x) = 0. Which step of the proof below is overly restrictive i.e., a weaker statement would suffice? Necessity: Suppose f( x) 0; will show f( ) not maxed at x Let dx = ε f( x). f( x + dx) f( x) = f( x)dx + a remainder term If ε small enough, Local Taylor applies, expansion term dominates. f( x)dx > 0. f( x + dx) f( x) > 0. f not locally maximized at x. () September 23, / 17

3 Second order sufficiency conditions for a local max Theorem: If f : R n R is C 2 & Hf( x) is neg def, then f attains a strict local maximum at x R n iff f( x) = 0. Sufficiency: Suppose Hf( x) is negative definite Pick ε > 0 s.t. 2nd order Local Taylor applies if 0 < dx < ε. How do I know that such an ε > 0 exists? See notes for details. f( x + dx) f( x) = f( x)dx + 0.5dxHf( x)dx + a remainder term f( x + dx) f( x) = 0.5dxHf( x)dx + a remainder term (because Local Taylor applies,) 0.5dxHf( x)dx < 0 term dominates f( x + dx) f( x) < 0. f( x) > f( x + dx) for all dx s.t. 0 < dx < ε. f attains a strict local max at x. NOT TRUE: If f : R n R & f is C 2, then f attains attains a strict global maximum at x R n iff f( x) = 0 and Hf( x) is neg def. () September 23, / 17

4 Second order sufficiency conditions for a global max Theorem: If f : R n R is C 2 and Hf( ) is neg (semi) def, then f attains a strict (weak) global maximum at x R n iff f( x) = 0. Sufficiency: Suppose Hf( ) is globally negative (semi) definite Use global Taylor Pick dx arbitrarily, λ [0,1] s.t. f( x + dx) f( x) = f( x)dx + 0.5dxHf( x + λdx)dx = 0.5dxHf( x + λdx)dx < ( )0. f is globally (weakly) maximized at x. global negative semi-definiteness buys you a weak global property; local semi-definiteness buys you nothing () September 23, / 17

5 Local Negative definiteness and the tangent plane Theorem: If f : R n R is C 2 & Hf( x) is neg def, then ε > 0 s.t. tangent plane to f at x lies above graph of f on the ε-ball around x. Proof: Use Global Taylor since f is continuously diff able ε > 0 s.t. dx s.t. dx < ε, Hf( x + dx) is neg. def. λ [0,1] s.t. f( x + dx) f( x) = f( x)dx + 0.5dxHf( x + λdx)dx ( ) f( x + dx) f( x) + f( x)dx = 0.5dx Hf( x + λdx)dx }{{}}{{} the height of f at x + dx the height of the tangent plane at x + dx Since λdx < ε, Hf( x + λdx) is neg def graph of f lies below tangent plane on ε-ball around x () September 23, / 17

6 Global Negative (semi) definiteness and the tangent plane Theorem: If f : R n R if C 2 and Hf( ) is neg (semi) def., then tangent plane to f at x lies (weakly) above the graph of f. Proof: Use Global Taylor Pick dx arbitrarily, λ [0,1] s.t. Hf( x + λdx) is neg (semi) def and f( x + dx) f( x) = f( x)dx + 0.5dxHf( x + λdx)dx ( ) f( x + dx) f( x) + f( x)dx = 0.5dx Hf( x + λdx)dx }{{}}{{} the height of f at x + dx the height of the tangent plane at x + dx graph of f lies everywhere (weakly) below tangent plane global negative semi-definiteness buys you a weak global property; local semi-definiteness buys you nothing () September 23, / 17

7 1 Quasi-concavity in one dimension y y y x x x Concave Quasi-concave Not quasi-concave () September 23, / 17

8 Concavity vs Quasi-concavity: concave () September 23, / 17

9 Concavity vs Quasi-concavity: quasi-concave () September 23, / 17

10 Concavity vs Quasi-concavity: cross-sections Concave X section: orthogonal to gradient Convex X section: collinear with gradient () September 23, / 17

