Chapter 8 Nonlinear programming Lectures in Mathematical Economics L Cagandahan Abueg De La Salle University School of Economics Extreme value theorem Maxima and minima Definition. Let f: D R D R A point x* in Dis said to be a global maximizer of f iff f( x) f( x*), x D The value f(x*) is called the global maximum of f. n chapter 8: nonlinear programming 1
Maxima and minima Definition. Let f: D R D R A point x # in Dis said to be a global minimizer of f iff f( x) f( x ), x D # The value f(x # ) is called the global minimum of f. n Maxima and minima Definition. Let f: D R D R A point x* in Dis said to be a local maximizer of f iffthere is a neighborhood V r (x*)(with r> 0) such that f( x) f( x*), x D V r ( x*) The value f(x*) is called the local maximum of f. n Maxima and minima Definition. Let f: D R D R A point x # in Dis said to be a local minimizer of f iffthere is a neighborhood V r (x # )(with r> 0) such that f( x) f( x ), x D V r ( x ) # # The value f(x # ) is called the local minimum of f. n chapter 8: nonlinear programming 2
Maxima and minima f(x) a x* c x m x M d b global maximizer local minimizer global minimizer local maximizer Maxima and minima Definition.A maximizer or a minimizer of a function f is in general, referred to as an optimizer(or an optimum point of f). The value of fat the optimum point is called an optimum of f. Suprema and infima Definition. Let S R A number v is said to be an upper boundof S if s v, s S A number w is said to be a lower boundof S if w s, s S chapter 8: nonlinear programming 3
Suprema and infima Definition. Let S R Sis said to be bounded above, if S has an upper bound. Sis said to be bounded below, if Shas a lower bound. Sis said to be bounded, if S has both an upper bound and a lower bound. Sis said to be unbounded, if Shas no upper bound nor a lower bound. Suprema and infima Definition. Let S R If Sis bounded above, then an upper bound u is said to be a supremum(or a least upper bound) of Sif no number smaller than uis an upper bound of S. We denote u: = sups Suprema and infima Definition. Let S R If Sis bounded below, then a lower bound z is said to be an infimum(or a greatest lower bound) of Sif no number bigger than zis a lower bound of S.We denote z: = infs chapter 8: nonlinear programming 4
Suprema and infima lower bounds of S upper bounds of S R w S v inf S sup S Suprema and infima In symbols, we have u = sups { v s v s S} = min :, and z = infs { w s w s S} = max :, Suprema and infima Example. Consider the interval [ 1,2] R We have sup[ 1,2] = 2 and inf[ 1,2] = 1. Now consider the interval ( 1,2) R We have sup( 1,2) = 2 and inf( 1,2) = 1. chapter 8: nonlinear programming 5
Suprema and infima Example. The intervals [ a, ) ( a, ) have infimaequal to a, but no suprema exist. Also, the intervals (, b) (, b] have suprema equal to b, but no infima exist. Suprema and infima Remark. Let S R If Shas a lower bound or an upper bound (or both), it must be finite. Moreover, the supremum or infimum of a set may not be an element of the set. For intervals, the sup and inf are always its finite endpoints. Extreme value theorem Theorem 8.1.[B. Bolzano (1830)] If Iis a closed and bounded interval and f: I R is continuous on I, then fis bounded on I. Moreover, if then M = sup( f x) m = inf f( x) x I x I ( ) x, x I f( x ) = m f( x ) = M m M m M chapter 8: nonlinear programming 6
Extreme value theorem f(x M ) = M f(x) f(x m ) = m a x M x m b Extreme value theorem Theorem 8.2.[K. Weierstrass (1860)] Let Kbe a compact n subset of R. Suppose that f: K R is continuous. Then fhas a maximizer and a minimizer in K. Unconstrained optimization chapter 8: nonlinear programming 7
