MET 503 Applied Optimization – Spring 2017

 Instructor:    Mark French

                      121 Knoy Hall

                      desk:         765-494-7521

                      mobile:      765-714-9382






Homework (10 pts/problem)


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

Week 1


Intro to optimization

Lifeguard problem







Binary Search


Sample 1-D Functions

Binary Search

HW #1:

Solve Lifeguard problem using binary search

·         Δx = 30

·         Exit criteria:

o    Δx < 1m


o    Change in T* < 1%

·         Repeat the solution with Δx = 18


You can do the problems by hand or with Matlab


Due:  1/20


Week 2


MLK Holiday



2-D Lifeguard Problem

Monte-Carlo Method




Successive Parabolic Approximations

Exit Criteria

Solving Simultaneous Equations on a TI-89

Solving Matrix Equations on a TI-89

Tutorial on Solving Simultaneous Equations


Week 3



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

Optimization in Practice: 

Write a two page (with pictures) description of some application of optimization.  Be sure to identify the objective function and the design variables.  Most practical optimization problems have constraints, so be sure to describe these as well.

Due: 1/30



Local vs. Global Minima

Marching Grid: 2-D analogy to binary search


Two Variable Monte Carlo Example


Marching Grid Example





Marching Grid Class Example


Week 4


Least Squares Curve Fits





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)

·         Find minimum by setting grad f = 0 and solving with whatever Matlab function you want to use

·         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



Steepest Descent Using Binary Search for 1-D Minimization



Week 5




HW #3

Several Methods for Single Variable Problems


Find the minimum values of test function 2 on page 197 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: 2/22


Note – This dues date was moved back from 2/17



Exam 1 Review


1-D Review Problems 




Exam 1


Spring 2007 Exam 1

Spring 2007 Exam 1 Answer Key


Spring 2008 Exam 1

Spring 2008 Exam 1 Answer Key


Due 2/18

Week 6


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: 2/27








Week 7


Sample 2-D functions


Constrained Optimization Problems


Constrained Lifeguard Problem


Heaviside Step 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: 3/4









Week 8


Structural Optimization


Maximizing Volume of a Box



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%



Due: 3/11



Exterior Penalty Function






Exterior Penalty Function


Week 9


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


Due: 3/21



Interior Penalty Function







Week 10





Spring Break




Week 11


Conjugate Gradient Method

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 matlab ‘fminunc’ 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: 3/28















Week 12


No Class – Trip to NASA









HW #9: 

Constrained Optimization Problems


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

o    Use steepest descent with Matlab ‘fminunc’ function for 1-D search.  Initial design:  D=100mm, H=100mm

o    Exit criteria is satisfied when change in f* < 1% or change in both design variables is < 1%.



Due: 4/8

Week 13



No Class




HW #10:

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 Exterior Penalty Function (EPF) 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 – when any of these three requirements is satisfied:

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

o    ΔVol < 1%

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


Due:  4/18




Exam 2 Review


Week 14


Exam 2

Exam 2 – Spring 2007    Answer Key

Exam 2 – Spring 2008    Answer Key

Exam 2 – Spring 2009    Answer Key












Week 15



Solving boundary value problems using optimization – The Ritz method







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


Project Presentations





Project Presentations





Project Presentations

Last Day of Class










No Final




Wikipedia Article on Optimization

e-optimization community

Wikipedia Article Least Squares Curve Fits

Wikipedia Article on Steepest Descent (Gradient Descent)



Homework        15%

Exam 1             25%

Exam 2             25%

Project              35%