1. Write short notes on the following:

- Nonlinear programming
- Quadratic programming
- Separable programming

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

Solve by quadratic programming method

**1. Maximize f(x) = 2x _{1} + 4x_{2} –x_{1}^{2} – x_{2}^{2}**

subject to

x_{1} + 4x_{2} ≤ 5

2x_{1} + 3x_{2} ≤ 6

x_{1, }x_{2} ≥ 0.

**2. Maximize f(x) = x _{1} + x_{2}+ x_{3} - 1/2(x_{1}^{2} + x_{2}^{2 }+ x_{3}^{2})**

subject to

x_{1} + x_{2} + x_{3} ≤ 1

4x_{1} + 2x_{2} ≤ 7/3

x_{1, }x_{2}, x_{3} ≥ 0.

**3. Maximize 2x _{1}x_{2}+ 4x_{2} - x_{2}^{2 }- 2x_{1}^{2}**

subject to

x_{1} + 4x_{2} ≤ 12

x_{1, }x_{2} ≥ 0.

**4.** ** Maximize 6x _{1} + 4x_{2}+ 2x_{3} - 3x_{1}^{2} - 2x_{2}^{2 }- 1/3x_{3}^{2}**

subject to

x_{1} + 2x_{2} + x_{3} ≤ 4

x_{1, }x_{2}, x_{3} ≥ 0.

Solve the following nonlinear programming problem by separable programming method:

**1. Maximize f _{0} = 2x_{1} – x_{1}^{2} + x_{2}**

subject to

f_{1} = 2x_{1}_{}^{2} + 3x_{2}^{2} ≤ 6

f_{2} = x_{1} ≤ 2

f_{3} = x_{2} ≤ 2

x_{1, }x_{2} ≥ 0.

Take the break points of both x_{1} and x_{2} as 0,
1, 2, 3, 4.

**2. Minimize ****x _{1}^{2}- 4x_{1} + x_{2}^{2 }- 2x_{3}**

subject to

x_{1} + x_{2} + x_{3} ≤ 2

(x_{1} + 1) x_{2} ≥ 2

x_{1, }x_{2}_{, }x_{3 } ≥ 0.

Take breakpoints of x_{1} and x_{2} as 0, 1, 2. Keep
x3 unchanged.