Optimization using Calculus. Optimization of Functions of Multiple Variables subject to Equality Constraints

Transcription

1 Optimization using Calculus Optimization of Functions of Multiple Variables subject to Equality Constraints 1

2 Objectives Optimization of functions of multiple variables subjected to equality constraints using the method of constrained variation, and the method of Lagrange multipliers.

3 Constrained optimization A function of multiple variables, f(x), is to be optimized subject to one or more equality constraints of many variables. These equality constraints, g j (x), may or may not be linear. The problem statement is as follows: where Maximize (or minimize) f(x), subject to g j (X) = 0, j = 1,,, m X x1 x M x n = 3

4 Constrained optimization (contd.) m n With the condition that ; or else if m > n then the problem becomes an over defined one and there will be no solution. Of the many available methods, the method of constrained variation and the method of using Lagrange multipliers are discussed. 4

5 Solution by method of Constrained Variation For the optimization problem defined above, let us consider a specific case with n = and m = 1 before we proceed to find the necessary and sufficient conditions for a general problem using Lagrange multipliers. The problem statement is as follows: Minimize f(x 1,x ), subject to g(x 1,x ) = 0 For f(x 1,x ) to have a minimum at a point X * = [x 1*,x * ], a necessary condition is that the total derivative of f(x 1,x ) must be zero at [x 1*,x * ]. 5 f f df = dx + dx = x 1 1 x 0 (1)

6 Method of Constrained Variation (contd.) Since g(x 1*,x * ) = 0 at the minimum point, variations dx 1 and dx about the point [x 1*, x * ] must be admissible variations, i.e. the point lies on the constraint: g(x 1* + dx 1, x * + dx ) = 0 () assuming dx 1 and dx are small the Taylor series expansion of this gives us * gx ( * + dx, x + dx) 1 1 g g = gx (, x ) + (x,x ) dx+ (x,x ) dx= 0 * * * * * * x1 x (3) 6

7 Method of Constrained Variation (contd.) g g or dg = dx at [x 1*,x * ] (4) 1+ dx = 0 x x 1 which is the condition that must be satisfied for all admissible variations. Assuming, (4) can be rewritten as g/ x 0 g/ x dx = ( x, x ) dx 1 * * 1 1 g/ x (5) 7

8 Method of Constrained Variation (contd.) (5) indicates that once variation along x 1 (dx 1 ) is chosen arbitrarily, the variation along x (dx ) is decided automatically to satisfy the condition for the admissible variation. Substituting equation (5) in (1) we have: f g/ x1 f (6) df = dx1 = 0 x1 g/ x x * * (x 1, x ) The equation on the left hand side is called the constrained variation of f. Equation (5) has to be satisfied for all dx 1, hence we have 8 f g f g = 0 x x x x 1 1 (x, x ) This gives us the necessary condition to have [x 1*, x * ] as an extreme point (maximum or minimum) * * 1 (7)

9 Solution by method of Lagrange multipliers Continuing with the same specific case of the optimization problem with n = and m = 1 we define a quantity λ, called the Lagrange multiplier as f / x λ = g/ x (8) (x, x ) * * 1 Using this in (5) f x g + λ = 0 x 1 1 (x, x ) * * 1 (9) And (8) written as f x g + λ = 0 x (x, x ) * * 1 (10) 9

10 Solution by method of Lagrange multipliers contd. Also, the constraint equation has to be satisfied at the extreme point gx (, x ) = 0 1 ( x, ) * * 1 x (11) Hence equations (9) to (11) represent the necessary conditions for the point [x 1 *, x *] to be an extreme point. Note that λ could be expressed in terms of g / x1 as well and g / x1 has to be non-zero. Thus, these necessary conditions require that at least one of the partial derivatives of g(x 1, x ) be non-zero at an extreme point. 10

11 Solution by method of Lagrange multipliers contd. The conditions given by equations (9) to (11) can also be generated by constructing a functions L, known as the Lagrangian function, as Lx ( 1, x, λ) = f( x1, x) + λgx ( 1, x) (1) 11 Alternatively, treating L as a function of x 1,x and λ, the necessary conditions for its extremum are given by L f g ( x1, x, λ) = ( x1, x) + λ ( x1, x) = 0 x x x L f g ( x, x, λ) = ( x, x ) + λ ( x, x ) = 0 x x x L ( x 1, x, λ) = g ( x 1, x ) = 0 λ (13)

12 Necessary conditions for a general problem For a general problem with n variables and m equality constraints the problem is defined as shown earlier Maximize (or minimize) f(x), subject to g j (X) = 0, j = 1,,, m where X x1 x M x n = In this case the Lagrange function, L, will have one Lagrange multiplier λ j for each constraint as Lx (, x,..., xλ, λ,..., λ ) = f( X) + λ g( X) + λ g( X) λ g ( X) 1 n, 1 m 1 1 m m (14) 1

13 Necessary conditions for a general problem contd. L is now a function of n + m unknowns, x, and the 1, x,..., xn, λ1, λ,..., λm necessary conditions for the problem defined above are given by L f g = ( ) + ( X) = 0, i = 1,,..., n j = 1,,..., m x x x L λ m j X λ j i i j= 1 i j = g ( X) = 0, j = 1,,..., m j (15) 13 which represent n + m equations in terms of the n + m unknowns, x i and λ j. The solution to this set of equations gives us * * x 1 λ 1 (16) * * x and * λ X = M * x n λ = M * λ m

14 Sufficient conditions for a general problem A sufficient condition for f(x) to have a relative minimum at X * is that each root of the polynomial in Є, defined by the following determinant equation be positive. 14 L L L L g g L g n 11 1 m1 L L L g g g 1 n 1 m M O M M O M L L L L g g L g n1 n nn 1n n mn g g L g 0 L L n g g g 1 n g g L g 0 L L 0 m1 m mn M O M M O M M M = 0 (17)

15 Sufficient conditions for a general problem contd. where L * * Lij = ( X, λ ), for i = 1,,..., n and j = 1,,..., m x x i j g gpq = X p= m q= x p * ( ), where 1,,..., and 1,,..., q n (18) Similarly, a sufficient condition for f(x) to have a relative maximum at X * is that each root of the polynomial in Є, defined by equation (17) be negative. If equation (17), on solving yields roots, some of which are positive and others negative, then the point X * is neither a maximum nor a minimum. 15

16 Example Minimize f ( X) = 3x 6x x 5x + 7x + 5x 1 1 1, Subject to x 1+ x = 5 or 16

17 L x = 6x 10x + 5+ λ = => 3x1+ 5 x = (5 + λ1) 1 => 3( x1+ x) + x = (5 + λ1) x = 1 x = X* =, ; * = [ 3] λ L11 L1 g11 L1 L g1 = g11 g1 0 0

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19 Thank you 19