MET 503 Applied Optimization – Fall 2015

 Instructor:    Mark French

                      121 Knoy Hall

                      desk:         765-494-7521

                      mobile:      765-714-9382

                      e-mail:        rmfrench@purdue.edu

Syllabus:

 

Date

Topic

Homework (10 pts/problem)

Due

Assignments are due by 5:00 on the day listed, either in class or at my office (121 Knoy)

Week 1

1/11

Intro to optimization

Lifeguard problem

 

 

1/13

Binary Search

 

 

1/15

Sample 1-D Functions

Binary Search

HW #1:

Solve Lifeguard problem using binary search

·         Δx = 30

·         Exit criteria:

o    Change in Δx < 1m

and

o    Change in T* < 1%

·         Repeat the solution with Δx = 18

 

You can do the problems by hand or with MathCad

 

Due: 

 

Week 2

1/18

MLK Holiday

 

1/20

2-D Lifeguard Problem

Monte-Carlo Method

 

 

1/22

Successive Parabolic Approximations

Exit Criteria

Solving Simultaneous Equations on a TI-89

Solving Matrix Equations on a TI-89

Tutorial on Solving Simultaneous Equations

 

Due:

Week 3

1/25

 

Snap-Through Spring problem statement

Snap-Through Spring Solution

 

Physical Problems:

o      Snell’s Law of Refraction

o      Two-Bar Truss Problem

o      Weight on a Spring

o      Maximizing Range of a Projectile

 

 

1/27

Local vs. Global Minima

Marching Grid: 2-D analogy to binary search

 

Two Variable Monte Carlo Example

 

Marching Grid Example

                   

 

Due:

 

1/29

Marching Grid Class Example

 

Week 4

2/1

Least Squares Curve Fits

 

 

 

2/3

Steepest Descent

Gradient of Objective Function

HW #2:

 

1 - Given the two variable function:

 

 

·         Find the minimum value of this function and its location graphically (3 significant figures)

·         Find minimum by setting grad f = 0

·         Find minimum using evolution algorithm with Matlab

·         Find the minimum using the fminunc function in Matlab

 

2 – Solve shopping mall problem

·         Graphically (4 significant figures)

Using minimize function in Matlab

·         Find minimum by setting grad f = 0

·         Find minimum using evolution algorithm with Matlab

·         Find the minimum using the fminunc function in Matlab

 

 

The Shopping Mall Problem

 

Find parking place that minimizes distance walked between four stores.  You take a package back to the car after visiting the third and fourth stores.  All distances are in meters.

Due: 2/10

 

2/5

Steepest Descent Using Binary Search for 1-D Minimization

 

 

Week 5

2/8

 

 

HW #3

Several Methods for Single Variable Problems

 

Find the minimum values of test function 2 on page 109 of the notes

·         Set the derivative equal to zero and solve for x* using ‘fzero’

·         Find the minimum using Matlab function ‘fminunc’

·         Find minimum using sequential quadratic approximation

·         Use Monte Carlo / Quadratic hybrid method

·         Use binary search / quadratic hybrid method

 

Due: 2/17

 

2/10

Exam 1 Review

 

1-D Review Problems 

 

 

2/12

Exam 1

 

Spring 2007 Exam 1

Spring 2007 Exam 1 Answer Key

 

Spring 2008 Exam 1

Spring 2008 Exam 1 Answer Key

 

Week 6

2/15

Conjugate Gradient Method

HW #4:

Solve the Two Variable Lifeguard Problem using marching grid.

 

·         Starting point:  x1=50m, x2=50

·         Initial step size: Δx=30m

·         Exit criteria:  

o    Δx < 0.5m

o    Change in T < 0.5%

 

Due: 2/22

 

2/17

Conjugate Gradient Method

 

 

 

2/19

Constraints

Exterior Penalty Function

HW #5:

Least Squares Curve Fitting

 

1)      Do a parabolic curve fit of the following data using Matlab:

                            x   y

                            0   0

                            1   1

                            2   1

                            3   1

 

Find the curve fit by setting the gradient equal to zero and solving the resulting three equations.

 

2)     Do an exponential curve fit of the following data

                            x   y

                            1   1

                            2   2

                            3   3

                            4   4

 

Find the curve fit by setting the gradient equal to zero and solving the resulting two equations.

