Exam 9: Nonlinear Optimization Models

In a constrained, nonlinear, optimization problem with a maximize objective function and a single equality constraint, if the value of $\lambda$ corresponding to the optimal solution is:

The local maximum for the function $f(x)=8 X_{3}+2 X_{2}-4 X-3$ is obtained when $X$ is equal to

In order to find whether a local maximum is a global maximum, one must

In a wax business two types are made, jar wax and tube wax. The selling price of jar wax is $\$ 4.00$ per unit and tube wax is $\$ 3.00$ per unit. Let $X_{1}$ be the number of units of jar wax sold and $X_{2}$ the number of units of tube wax sold. Direct costs for each type of wax are $\$ 0.025$ times the square of the number of units sold. Profit contribution for this business is given by

The optimal solution to the problem: Minimize : $f\left(X_{1}, X_{2}\right)=3 X_{1}^{2}-30 X_{1}-2 X_{1} X_{2}+6 X_{2}^{2}-24 X_{2}+236$ is

Suitability of linear approximations for nonlinear problems can be found by analytical methods and does not involve judgment or experience.

In a paper business two types of paper are being made-sponge and cloth. Sponge sells for $\$ 5.00$ per package, and cloth sells for $\$ 3.00$ per package. Let $X_{1}$ is the number of packages of sponge sold and $X_{2}$ be the number of packages of cloth sold. Direct costs are $\$ 0.25$ times the square of the number of units of sponge sold and $\$ 0.05$ times the square of the number of units of cloth sold. Fixed costs for the two paper types together are $\$ 10.00$ for all the volumes contemplated or possible. There are no other costs or revenues. Find the net profit maximizing combination of sponge and cloth to be sold. (It may be assumed that the firm can sell whatever volume they produce.)

For the function $\mathrm{f}(\mathrm{x})=6 \mathrm{X}_{2}+12 \mathrm{X}+30$ , the slope of the function is 0 at $\mathrm{X}$ equal to

In order to find whether a local minimum is a global minimum, one must

Nonlinear optimization models are more popular in usage than linear optimization models because real life applications have more nonlinearity.

In a paper business two types of paper are being made-sponge and cloth. Sponge sells for $\$ 5.00$ per package, and cloth sells for $\$ 3.00$ per package. Let $X_{1}$ be the number of packages of sponge sold and $\mathrm{X}_{2}$ the number of packages of cloth sold. Direct costs are $\$ 0.25$ times the square of the number of units of sponge sold and $\$ 0.05$ times the square of the number of units of cloth sold. Fixed costs for the two paper types together are $\$ 10.00$ for all the volumes contemplated or possible. There are no other costs or revenues. However, all paper rolls have to be labeled by a single labeling machine which can handle at most 50 rolls of both types put together. Find the net profit maximizing combination of sponge and cloth to be sold subject to the constraint. (It may be assumed that the firm can sell whatever volume they produce.)

If a continuous function $\mathrm{f}(\mathrm{x})$ has only one local maximum, then it must also be its global maximum.

A point $\mathrm{x} 1$ is a local minimum of a function $\mathrm{f}(\mathrm{x})$ if the value of $\mathrm{f}(\mathrm{x})$ is more for all points around a fixed distance, usually $\pm 0.1 \mathrm{x}^{1}$ around $\mathrm{x}^{1}$ .

the function $\mathrm{f}$ . $D=\frac{\partial^{2} f}{\partial X_{1}^{2}} \cdot \frac{\partial^{2} f}{\partial X_{2}^{2}}-\left(\frac{\partial^{2} f}{\partial X_{1} \partial X_{2}}\right)^{2}>0$

Like linear programs, nonlinear programs will have a maximization or minimization objective function.

Which of the following assertions concerning the Lagrangian multiplier is true?

Linear models are more popular than nonlinear models because

In an unconstrained two-variable problem with a quadratic objective function, the constant does not have a role in the problem and hence may be dropped.

The optimal solution to the problem: Minimize : $f\left(X_{1}, X_{2}\right)=2 X_{1}^{2}-12 X_{1}+2 X_{1} X_{2}+X_{2}^{2}-4 X_{2}+160$ is

XYZ Inc. manufactures a critical vaccine needed for homeland security, and its cost of production is given by the following function: $C=10 X^{2}-50 X+50$ , where $\mathrm{X}$ is the number of thousands of vaccines produced. The government wants to go in for a cost plus contract and wants XYZ Inc. to produce as many vaccines as possible at the lowest total cost. What is the correct value of $\mathrm{X}$ that would minimize the cost of the contract?