Critical points Definition. Let f: D R D R A point x 0 in Dis said to be a critical point of f iff f ( x ) = 0 f ( x ) = 0 0 A critical point x 0 in Dthat satisfies f (x 0 ) = 0 is called a stationary point. n Univariate case Theorem 8.3.[Necessary condition for local optima] Let f: D R D R and suppose further that f is differentiable at x*, where x* intd If x* is a local optimizer of f, then f ( x*) = 0 Univariate case Example.Find the stationary point(s) of the function y = exp( x) x + 2 chapter 8: nonlinear programming 8
Univariate case Remark.The converse of Theorem 8.3 is not true. Consider 3 y = x Univariate case Theorem 8.4.[Sufficient condition for local optima] Let f: D R D R and suppose further that f is twice continuously differentiable on intdcontaining x* and that f (x*) = 0. Then Univariate case ( i) f ( x*) > 0 f( x*) < f( x), x V ( *) r x D ( ii) f ( x*) < 0 f( x*) > f( x), x V ( *) r x D chapter 8: nonlinear programming 9
Univariate case Remark.The condition that f( x*) < f( x), x V( x*) D means that x* is a strict local minimizer of f. Similarly, the condition that f( x*) > f( x), x V( x*) D means that x* is a strict local maximizer of f. r r Univariate case Example. Consider the function 2 y = f( x) = ax + bx + c, a 0 Univariate case Example.Consider the cubic polynomial f x x x x 3 2 ( ) = 2 3 12 + 1 chapter 8: nonlinear programming 10
Univariate case Example.Consider the cubic polynomial 1 3 y = x x 3 Univariate case Remark.Consider again the function 3 y = x Univariate case Theorem 8.5.Let f: D R D R and suppose further that f have a continuous n th order derivative on V r (x*) contained in D. Suppose that ( k) f ( x*) = 0, k = 1,..., n 1 ( n) f ( x*) 0 chapter 8: nonlinear programming 11
Univariate case i. If n is even and f (n) (x*) > 0, then x* is a strict local minimizer of f. ii. If n is even and f (n) (x*) < 0, then x* is a strict local maximizer of f. iii. If n is odd, then x* is horizontal inflection point of f. Univariate case Example.Consider the quintic polynomial 1 1 g( x) = x x 5 3 5 3 Univariate case Example.Consider the quartic polynomial h x x x x 4 2 ( ) = 6 8 + 10 chapter 8: nonlinear programming 12
Univariate case Example.Consider a perfectly competitive firm with revenue function R(q) and cost function C(q). The profit function of the firm is given by Π ( q) = R( q) C( q) Univariate case p MC( x) AC( x) q* q Univariate case y Total cost Total revenue MR(q 1 *) = MC(q 1 *) MR(q 2 *) = MC(q 2 *) q 1 * q 2 * q chapter 8: nonlinear programming 13
Univariate case Exercise. Consider the function x 3 f( x) = 2 x 3x + 2 State the necessary restrictions for x(if necessary, since this is a rational function). Determine all critical points. Among those, which are stationary? Univariate case Exercise. Let y = F( z), z = f( x) be differentiable functions and let ybe increasing in z. Prove that x*is a local maximum of fiff x*is a local maximum of F. Review: matrix calculus Definition. Consider the function y = f x = f x x x 1 2 n ( ) (,,..., ) If fis continuously differentiable, then the vector y y y f = x x x 1 2 n is called the gradientof f, read del f. T chapter 8: nonlinear programming 14
Review: matrix calculus Definition. Consider the function y = f x = f x x x 1 2 n ( ) (,,..., ) If f is twice continuously differentiable, then the matrix where H = H( x, x,..., x ) 1 2 n Review: matrix calculus 2 2 2 f f f 2 x x x x x 1 1 2 1 n 2 2 2 f f f 2 H() f = x2 x1 x x 2 2 xn 2 2 2 f f f 2 x x x x n 1 n 2 xn Review: matrix calculus is called the Hessian matrix, and its determinant is the Hessian determinant. We may also denote the Hessian matrix as H() f = H = 2 f f chapter 8: nonlinear programming 15