 

Note:  exponential curve is

 

Due: 2/26

Week 7

2/22

Sample 2-D functions

 

Constrained Optimization Problems

 

Constrained Lifeguard Problem

 

Heaviside Step Function

HW #6:

Three variable function

 

 

 

·         Starting Point, x=4, y=8, z=4

·         Given Δx = Δy = Δz = 1, do 2 steps of marching grid search

·         Find minimum using steepest descent

o    Use mathcad minimize function for 1-D search

o    Exit criteria (satisfy one)

1.     15 iterations

2.     ΔF < 1%

3.     Δx, Δy and Δz all less than 1%

 

 

2/24

 

 

 

2/26

HW #7:

Rosenbrock Banana Function

·         Starting point x=2, y=2

·         Do four iterations of steepest descent

·         Plot F(d) to find d* for each iteration

·         Plot path through design space

What is the angle between successive search directions

 

 

Week 8

2/29

Structural Optimization

 

Maximizing Volume of a Box

 

 

 

 

 

3/2

Exterior Penalty Function

 

 

 

3/4

HW #8:

Conjugate Gradient Method

 

Estimate the minimum value of the bug splatter function (page 231)

·         Use steepest descent starting at x=4, y=1

·         Use the conjugate gradient method starting at x=4, y=1

 

Use the mathcad minimize function for the 1-D search in both cases.

 

Do six iterations of each method.  Plot the path through design space of each method.  Please plot both on the same axes for comparison.

 

Week 9

3/7

 

 

 

3/9

Interior Penalty Function

HW #9: 

Constrained Optimization Problems

 

·         Minimize the surface area of a cylindrical can that hold 500 mL of liquid using both interior and exterior penalty function methods

o    Use steepest descent with Mathcad minimize function for 1-D search.  Initial design:  D=100mm, H=100mm

 

 

Due:

 

3/11

 

Week 10

3/14

 

 

3/16

Spring Break

 

3/18

 

Week 11

3/21

Design Variable Linking

Semester Projects

Finite Difference Gradients

 

 

 

 

 

3/23

Structural Optimization

Finite Element Analysis

HW #9:

Maximize the volume of a box as shown in class.  The material from which the box is to be cut is 1m wide and 2m long.

 

 

1.     Assume the box is square when viewed from the top( x1 = x2).  Thus, there are two design variables.

2.     Allow all three dimensions of the box to vary, so there will be three design variables.

 

 

·         Use the Interior Penalty Function (IPF) to find the constrained maximum values for volume.

·         Write out the formal problem statement as we did in class.

·         Use binary search for the 1-D search.  Choose an appropriate Δd.

·         Make sure to scale the search direction S using the magnitude of S as shown in class.

·         Choose an appropriate value for R

·         Exit criteria:

o    Δx1, Δx2 and Δx3 are all less than 1%

o    ΔVol < 1%

o    25 iterations.  Every 1-D search counts as an iteration.

 

Due:

 

 

3/25

 

 

Week 12

3/28

.

 

 

3/30

 

 

 

 

4/1

 

 

 

Week 13

4/4

 

 

 

 

4/6

Exam 2 Review

 

 

 

4/8

Exam 2

Exam 2 – Spring 2007    Answer Key

Exam 2 – Spring 2008    Answer Key

Exam 2 – Spring 2009    Answer Key

Week 14

4/11

 

 

 

 

4/13

 

 

 

 

4/15

 

 

 

Week 15

4/18

HW #10

Solve the two variable box problem from HW #8 using finite difference gradients

 

4/20

 

 

4/22

 

HW #11:

 

Boundary Value Problem from Class using cubic polynomial as test function.  a) Use Mathcad minimize function to find unknown parameters.

b) Verify using steepest descent.  You can use either analytical gradients or finite different gradients.  Use Mathcad minimize function for 1-D search

 

 

Week 16

4/25

Project Presentations

 

 

 

4/27

Project Presentations

 

 

 

4/29

Project Presentations

Last Day of Class

 

 

 

 

 

 

 

 

 

No Final

 

 

Links:

Wikipedia Article on Optimization

e-optimization community

Wikipedia Article Least Squares Curve Fits

Wikipedia Article on Steepest Descent (Gradient Descent)

 

Grading

Homework        15%

Exam 1             25%

Exam 2             25%

Project              35%