Questions from the course textbook by Rao 4th Edition (freely accessible online at UW Library Website):

Problem 2.23

Problem 2.48- parts (a), (c)

Problem 2.50

Problem 6.23

Problem 6.47

Questions from the course textbook by Belegundu & Chandrupatla 2nd Edition (freely accessible online at UW Library Website):

Problem 3.2-part (iii)

A question from the course textbook by Arora 4th Edition (freely accessible online at UW Library Website):

Problem 4.140

Problem 8.52

–

Important Notes:

• The written answers, calculations, etc. must be your own work, prepared by each student individually.

• The use of Matlab, Python, or similar packages (by permission) is allowed, but related codes/files should be submitted.

• The submission deadline is December 7, 2020 and it will be through a Dropbox on LEARN.

Problems 109

• Determine whether the following matrix is positive definite:
• The potential energy of the two-bar truss shown in Fig. 2.11 is given by

where E is Youngs modulus, A the cross-sectional area of each member, l the span of the truss, s the length of each member, h the height of the truss, P the applied load, the angle at which the load is applied, and x₁ and x₂ are, respectively, the horizontal and vertical displacements of the free node. Find the values of x₁ and x₂ that minimize the potential energy when E = 207 10⁹ Pa, A = 10⁵ m²,l = 1.5 m, h = 4.0 m, P = 10⁴ N, and = 30.

• The profit per acre of a farm is given by

where x₁ and x₂ denote, respectively, the labor cost and the fertilizer cost. Find the values of x₁ and x₂ to maximize the profit.

• The temperatures measured at various points inside a heated wall are as follows:

Distance from the heated surface as

a percentage of wall thickness, d 0 25 50 75 100

Temperature, t(C) 380 200 100 20 0

It is decided to approximate this table by a linear equation (graph) of the form t = a + bd , where a and b are constants. Find the values of the constants a and b that minimize the sum of the squares of all differences between the graph values and the tabulated values.

Figure 2.11 Two-bar truss.

112 Classical Optimization Techniques

2.42 Find the dimensions of an open rectangular box of volume V for which the amount of material required for manufacture (surface area) is a minimum.

2.43 A rectangular sheet of metal with sides a and b has four equal square portions (of side d) removed at the corners, and the sides are then turned up so as to form an open rectangular box. Find the depth of the box that maximizes the volume.

2.44 Show that the cone of the greatest volume that can be inscribed in a given sphere has an altitude equal to two-thirds of the diameter of the sphere. Also prove that the curved surface of the cone is a maximum for the same value of the altitude.

2.45 Prove Theorem 2.6.

2.46 A log of length l is in the form of a frustum of a cone whose ends have radii a and b(a >b). It is required to cut from it a beam of uniform square section. Prove that the beam of greatest volume that can be cut has a length of al/[3(a b)].

2.47 It has been decided to leave a margin of 30mm at the top and 20mm each at the left side, right side, and the bottom on the printed page of a book. If the area of the page is specified as 5 10⁴ mm², determine the dimensions of a page that provide the largest printed area.

2.48

subject to

Minimize f = 9 8x1 6x2 4x3 + 2x12+ 222 + x32 + 2x1x2 + 2x1x3x1 + x2 + 2x3 = 3

by (a) direct substitution, (b) constrained variation, and (c) Lagrange multiplier method.

2.49 Minimize

subject to

g₁(X) = x₁ x₂ = 0 g₂(X) = x₁ + x₂ + x₃ 1 = 0

by (a) direct substitution, (b) constrained variation, and (c) Lagrange multiplier method.

2.50 Find the values of x,y, and z that maximize the function

when x,y, and z are restricted by the relation xyz = 16.

2.51 A tent on a square base of side 2a consists of four vertical sides of height b surmounted by a regular pyramid of height h. If the volume enclosed by the tent is V , show that the area of canvas in the tent can be expressed as

Also show that the least area of the canvas corresponding to a given volume V , if a and h can both vary, is given by

5h

a = and h = 2b

2

Problems 375

6.15 Find a suitable transformation or scaling of variables to reduce the condition number of the Hessian matrix of the following function to one:

6.16 Determine whether the following vectors serve as conjugate directions for minimizing the function

6.17 Consider the problem: Minimize

Find the solution of this problem in the range 10 x_i 10, i = 1,2, using the random jumping method. Use a maximum of 10,000 function evaluations.

6.18 Consider the problem: Minimize

Find the minimum of this function in the range 5 x_i 5, i = 1,2, using the random walk method with direction exploitation.

6.19 Find the condition number of each matrix.

6.20 Perform two iterations of the Newtons method to minimize the function

from the starting point

6.21 Perform two iterations of univariate method to minimize the function given in Problem 6.20 from the stated starting vector.

6.22 Perform four iterations of Powells method to minimize the function given in Problem

6.20 from the stated starting point.

6.23 Perform two iterations of the steepest descent method to minimize the function given in Problem 6.20 from the stated starting point.

6.24 Perform two iterations of the FletcherReeves method to minimize the function given in Problem 6.20 from the stated starting point.

6.25 Perform two iterations of the DFP method to minimize the function given in Problem

6.20 from the stated starting vector.

6.26 Perform two iterations of the BFGS method to minimize the function given in Problem 6.20 from the indicated starting point.