Multivariate case Theorem 8.6.[Necessary condition for local optima] Let n f: D R D R and suppose further that f have partial derivatives at x*, where x* intd If x* is a local optimizer of f, then f( x*) = θn Multivariate case Definition. A function f(x,y) is said to have a saddle point (x #,y # ) iff f x y f x y f x y # # # # (, ) (, ) (, ), ( x, y) Df Remark.A saddle point is an n dimensional analogue of the horizontal point of inflection. Multivariate case Theorem 8.7.[Sufficient condition for local optima] Let n f: D R D R and suppose further that fhave continuous second order partials on V ( x *) D r and that f( x*) = θn chapter 8: nonlinear programming 16
Multivariate case If 2 f( x*) is positive definite (resp., negative definite, indefinite), then x* is a strict local minimizer (resp., strict local maximizer, saddle point) of f. Multivariate case Example. Consider 2 2 f( x, y) = x y + x y Multivariate case Example.Classify the stationary points of the following functions: 2 2 ( i) f( x, y) = x y + xy ( ) 2 = + 2 + + + ( iii) 2 2 x y h( x, y) = e ii g( x, y) x y xy x 5y chapter 8: nonlinear programming 17
Multivariate case f( x, y) = x 2 y 2 + xy Multivariate case g( x, y) = x 2 + y 2 + xy + x + 5y Multivariate case 2 2 h( x, y) = exp( x y ) chapter 8: nonlinear programming 18
Multivariate case Exercise. Consider the function 2 3 z = f( x, y) = xy + x y xy Show that there are six optimizers for this function: (0,0),(1,0),( 1,0), 5 2 5 2 (0,1),,,, 2 5 2 5 Multivariate case Exercise. Verify that 2 2 2 g( x, y, z) = x + y + z xyz has four saddle points: { ( x*, y*, z *) = (2,2,2),( 2, 2,2), ( 2,2, 2),(2, 2, 2) } Multivariate case Remark.In the case of a function with two variables z = f(x,y), we can restate the sufficiency condition: let 2 f: D R D R and suppose further that have continuous second order partials on V ( x*, y*) D r ( ) chapter 8: nonlinear programming 19
Multivariate case and Define f f ( x*, y*) = 0, ( x*, y*) = 0 x y D( x, y) = f ( x, y) f ( x, y) f ( x, y) xx yy xy assumes that for this class of multivariate functions, Young s Theorem holds (Theorem 3.15) 2 Multivariate case Then ( i) D( x*, y*), f ( x*, y*), f ( x*, y*) > 0 ( ii) D x y > xx yy ( x*, y*) local minimum ( *, *) 0, f ( x*, y*), f ( x*, y*) < 0 xx yy ( x*, y*) local maximum Multivariate case ( iii) D( x*, y*) < 0 ( x*, y*) saddle point ( iv) D( x*, y*) = 0 no conclusion can be made about ( x*, y*) chapter 8: nonlinear programming 20
Multivariate case Example.Consider a perfectly competitive firm with revenue function R(x,y) and cost function C(x,y), where xand y are the two goods produced by the said firm. The firm s objective is to maximize profit. Find the necessary and sufficiency conditions that the firm must satisfy, if x* and y* exist. Multivariate case Solution: Formulate the profit function Π(x,y) as follows: Π ( x, y) = R( x, y) C( x, y) Multivariate case Exercise. Consider the function z = f( x, y) = (1 + y) x + y 3 2 2 defined over the Cartesian plane. Show that fhas a unique local minimum at the origin, but it does not have a global minimum. chapter 8: nonlinear programming 21
Convex sets Definition. A set Cis said to be convexiff x, y C r [0,1] rx + (1 r) y C We call rx+ (1 r)ya convex combinationof xand y. If the above is true for all r in (0,1), we then call Ca strictly convex set. Convex sets convex set the line segment (i.e., the convex combination) is contained in the set, possibly at the border/edge/ boundary of the set) strictly convex set the line segment (i.e., the convex combination) cannot be placed at the boundary of the set (at most only one point of the line segment) chapter 8: nonlinear programming 22
Convex sets Example. Consider the [real] Cartesian plane R 2 {( a, b): a, b R} = Define the following sets: S = [0,1] [0,1] 1 2 2 {(, ): 1} C = x y x + y 1 S 1 is called the unit square and C 1 is called the unit circle. Convex sets y (0,1) y (0,1) (1,1) ( 1,0) (0, 1) (0,0) (1,0) x (0,0) (1,0) x unit circle (C 1 ) unit square(s 1 ) a strictly convex set a convex set Convex sets Example. In microeconomics, the set defined by the budget constraint p x + p y m x y is called the budget set of the consumer, which is a convex set. The property of convexity of the budget set implies perfect divisibilityof x and y: possibility of consuming these in fractional units. chapter 8: nonlinear programming 23
