Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Optimization techniques, focusing on nonlinear programming. It covers the principles of lagrange multipliers, equality and inequality constrained problems, and applications to linear programming and calculus of variations. Nonlinear programming is a powerful tool for solving complex optimization problems.
Typology: Study notes
1 / 23
This page cannot be seen from the preview
Don't miss anything!
The purpose of this chapter is to provide some examples of optimization where the previously mentioned basic principles can be applied. Each of the following subjects has been a major mathematical study. Only the ideas are outlined in this chapter. More details will be discussed later.
34 Applications
A main tool used in nonlinear programming is the theory of Lagrange multipliers.
36 Applications
¶ Consider the artificial function H : Rn^ × R → R × Rm,
H(x, u) := (f (x) − f (x 0 ) − u, g(x)).
¶ Obviously H(x 0 , 0) = 0. ¶ Note that ∂H ∂x
∣(x 0 ,0)
= [∇f (x 0 ), ∇g 1 (x 0 ),... , ∇gm(x 0 )].
¶ If the columns are linear independent, then by the implicit function theorem there exists a neighborhood of (x 0 , 0) in which H(x, u) ≡ 0. ¶ In particular, there exist x and u < 0 such that f (x) = f (x 0 ) + u < f (x 0 ).
Nonlinear Programming 37
Minimize f (x) subject to g(x) = 0, h(x) ≤ 0 ,
¶ f : D ⊂ Rn^ −→ R. ¶ g : D ⊂ Rn^ −→ Rm, m < n. ¶ h : D ⊂ Rn^ −→ Rp.
¶ ∇f (x 0 ) + λ>∇g(x 0 ) + μ>∇h(x 0 ) = 0. ¶ μ>h(x 0 ) = 0.
. If hj (x 0 ) < 0, then μj = 0. . If μj > 0, then hj (x 0 ) = 0. ¶ μj ≥ 0, j = 1,... , m.
Linear Programming 39
¶ xj = the level of activity j. ¶ cj = the “increase” in profit Z resulted from each unit increase in xj. ¶ bi = the amount of resource i available for allocation. ¶ aij = the amount of resource i consumed by each unit of activity j.
Maximize Z = c 1 x 1 +... cnxn subject to a 11 x 1 + a 12 x 2 +... + a 1 nxn ≤ b 1 a 21 x 1 + a 22 x 2 +... + a 2 nxn ≤ b 2 .. . am 1 x 2 + am 2 x 2 +... amnxn ≤ bm x 1 , x 2 ,... , xn ≥ 0.
40 Applications
¶ The optimal solution occurs necessarily at “corner-point” feasible solutions. ¶ There are only finitely many corner-point feasible solutions. ¶ If a corner-point feasible solution is better (as measured by Z) than all its adjacent corner-point feasible solutions, then it is better than all other corner-point feasible solutions, i.e., it is optimal.
¶ Details of the simplex method will not be reviewed in this note.
42 Applications
Minimize y 1 b 1 +... + ymbm (Total value of resources) subject to y 1 a 11 +... + ymam 1 ≥ c 1 y 1 a 12 +... + ymam 2 ≥ c 2 .. . y 1 a 1 n +... + ymamn ≥ cn y 1 ,... , ym ≥ 0
¶ One unit of activity j will consume, respectively, a 1 j ,... , amj units of resources. ¶ The expense of one unit of activity j, i.e.,
∑m i=1 yiaij^ must generate at least the value cj in profit Z.
Linear Programming 43
¶ The minimal daily requirement of calories and vitamins for an adult are 750 calories and 400 units of vitamins. ¶ There are 5 categories of food to choose from with the following nutrition content:
Calories Vitamins Market Prices A 1 0 2 B 0 1 20 C 1 0 3 D 1 1 11 E 1 1 12
Linear Programming 45
¶ Let xi, i = 1,... , 5, denote the respective amount of food in a diet. ¶ Want to minimize W = 2x 1 + 20x 2 + 3x 3 + 11x 4 + 12x 5 , subject to
x 1 + x 3 + x 4 + x 5 ≥ 750 , x 2 + x 4 + x 5 ≥ 400 , x 1 ,... , x 5 ≥ 0.