378 Nonlinear Programming II: Unconstrained Optimization Techniques

6.45 Minimize 5x₁x₂ 8x₁ starting from point (0, 0) using Powells method.

Perform four iterations.

6.46 Minimize 1 by the simplex method. Perform two steps of reflection, expansion, and/or contraction.

6.47 Solve the following system of equations using Newtons method of unconstrained minimization with the starting point

X

0

2x₁ x₂ + x₃ =1, x₁ + 2x₂ = 0, 3x₁ + x₂ + 2x₃ = 3

6.48 It is desired to solve the following set of equations using an unconstrained optimization method:

x² + y² = 2, 10x² 10y 5x + 1 = 0

Formulate the corresponding problem and complete two iterations of optimization using the DFP method starting from X.

6.49 Solve Problem 6.48 using the BFGS method (two iterations only).

6.50 The following nonlinear equations are to be solved using an unconstrained optimization method:

2xy = 3, x² y = 2

Complete two one-dimensional minimization steps using the univariate method starting from the origin.

6.51 Consider the two equations

7x³ 10x y = 1, 8y³ 11y + x = 1

Formulate the problem as an unconstrained optimization problem and complete two steps of the FletcherReeves method starting from the origin.

6.52 Solve the equations 5x₁ + 3x₂ = 1 and 4x₁ 7x₂ = 76 using the BFGS method with the starting point (0, 0).

6.53 Indicate the number of one-dimensional steps required for the minimization of the function 5 according to each scheme:

• Steepest descent method
• FletcherReeves method
• DFP method
• Newtons method
• Powells method
• Random search method
• BFGS method
• Univariate method

124 Unconstrained Optimization

For instance, we can supply the gradient in the user subroutine and avoid the possibly expensive automatic divided difference scheme (the default) by switching on the corresponding feature as

options=optimset(GradObj, on)

fminunc is then executed using the command

[Xopt,fopt,iflag,output] = fminunc(testfun, X, options)

with a subroutine getfun that provides the analytical gradient in the vector DF as

function [f, Df] = getfun(X)

f =

Df(1) = ; Df(2) = ; Df(N) = ;

COMPUTERPROGRAMS

STEEPEST, FLREEV, DFP

PROBLEMS

P3.1. Plot contours of the function f 8, in the range 0 < x₁ < 3,0 < x₂ < 10. You may use Matlab or equivalent program.

P3.2. For the functions given in the following, determine (a) all stationary points and (b) check whether the stationary points that you have obtained are strict local minima, using the sufficiency conditions:

100

• f = 3x₁ + + 5x₂

x1x2

• f = (x₁1)² + x₁x₂ + (x₂1)²
• f = x¹ + ^x2

P3.3. (a) What is meant by a descent direction? (Answer this using an inequality.)

(b) If d is a solution of W d =f, then state a sufficient condition on W that guarantees that d is a descent direction. Justify/prove your statement.

exeRCises fOR ChapteR 4 205

4.125 exercise 4.72

4.126 exercise 4.73

4.127 exercise 4.74

4.128 exercise 4.75

4.129 exercise 4.76

4.130 exercise 4.77

4.131 exercise 4.78

Section 4.8 Global Optimality

4.132 Answer true or false.

1. a linear inequality constraint always defines a convex feasible
2. a linear equality constraint always defines a convex feasible
3. a nonlinear equality constraint cannot give a convex feasible
4. a function is convex if and only if its hessian is positive definite
5. an optimum design problem is convex if all constraints are linear and the cost function is
6. a convex programming problem always has an optimum
7. an optimum solution for a convex programming problem is always
8. a nonconvex programming problem cannot have global optimum
9. For a convex design problem, the hessian of the cost function must be positive semidefinite

• Checking for the convexity of a function can actually identify a domain over which the function may be
  • Using the definition of segment given in (4.71), show that the following set is convexS
  • Find the domain for which the following functions are convex: (1) sin x, (2) cos x.

Check for convexity of the following functions. If the function is not convex everywhere, then determine the domain (feasible set S) over which the function is convex.

• f (x₁, x₂) = 3x₁² + 2x₁x₂ + 2x²₂ + 7
• f (x₁, x₂) = x₁² + 4x₁x₂ + x₂² + 3
• f x( ₁, x₂) = x₁³ +12x x_{1 2}² + 2x₂² + 5x₁² + 3x₂

1 ^{2 2} + 1 x₂²

4.138 f x( ₁, x₂) = 5x₁ x x_{1 2}

16 4x₁

• f (x₁, x₂) = x²₁ + x₁x₂ + x²₂
• U = (21.9V C2107) +(3.9106)C +1000 V
• Consider the problem of designing the can formulated in Section 2. Check convexity of the problem. Solve the problem graphically and check the KKT conditions at the solution point.

Formulate and check convexity of the following problems; solve the problems graphically and verify the KKT conditions at the solution point.

• exercise 1

1. The BaSIC COnCepTS

380 8. Linear PrOgraMMing MethODs fOr OPtiMuM Design

8.45 Minimize f subject

x1 x2

x₁, x₂ 0

8.46 Minimize f = x₁ x₂

8.47 Maximize z subject

x₁ x₂

x x

8.48 Maximize

8.49

8.50

8.51 Minimize f = 2x₁ x₂

8.52 Maximize

8.53 Maximize z

1 2