MET 503 Applied Optimization

 Instructor:    Mark French

                      138 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 4:30 on the day listed at the homework drop box in Knoy Hall

1/13

Intro to optimization

Lifeguard problem

 

1/15

Binary Search

 

1/17

Sample 1-D Functions

Binary Search

HW #1:

Solve Lifeguard problem using binary search

·         Δx = 30

·         Exit criteria:

o    Change in x* < 1%

o    Change in T* < 1%

·         Repeat the solution with Δx = 18

 

You can do the problems by hand or with MathCad

 

Due:  1/24

 

1/20

MLK Day

1/22

2-D Lifeguard Problem

Monte-Carlo Method

 

1/24

Successive Parabolic Approximations

Exit Criteria

Solving Simultaneous Equations on a TI-89

Solving Matrix Equations on a TI-89

Tutorial on Solving Simultaneous Equations

HW #2:

 

1 - Given the 2-D function:

 

 

 

·         Find the minimum value of this function and its location graphically (4 significant figures)

·         Find the minimum using the minimize function in Mathcad

 

2 – Solve shopping mall problem

·         Graphically (4 significant figures)

Using minimize function in Mathcad

The Shopping Mall Problem

Find parking place that minimizes distance walked between three stores.  You take a package back to the car after visiting the second store.  All distances are in meters.

Due: 1/31

1/27

 

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

 

1/29

Local vs. Global Minima

Marching Grid: 2-D analogy to binary search

 

Two Variable Monte Carlo Example

 

Marching Grid Example

HW #3:

Solve the 2-D Lifeguard Problem using marching grid.

 

·         Starting point:  x0=50m

·         Initial step size: Δx=30m

·         Exit criteria:  

o    Δx < 0.5m

o    Change in T < 0.25%

                   

Due:  2/5

1/31

Marching Grid Class Example

 

2/3

Least Squares Curve Fits

HW #4:

Least Squares Curve Fitting

 

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

                            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: 2/10

2/5

Steepest Descent

Gradient of Objective Function

HW #5:

Three variable function

 

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

·         If 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%

 

Due 2/24

2/7

Steepest Descent Using Binary Search for 1-D Minimization

 

 

2/10

 

HW #6:

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?

 

Due 2/24

 

2/12

Exam 1 Review

 

1-D Review Problems 

 

2/14

Exam 1

 

Spring 2007 Exam 1

Spring 2007 Exam 1 Answer Key

 

Spring 2008 Exam 1

Spring 2008 Exam 1 Answer Key

 

2/17

Conjugate Gradient Method

 

2/19

Conjugate Gradient Method

HW #7: 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.

 

Due: 2/26

 

2/21

Constraints

Exterior Penalty Function

 

2/24

Sample 2-D functions

 

Constrained Optimization Problems

 

Constrained Lifeguard Problem

 

Heaviside Step Function

 

 

2/26

 

No Class

 

2/28

 

 

 

3/3

Structural Optimization

 

Maximizing Volume of a Box

 

 

HW #8:  Constrained Optimization Problems

 

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

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

 

Due:  3/10

 

 

3/5

Exterior Penalty Function

 

 

3/7

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:  3/14

3/10

 

 

3/12

Finite Difference Gradients

HW #10

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

 

Due:  3/14

 

3/14

 

3/17

 

3/19

Spring Break

3/21

 

3/24

Design Variable Linking

Semester Projects

 

 

3/26

Structural Optimization

Finite Element Analysis

 

3/28

Adding buckling constraints

HW #11:

 

Find the minimum weight truss that supports 1000N as shown.  The only design variable is wall thickness, t.  Use any method you wish for the 1-D search.  Plot pseudo-objective function vs. t.  E=70 GPa and σmax = 100 MPa.  The outer diameter of the tubes is 75mm.

 

Use finite elements.  The only constraint is maximum stress

 

Due:  4/4.

 

Due:  4/18

3/31

Literature Review

 

 

4/2

Literature Review

 

4/4

Literature Review

 

 

4/7

 

 

 

4/9

 

 

 

4/11

 

 

4/14

 

 

 

4/16

Structural Optimization

 

 

4/18

Structural Optimization

 

Review of Constraints

HW #12:

 

Find the minimum weight truss that supports 1000N as shown.  Use any method you wish for the 1-D search.  Plot pseudo-objective function vs. t.  E=70 GPa and σmax = 100 MPa.  The outer diameter of the tubes is 75mm.

 

Use finite elements.

 

The design variables are wall thickness, t, and vertical position of the joint between the two elements.

 

Use both stress and buckling constraints.

 

Due:  4/25

 

Due:  4/20

4/21

Interior Penalty Function

4/23

Interior Penalty Function

 

4/25

 

Exam 2 – Spring 2007    Answer Key

Exam 2 – Spring 2008    Answer Key

Exam 2 – Spring 2009    Answer Key

Exam 2 will be a take-home exam.  I will e-mail it to the class on 4/20.  The completed exam will be due at the beginning of class on 4/23

4/28

Project Presentations

 

 

4/30

Project Presentations

 

 

5/2

Project Presentations

 

 

 

 

 

 

 

 

 

 

 

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%