Deck 9: Nonlinear Optimization Models

Question 1

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

Answer 1

True

Question 2

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

Answer 2

False

Question 3

Nonlinear models involve more computing burden than linear models for problems of comparable size.

Answer 3

True

Question 4

A necessary condition for a one-variable decision problem with an objective function, which is continuous, is that the first derivative of the function be 0 at a local maximum or minimum point.

Question 5

Necessary and sufficient conditions for the existence of a local maximum in a single- variable, unconstrained, nonlinear optimization problem are that the first derivative be 0 at a point and the second derivative be negative at the same point.

Question 6

Necessary and sufficient conditions for the existence of a local minimum in a single- variable, unconstrained, nonlinear optimization problem are that the second derivative be negative at a point and at the same point the slope of the function be 0 .

Question 7

In general, the values of global maximums of the objective function for unconstrained, nonlinear optimization problems will be greater than all local maximums.

Question 8

The local minimum of a convex function will also be the global minimum.

Question 9

Partial derivatives are used to find extreme values of decision variables in a nonlinear model with 2 or more variables.

Question 10

In single-variable, constrained minimization problems, the optimal solution will always be at a local minimum.

Question 11

In single-variable, unconstrained minimization problems, if there is only one local minimum, then it must be the global minimum.

Question 12

In single-variable, constrained minimization problems, the optimal solution may be at one of the extreme PPints, thatsejs, a point where the function intersects a constraint.

Question 13

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

Question 14

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 $\mathrm{x}^{1}$ .

Question 15

As the number of turning points of a continuous function $\mathrm{f}(\mathrm{x})$ increases, the number of local maximums and minimums will increase.

Question 16

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

Question 17

A quadratic function $\mathrm{f}(\mathrm{x})$ has one local and global maximum or one local and global minimum.

Question 18

Lagrangian method for optimization may be used for even unconstrained two variable optimization problems.

Question 19

Lagrangian method with a single $\lambda$ may be used to find optimal solutions for problems with, at most, two constraints.

Question 20

A two-variable problem with an inequality constraint may, in some instances, be solvable by ignoring the constraint because it may not be binding.

Question 21

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

Question 22

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

Question 23

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

Question 24

If a local optimal solution is found for a two-variable maximization problem, the maximum value of the

Question 25

In an unconstrained two-variable problem with a quadratic objective function, if there is a local optimum, it must also be the global optimum solution.

Question 26

In an unconstrained two-variable problem with a quadratic objective function, there will always be a local optimal solution, though global optimal solution may not be available even in such problems.

Question 27

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.

Question 28

In an unconstrained two-variable problem with a quadratic objective function, the constant affects the value of the objective function corresponding to the optimal solution, if any, but does not affect the optimal value of the variables.

Question 29

In an unconstrained two-variable problem with a quadratic objective function, the saddle point is where one variable reaches a local maximum and the other variable reaches a local minimum.

Question 30

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

A) set the second derivative to 0 and solve for $\mathrm{x}$ and check whether the first derivative is negative at that $\mathrm{x}$
B) check whether the second derivative is negative at the value of $x$ corresponding to the local minimum
C) set the second derivative to 0 and solve for $\mathrm{x}$ and check whether the first derivative is positive at that $\mathrm{x}$
D) check whether the second derivative is positive at the value of $x$ corresponding to the local minimum

Question 31

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

A) set the second derivative to 0 and solve for $x$ and check whether the function is negative at that $x$
B) check whether the second derivative is negative at the value of $x$ corresponding to the local maximum
C) set the second derivative to 0 and solve for $\mathrm{x}$ and check whether the function is positive at that $\mathrm{x}$
D) check whether the second derivative is positive at the value of $x$ corresponding to the local maximum

Question 32

Linear models are more popular than nonlinear models because

A) they are relatively easier to formulate and solve
B) most practical problems are linear
C) they involve proportionate relationship between variables
D) are more accurate than non-linear models

Question 33

Causes of nonlinearity include all of the following except

A) additivity may be absent because mixing one cup each of two ingredients may produce less than 2 cups of products
B) increasing sales may result in less than proportionate total sales revenue because of price breaks
C) doubling of sales volume doubles sales revenue
D) economies produced in consumption of raw materials, such as making a 4-pack, uses less materials than individual wraps

Question 34

Which of the following is true for a strictly convex function?

A) It is easier to plot
B) Local maximum is less than global maximum
C) Local minimum is equal to global minimum
D) Local minimum is greater than global minimum

Question 35

In a small, specialty door-locks business, demand is given by $\mathrm{d}=100-2 \mathrm{p}$ , where $\mathrm{d}$ is the monthly demand in units and $\mathrm{p}$ is the price per unit. Identify the expression that correctly captures

A) total sales revenue
B) It is easier to plot
C) Local maximum is less than global maximum
D) Local minimum is equal to global minimum
E) Local minimum is greater than global minimum

Question 36

The local minimum for the function $f(X)=8 X_{2}-4 X-3$ occurs at $X$ equal to

A) 2
B) 5
C) 0.25
D) 4

Question 37

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

A) 8
B) 4
C) 0.75
D) 0.5

Question 38

Which of the following is true of the function $f(X)=16+4 X-2 X_{2}$ ?