Convex sets Budget set, which is [always] convex Exercise. True or false. i. If Aand Bare convex sets, then both their union and intersection are convex. ii. Let Uand Vbe convex subsets of the reals. Define { u v: u U, v V} U + V = + Then U + Vis convex. iii. Consider the linear programming problem max T z = c x s.t. Ax b, x θn Then the set of feasible [and optimal] solutions is convex. Hint: Consider Exhibit A, Special Cases in 6. chapter 8: nonlinear programming 24
Convex functions Definition. Let f: C R where Cis a convex subset in R. We say that fis a convex function iff x, y C r [0,1] ( (1 r) y) r ( f( x) ) (1 r) ( f( y) ) f rx + + Convex functions Definition. Let f: C R where Cis a convex subset in R. We say that fis a strictly convex function iff x, y C r (0,1) ( (1 r) y) r ( f( x) ) (1 r) ( f( y) ) f rx + < + Convex functions f( x ) 2 f( x) 1 f( rx + (1 r) x ) 1 2 x rx + (1 r) x x 1 1 2 2 chapter 8: nonlinear programming 25
not strictly convex function strictly convex function Concave functions Definition. Let g: C R where Cis a convex subset in R. We say that gis a concave function iff x, y C r [0,1] ( (1 r) y) r ( g( x) ) (1 r) ( g( y) ) g rx + + Concave functions Definition. Let g: C R where Cis a convex subset in R. We say that gis a strictly concave function iff x, y C r (0,1) ( (1 r) y) r ( g( x) ) (1 r) ( g( y) ) g rx + > + chapter 8: nonlinear programming 26
Concave functions f( rx + (1 r) x ) 1 2 f( x) 1 f( x ) 2 rx r x x + (1 ) x 1 1 2 2 strictly concave function not strictly concave function Example. The function exp(x) is convex on the whole real line, and ln(x) is concave on (0, ). y = exp( x) 2-1 -0.5 0.5 1-2 -4 y = ln( x) -6 chapter 8: nonlinear programming 27
Concave functions Remark. In general, the quadratic function y ax 2 = + bx + c a, 0 is concave on R if a < 0, and convex on R if a >0. In particular, the function y = x 2 is convex on the whole real line, while the function y = x 2 + 2xis concave on the real line. 2 2 y = x -3-2 -1 1 2 3 2 y = x + 2x -2-4 Example. The cubic function y = x 3 is concave on (,0] and is convex on [0, ). In the whole of the real line, it is neither concave nor convex. 5 0.02 0.01-0.01-0.02 chapter 8: nonlinear programming 28
Example. The hyperbola y = 1/x is concave on (,0) and is convex on (0, ). In the whole of the real line it is neither concave nor convex. 75 50 25-1 -0.5 0.5 1-25 -50-75 -100 Example. The absolute value function y = x is convex on the whole of R, but, it is both concave on convex on the intervals (,0] and [0, ). 1 0.8 0.6 0.4 0.2-1 -0.5 0.5 1 Exercise. i. Prove: a line [in the Cartesian plane] is both convex and concave. ii. Is the requirement of convexity of domain essential for the class of concave and convex functions? Explain why or why not. chapter 8: nonlinear programming 29
Remark. In either a concave function g(x)or a convex function f(x),graphically, there are no breaks or jumps that occur in the respective graphs of gand f, because the respective domains are convex sets. The nonconvexity of the domainwill not make fa valid convex function f( x ) 2 f( x) 1 x x 1 2 f( x ) 2 f( x) 1 The presence of the jump discontinuity will not make fa valid concave function x x 1 2 chapter 8: nonlinear programming 30
To avoid confusion with the word convex, old terminology used the word convex to denote a convex set and the word concave to denote the following: concave upward (for convex functions), and concave downward (for concave functions). Theorem 8.8. A concave function is continuous in the interior of its domain. Theorem 8.9. A convex function is continuous in the interior of its domain. Remark. If fis concave (resp., convex), then fis convex (resp., concave). Theorem 8.10. Let f(x) be twice continuously differentiable on ain open interval (a,b). i. fis concave on (a,b) iff f (x) 0, for every xin (a,b). ii. fis convex on (a,b) iff f (x) 0, for every xin (a,b). chapter 8: nonlinear programming 31