11 Quasi-concavity and the tangent plane Definition: f : X R is quasi-concave if all of its upper contour sets are convex X is convex if x,y X, λ (0,1), λx+(1 λ)y X. f is quasi-concave if x,y s.t. f(y) f(x), λ (0,1), f(λx+(1 λ)y) f(x). Theorem: f is quasi concave iff tangent planes in domain lie below level sets. Definition: f is strictly quasi-concave (sqc) if all of its upper contour sets are strictly convex and if there is no open nbd of X on which f is constant a closed set X is strictly convex if x,y X, λ (0,1), λx+(1 λ)y int(x). f is sqc if x,y s.t. f(y) f(x), λ (0,1), f(λx+(1 λ)y)>f(x). Thing on next slide has strictly convex upper contour sets, but isn t sqc local non-satiation implies there s no open nbd of X on which f is constant Theorem: A sufficient condition for f : R n R, f is C 2, to be strictly quasi-concave is that for all x, Hf(x) is negative definite on the subspace of R n which is orthogonal to the gradient of f, i.e., for all x and all dx 0 such that f(x) dx = 0,dx Hf(x)dx < 0. why is this condition sufficient but not necessary? () September 23, / 17

12 Metcalf s Giant Screw-like Ziggurat Thing () September 23, / 17

13 Definiteness on a subspace and Local/Global Taylor Assume for x R n, f(x)dx = 0 (i.e., dx lives in tangent plane) implies dxhf(x)dx < 0 (suff. cond. for strict quasi-concavity) Local Taylor now implies a local relationship b/n level set & tangent plane = for dx in tangent plane, f(x + dx) f(x) 0.5dxHf(x)dx }{{} < 0 = locally, tangent plane is below the level set Local Taylor f on left is s. quasi-concave ; f in right panel is s quasi-convex x2 x2 f(x+dx) f(x)+0.5dx Hf(x)dx<f(x) f(x+dx) f(x)+0.5dx Hf(x)dx>f(x) f(x) f(x) x x + dx x x + dx subspace f(x) subspace f(x) dx x1 dx x1 But this local analysis isn t enough to get us to convex upper contour sets Why not? Global taylor is completely useless for this particular job () September 23, / 17

14 x x + dx Obviously f(x + dx) > f(x): why doesn t the following reasoning apply? Since f(x)dx = 0, λ [0, 1] s.t. f(x + dx) f(x) = 0.5dxHf(x + λdx)dx < 0 Global Taylor does not imply: dx 0 s.t. f(x) dx = 0,f(x + dx) = f(x) + 0.5dxHf(x + λdx)dx < f(x) In general, dx will not be orthogonal to (x + λdx) () September 23, / 17

15 The answer: this f not sqc z = argmin{f(x + αdx) : α [0, 1]} x f(z) x + dx f( ) attains an interior min on {x + αdx : α [0, 1]} (Why interior?) Hf( ) is necessarily quasi-convex at z Conclude: upper contour sets of f not convex implies f not globally strictly q-concave () September 23, / 17

16 Strict quasi-concavity, convexity, local and global maxima Theorem: If f : X R is strictly quasi-concave, X is convex, and f is locally maximized at x, then f is globally maximized at x. Proof: Fix x,y X such that f(y) > f(x), so that x is not a global max on X. we will show that x is not a local max on X. Upper contour set of f corresponding to f(x) is a convex set (s.q.c.); X is a convex set (by assumption); Z = X ( upper contour set of f corresponding to f(x) ) is convex The intersection of convex sets is convex line segment L joining y and x belongs to Z. for z x on L, f(z) > f(x) (property of s.q.c) every neighborhood of x contains a bit of L x isn t a local max on f on X. Theorem would be false if f were just quasi-concave Ziggurats have many local maxes that aren t global maxes Where does above proof break down? () September 23, / 17

17 A Ziggurat: local maxes that aren t global maxes () September 23, / 17