A) It reaches global maximum at $X=4$
B) It reaches global minimum at $X=1$
C) It reaches local maximum at $X=1$
D) It reaches local minimum at $\mathrm{X}=1$

Question 39

The optimal solution to the nonlinear model : Maximize: $f(x)=5 X_{2}-30 X+50$ , subject to $X^{3} 0$ and $X \pounds$ 4 , is given by $X$ equal to

A) 4
B) 0
C) 2
D) 1

Question 40

The optimal solution to the nonlinear model : Minimize: $f(x)=10 X_{2}+40 X+1$ , subject to $X^{3} 3$ , is given by $\mathrm{X}$ equal to

A) -29
B) -39
C) 0
D) 1

Question 41

The optimal solution to the nonlinear model : Maximize: $f(x)=3 X_{3}-15 X_{2}+24 X$ , subject to $X \pounds 2$ , is given by $\mathrm{X}$ equal to

A) 2
B) 1.5
C) 1.333
D) 1.867

Question 42

The optimal solution to the nonlinear model : Minimize: $f(x)=3 X_{3}-20 X_{2}+150$ , subject to $X^{3} 0$ and $X$ $\pounds 5$ , is given by $X$ equal to

A) 0
B) 4.4444
C) 2.828
D) 5

Question 43

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

A) 1
B) -1
C) $5 / 6$
D) $1 / 6$

Question 44

The value of the function $f(x)=4 X_{3}+2 X_{2}-4 X-10$ at its local maximum is

A) -8
B) 8
C) 1
D) -1

Question 45

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

A) 6
B) 12
C) 3
D) -1

Question 46

If the slope of the function $f(x)=4 X_{2}-24 X+32$ is 0 at $X=3$ , then it follows that $X=3$ is a

A) local maximum
B) local minimum
C) global maximum
D) saddle point

Question 47

The necessary condition for optimality in a two-variable unconstrained function is that

A) both partial derivatives must be non-negative
B) at least one of the partial derivatives must be negative
C) the partial derivatives must be equal and negative
D) both partial derivatives must be negative

Question 48

Which of these factors is essential in deciding when to use linear approximation of nonlinear problems?

A) Feasibility
B) Experience
C) Efficiency
D) Performance

Question 49

A point on a complex curve of a two-variable unconstrained function where two partial derivatives are zero is a local maximum if

A) the second partial derivatives and mixed partial derivative are zero
B) the second partial derivative is negative
C) the second partial derivatives are negative, and the mixed partial derivative is positive
D) the second and mixed partial derivatives are positive

Question 50

If the first partial derivative of a Lagrangian function of two decision variables are equal to zero at a point, then the point is

A) a saddle point maximum if the mixed partial is negative
B) a local maximum if the second partials are negative and $\mathrm{D}>0$
C) a global maximum if the second and mixed partials are all positive
D) a global minimum if the second partials are positive and $\mathrm{D}<0$

Question 51

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

A)If it is negative, then a one-unit increase in RHS value will decrease the objective function by the value of the multiplier
B) It has the same interpretation as the dual variables in linear programming
C) It represents the slope of the objective function
D) A and B

Question 52

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

A) $X_{1}=2, X_{2}=0$
B) $X_{1}=4, X_{2}=-4$
C) $X_{1}=4, X_{2}=-2$
D) $X_{1}=-3, X_{2}=4$

Question 53

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

A) $X_{1}=6, X_{2}=-3$
B) $X_{1}=-3, X_{2}=6$
C) $X_{1}=3, X_{2}=-6$
D) $X_{1}=6, X_{2}=3$

Question 54

The Lagrangian function corresponding to the following constrained optimization problem: Maximize: $f\left(X_{1}, X_{2}\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160$ , subject to $X_{1}+X_{2}=10$ is

A) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}\right)$
B) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda .+\left(X_{1}+X_{2}\right)$
C) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}-10\right)$
D) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160-\lambda \cdot\left(X_{1}+X_{2}+10\right)$

Question 55

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:

A) 5 , then the objective function will go up by 5 for each additional unit of the resource represented by RHS
B) 5 , then the objective function will go down by 5 for each additional unit of the resource represented by RHS
C) -5 , then the objective function will go up by 5 for each additional unit of the resource represented by RHS
D) -5 then the objective function will go down by 5 for each additional unit of the resource represented by RHS

Question 56

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

A) $4 X_{1}+3 X_{2}$
B) $4 X_{1}+3 X_{2}-0.025 X_{1}^{2}$
C) $4 X_{1}+3 X_{2}-0.025 X_{1}^{2}-0.025 X_{2}^{2}$
D) $4 X_{1}+3 X_{2}+0.025 X_{1}^{2}+0.025 X_{2}^{2}$

Question 57

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?

Question 58

We hire today a firm specializing in temporary employment opportunities in New Orleans, after the damages of Hurricane Katrina, which has developed a profit function for November 2005 given by $P=-0.25 X^{2}+100 X+5000$ , where $\mathrm{X}$ is the number of employees hired in November 2005 . What should be their target employment for November in order to maximize their profit for November? Their current facilities have sufficient capacity to accommodate, at most, 400 new-hires in November 2005.

Question 59

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.)

Question 60

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.)