Theorem 8.11. Let f(x) be twice continuously differentiable on ain open interval (a,b). i. If f (x) < 0, for every xin (a,b), then fis strictly concave on (a,b). ii. If f (x) > 0, for every xin (a,b), then fis strictly convex on (a,b). Remark. The converse of Theorem 8.11 is not true. For example, the function g(x) = x 4 is strictly convex on R. But f (0) = 0 Recall that we had this similar case when we considered the function h(x) = x 3. In here, x* = 0 is a horizontal point of inflection. Thus, the condition that f ( x*) = 0 does not always yield to the conclusion that the stationary point x* = 0 is indeed a horizontal point of inflection. chapter 8: nonlinear programming 32
4 y = x 2 y = x The point x* = 0 is the minimum point of y = x 4, but it does not satisfy the sufficiency condition. (0,0) Theorem 8.12. Let f(x) have continuous second order partial derivatives on an [open] convex n Cof R. i. fis concave on C iffthe Hessian matrix 2 f is negative semidefinite on C. ii. fis convex on C iffthe Hessian matrix 2 f is positive semidefinite on C. Theorem 8.13. Let f(x) have continuous second order partial derivatives on an [open] convex n Cof R. i. If the Hessian matrix 2 f is negative definite x C, then fis strictly concave on C. ii. If the Hessian matrix 2 f is positive definite x C, then fis strictly convex on C. chapter 8: nonlinear programming 33
Example. Consider the function f( x, y) = x y 2 2 f( x, y) = x y 2 2 By the remark after Theorem 8.9, g( x, y): = f( x, y) = x + y 2 2 is a strictly convex function. chapter 8: nonlinear programming 34
g( x, y) = x + y 2 2 Theorem 8.14. Let Cbe convex, f, g: C R, C R If fand gare (strictly) concave on Cthen i. f + gis (strictly) concave on C ii. rfis (strictly) concave on C for every r> 0. iii. sfis (strictly) convex on C for every s< 0. n Example. Consider the functions y = f( x) = x y = g( x) = x 2 and let r = 1, s = 2. 2 chapter 8: nonlinear programming 35
Example. Consider a firm in the perfectly competitive market producing output qwith revenue function R(q) and cost function C(q). The profit function is given by Π ( q) = R( q) C( q) If pis the market price of q, then R( q) = pq Theorem 8.15. Let C be convex, f, g: C R, C R If fand gare (strictly) convex on Cthen i. f + gis (strictly) convex on C ii. rfis (strictly) convex on C for every r> 0. iii. sfis (strictly) concave on C for every s< 0. n Example. Consider the functions y = f x = + 2 ( ) x 1 y = g( x) = (5/2) x and let r = 2, s = 1. chapter 8: nonlinear programming 36
Theorem 8.16. Let Cbe convex, f: C R, C R Let the function g: f( C) R be defined on f(c), and f(c) is convex. If both fand gare (strictly) concave (with respect to domains) with gincreasing on f(c), then g fis (strictly) concave on C. n Theorem 8.17. Let Cbe convex, f: C R, C R Let the function g: f( C) R be defined on f(c), and f(c) is convex. If both fand gare (strictly) convex (with respect to domains) with gincreasing on f(c), then g fis (strictly) convex on C. n Exercise. Consider the function a b z = f( x, y) = Cx y, a, b, C > 0 Show that i. zis concave iff a + b = 1. ii. z is strictly concave iff a + b < 1 iii. z is not concave iff a + b > 1. chapter 8: nonlinear programming 37