¶ W is the total price paid for food.
46 Applications
¶ Let
y 1 = the unit price of calorie, y 2 = the unit price of vitamins.
¶ Want to maximize Z = 750y 1 + 400y 2 , subject to
y 1 ≤ 2 , y 2 ≤ 20 , y 1 + y 2 ≤ 3 , y 1 + y 2 ≤ 11 , y 1 + y 2 ≤ 12 , y 1 , y 2 ≥ 0.
¶ Z is the total price paid for nutrition.
48 Applications
¶ x 1 (t) = rate of production. ¶ x 2 (t) = rate of reinvestment. ¶ x 3 (t) = rate of storage.
¶ x 1 (t) = x 2 (t) + x 3 (t). ¶ The reinvestment increases the production rate via ˙x 1 (t) = x 2 (t) with initial rate x 1 (0) = x 0 > 0. ¶ x 1 (t), x 2 (t), x 3 (t) ≥ 0 for all t ≥ 0.
Maximize
0
x 0 +
∫ (^) t 0 x^2 (τ^ )dτ^ −^ x^2 (t)
dt
subject to x 0 +
∫ (^) t 0 x^2 (τ^ )dτ^ ≥^ x^2 (t) x 2 (t) ≥ 0
¶ This is an infinite-dimensional “linear” programming problem. ¶ The feasible solutions are defined “implicitly” by the constraints. ¶ The problem can be handled by the theory of Lagrange multiplier.
Calculus of Variations 49
The earliest work of optimization, called the calculus of variations, began in 1969 with the Brachistochrone problem. It then developed into the optimal control theory, mainly by the governments for military usage, in the 1950’s. The theory of nonlinear programming, characterized by the use of Lagrange multiplier principles, came into play only in the last forty years.
Calculus of Variations 51
Minimize I(y) =
∫ (^) b a f^ (x,^ y(x),^ y
′(x))dx, subject to y(a) = ya, y(b) = yb,
where f : R × Rn^ × Rn^ −→ R is smooth and y : [a, b] −→ Rn^ is piecewise smooth.
∫ (^) b
a
f (x, y(x, ≤), y′(x, ≤))dx,
¶ y(x, ≤) := y(x) + ≤z(x). ¶ z(x) is piecewise continuous with z(a) = z(b) = 0.
∫ (^) b
a
z>(x)
∂f ∂y
∂f ∂y′
dx ︸ ︷︷ ︸ F ′(0)
∫ (^) b
a
z>(x)
∂^2 f ∂y^2
z(x) + 2z(x)>^
∂^2 f ∂y∂y′^
z′(x) + z′(x)>^
∂^2 f ∂y′∂y′^
z′(x)
dx ︸ ︷︷ ︸ F ′′(0)
52 Applications
¶ Upon integration by parts,
0 = F ′(0) = z(x)>
∫ (^) x
a
∂f ∂y
ds
b
a
∫ (^) b
a
z′(x)>
∫ (^) x
a
∂f ∂y
ds +
∂f ∂y′
dx
for all piecewise continuous function z(x) with zero boundary values. ¶ It follows that at the critical point y 0 , the Euler-Lagrange equation (in integral form) ∂f ∂y′^
(x, y 0 (x), y′ 0 (x)) =
∫ (^) x
a
∂f ∂y
(s, y 0 (s), y′ 0 (s))ds + c
must be satisfied. ¶ Equivalently, the Euler-Lagrange equation in differential form is
∂f ∂y
(x, y 0 (x), y 0 ′(x)) =
d dx
∂f ∂y′^
(x, y 0 (x), y′ 0 (x)).
