Nonlinear Programming

Self Test Questions

Theory

1. Write short notes on the following:

• Nonlinear programming
• Quadratic programming
• Separable programming

2. Give the mathematical formulation of a general nonlinear programming problem.

Practical

Solve by quadratic programming method

1. Maximize f(x) = 2x₁ + 4x₂ –x₁² – x₂²

subject to
x₁ + 4x₂ £ 5
2x₁ + 3x₂ £ 6

x_1, x₂ ³ 0.

2. Maximize f(x) = x₁ + x₂+ x₃ - 1/2(x₁² + x₂² + x₃²)

subject to
x₁ + x₂ + x₃ £ 1
4x₁ + 2x₂ £ 7/3

x_1, x₂, x₃ ³ 0.

3. Maximize 2x₁x₂+ 4x₂ - x₂² - 2x₁²

subject to
x₁ + 4x₂ £ 12

x_1, x₂ ³ 0.

4. Maximize 6x₁ + 4x₂+ 2x₃ - 3x₁² - 2x₂² - 1/3x₃²

subject to
x₁ + 2x₂ + x₃ £ 4

x_1, x₂, x₃ ³ 0.

Solve the following nonlinear programming problem by separable programming method:

1. Maximize f₀ = 2x₁ – x₁² + x₂

subject to
f₁ = 2x₁² + 3x₂² £ 6
f₂ = x₁ £ 2
f₃ = x₂ £ 2

x_1, x₂ ³ 0.

Take the break points of both x₁ and x₂ as 0, 1, 2, 3, 4.

2. Minimize x₁²- 4x₁ + x₂² - 2x₃

subject to
x₁ + x₂ + x₃ £ 2
(x₁ + 1) x₂ ³ 2

x_1, x_2, x₃ ³ 0.

Take breakpoints of x₁ and x₂ as 0, 1, 2. Keep x3 unchanged.