Theorem 8.18. Let Cbe an open convex set, f: C R, x* C R where fhave continuous first order partial derivatives on C. If f is (strictly) concave on C, then x* is a (strict) local maximizer of fiff x* is a stationary point of f. n Theorem 8.19. Let Cbe an open convex set, f: C R, x* C R where fhave continuous first order partial derivatives on C. If f is (strictly) convex on C, then x* is a (strict) local minimizer of fiff x* is a stationary point of f. n Theorem 8.20. Every local maximizer of a concave function is a global maximizer. Also, every local minimizer of a convex function is a global minimizer. Theorem 8.21. A local maximizer of a strictly concave function is unique. Also, a local minimizer of a strictly convex function is unique. chapter 8: nonlinear programming 38
A local minimum of a strictly convex function is unique, and also it is the global minimum f( x ) 2 f( x) 1 f( rx + (1 r) x ) 1 2 x rx + (1 r) x x 1 1 2 2 f( rx + (1 r) x ) 1 2 f( x) 1 A local maximum of a strictly concave function is unique, and also it is the global maximum f( x ) 2 (1 ) x rx + r x x 1 1 2 2 Example. Consider w = x + xz y + y + yz + 3z 2 2 2 chapter 8: nonlinear programming 39
Exercise. To make economic sense, what restrictions must be imposed to the parameters a, b, c, and dof the total cost function 3 2 C( q) = aq + bq + cq + d Justify your answer. Optimization Optimization Definition.The general optimization problem, also known as the mathematical programming problem (or a constrained optimization problem), is given by max(ormin) f( x) subject to x X chapter 8: nonlinear programming 40
Optimization where fis a real-valued function and Xis a subset of the Euclidean n space R. We call fthe objective functionand Xis called the feasible setor the feasible region. An element of the feasible set is called a feasible pointor feasible solution. The feasible set is described by equalities or inequalities which we call constraints. Optimization If the objective function and all the constraints are linear, we call the optimization problem a linear programming problem. If either the objective function or at least one of the constraints are nonlinear, we call the optimization problem a nonlinear programming problem. chapter 8: nonlinear programming 41
Definition. Consider the nonlinear programming problem max z = f( x, y) s.t. g( x, y) = b The above problem is also called as a constrained optimization problem. We call z= f(x,y)the objective function and g(x,y) = b the constraint. Geometric interpretation Remark. Consider the constrained problem max z = f ( x, y ) s.t. ax + by = c The constraint restricts the values of fto the points of the constraint; i.e., the intersection of the surface fand the plane ax+ by = c; thereby maximizing fon this plane. Geometric interpretation z f(x,y) x ax + by = c y chapter 8: nonlinear programming 42
Definition. Given the problem max(ormin) z = f( x, y) s.t. g( x, y) = b The Lagrangean function(or simply, Lagrangean) is defined by ( λ, x, y) = f( x, y) + λ b g( x, y) The scalar λis called the Lagrange multiplier. Remark. The gradient of from the above problem is given by ( λ, x, y) λ x(, x, y) / λ = λ y(, x, y) = / x λ λ / y (, x, y) The Hessian matrix of (with arguments suppressed) is given by λλ λx λy 2 = xλ xx xy y λ yx yy called the bordered Hessian. chapter 8: nonlinear programming 43
Theorem 8.22. [Necessary condition] Given the problem max(ormin) z = f( x, y) s.t. g( x, y) = b where fand g have continuous first order partial derivatives. Let (x*,y*) be an optimizer of fon the feasible set and suppose that g x( x, y) 0 g( x, y) = g ( x, y) 0 y Then, there is a scalar λ* such that ( λ*, x*, y*) = θ i.e., (λ*,x*,y*)is a stationary point of. 3 Theorem 8.23. [Sufficient condition] Given the problem max(ormin) z = f( x, y) s.t. g( x, y) = b where fand g have continuous second order partial derivatives. Let (x*,y*) be an optimizer of fon the feasible set and suppose that chapter 8: nonlinear programming 44
g ( x, y) 0 g ( x, y) 0 x and that λ* and x*, y* satisfy 2 ( i) ( λ*, x*, y*) = θ det[ ( λ*, x*, y*)] < 0 2 ( ii) y ( x*, y*)localmin of f det[ ( λ*, x*, y*)] > 0 ( x*, y*)localmax of f 3 Example. Consider the firm s problem max F( K, L) = 5KL s.t. K + L = 10 Example. Examine for maxima or minima: s.t. f( x, y) = kxy x 2 + y 2 = b where band kare positive constants. chapter 8: nonlinear programming 45
Remark. Given the problem max(ormin) f( x, y) subjectto g( x, y) = b It is possible to solve the above by graphical methodor to do a substitution method: to rewrite the constraint and substitute into the objective function as if it were an unconstrained problem. Example. What is the point on the line x+ 2y = 4 that is closest to the origin? l (0,2) P = (0,0) (4,0) chapter 8: nonlinear programming 46
Graphical method: A careful sketching of the objective function and the constraint will yield the point Substitution method: We can rewrite the constraint as x = 4 2y Lagrangean method: Recall that the constrained problem is given by min 2 2 x + y = z s.t. x + 2y = 4 chapter 8: nonlinear programming 47
Exercise. i. What is the maximum product of n positive numbers and whose sum is unity? ii. Show that of all rectangles with a fixed perimeter P, the square will have the largest area. Remark. In the previous slide (exercise 1), if there are two positive numbers, we have the nonlinear programming problem max xy = z s.t. x + y = 1, x, y > 0 Exercise. Consider again the problem max F( K, L) = 5KL s.t. K + L = 10 Verify the solution obtained previously using the Lagrangean method by using the graphical and substitution methods. chapter 8: nonlinear programming 48
Theorem 8.24. Consider the problem max(ormin) z = f( x, y) s.t. g( x, y) = b where fand ghave continuous second order partial derivatives and bis a parameter. Suppose that (x*,y*) solves the problem and λ* is the associated Lagrange multiplier. If f(x*,y*) is a differentiable function of b, then df( x*, y*) = λ* db Theorem 8.25. Consider the problem max(ormin) f( x) = z subject to g( x) = b Suppose that x*solves the problem and f(x*) is a differentiable function of b. Then df( x*) = λ* db chapter 8: nonlinear programming 49
Exercise. Consider the problem min z = f( x, y) = x + y 2 2 3 2 s.t. ( x 1) y = 0 Show that the method of Lagrange multipliers does not work in this case and explain why. Exercise. Examine for optima using the accompanying Lagrangean function: z = x + y 2 2 s.t. xy = 1 Verify the answers using graphical method. Does the substitution method apply? Definition. Consider the problem max f( x) s.t. g( x) = b m < n with Lagrangean k n x R, k = 1,..., m m ( λ, x ) = f( x ) + λ b g( ) k x k k k k= 1 chapter 8: nonlinear programming 50
A point (λ*,x*) is called a saddle pointof iff ( λ*, x) ( λ*, x*) ( λ, x*) Remark. The saddle point of the Lagrangean is defined similarly with that of a function f(x,y); i.e., a point (x #,y # ) is a saddle point of fiff f x y f x y f x y # # # # (, ) (, ) (, ), ( x, y) Df Theorem 8.26. Given the problem max f( x), s.t. g( x) = b, k = 1,..., m k x R where f,g 1,g 2,...,g k are real-valued functions defined on k n X R n m < n If (λ*,x*) is a saddle point of, then g( x*) b = 0, k = 1,..., m k k Remark. More generally, we have the next result, stated in a theorem. chapter 8: nonlinear programming 51
Theorem 8.27. Given the problem max f( x), s.t. g( x) = b, k = 1,..., m k x R where f,g 1,g 2,...,g k are real-valued functions defined on k n X R n m < n If (λ*,x*) is a saddle point of, then x* is a global maximizer of f, subject to the constraints g( x) = b, k = 1,..., m k k Exercise. Consider again the problem max F( K, L) = 5KL s.t. K + L = 10 Verify the stationary point (λ*,x*,y*) = (25,5,5) is a saddle point of the corresponding Lagrangean function. chapter 8: nonlinear programming 52
To end... It is true that a mathematician who is not also something of a poet will never be a perfect mathematician. Karl Weierstrass chapter 8: nonlinear programming 53