A very effective way to study the ideas and methods
in this book in a university course environment is
through student term projects. Several exercises are
aimed at such project work, and the checklists in
Chapter 8 should prove useful. Experienced designers
will also find this material helpful in their early
attempts at applying the ideas of the book in their
A term project should be undertaken as early in the semester as possible.
After studying the first two chapters, a student should
be able to formulate the optimization problem and
develop an initial mathematical model. As progress
is achieved through the book, the various methods
and ideas can be applied to modify, simplify, and
eventually solve the optimization problem. Students can use the available optimization codes as "black boxes" to obtain some early results. As their understaning of the algorithms increases they will be able to use the available codes more effectively, perform diagnostics, and interprete the numerical results properly.
Typical project milestones are: a project proposal
that contains the description on the design optimization
trade-offs and initial mathematical model; a progress
report that outlines efforts to analyze and simplify
the model based primarily on the material in Chapters
3 through 6; and a final report that contains the
final model statement, model reduction, numerical
solutions, parametric studies, and interpretation
of results based on material throughout the book.
Examples of course deliverables for such reports are
Project topics may be assigned by the instrutor or chosen by the students.
Both approaches have merit and in a typical class
a mixture is usually required. For the students who
do not have a project idea of their own, topics may
be selected from journal articles published in the
engineering literature. However, students should be
strongly encouraged to take full responsibility in
accepting a published model or problem. This usually
forces a more than perfunctory study of the problem
at hand and a familiarization with the model, its
sources, and limitations, which is necessary for an
eventually successful optimization study.
The project archive in the present website
is a valuable resource for project work. Instructor can obtain access information for their students by contacting the first author.
Modern mathematical software that combine modeling
and symbolic and numerical computation capabilities
are dramatically increasing the scope and ease of
formulating and solving optimal design problems. This
book offers many opportunities for the inspired reader
to implement or test the ideas and methods presented
using such software. These efforts are strongly recommended.
PROJECT PROPOSAL GUIDELINES
here for PDF] (43.3 KByte)
team will submit a single proposal about the system
to be designed. Each team member will be responsible
for an individual subsystem and the team as a whole
will study how the subsystem designs must be coordinated
to achieve an overall system optimum. Clearly, the
team must work together from the beginning but the
idea is to assume the viewpoint of individual designers
working concurrently on their portion of a larger
subsystem design problem should have at least 4-5
variables and about twice as many constraints. The
overall system should comprise at least two subsystems,
which will likely share some common variables.
student will be graded separately for their individual
project proposal must be formulated to have the
is a 200-word description of the design project,
the motivation for performing an optimization study,
and the anticipated results.
section introduces a qualitative statement of the
system design project. Describe the system design
problem, the anticipated trade-offs that motivate
the optimization study, and the previous work that
has been done by others. Identify the individual
subsystems and explain qualitatively how they are
linked. Specifically, explain if you expect that
improving the design of each subsystem independently
may not lead you to an overall optimal system design.
each subsystem identified in the introduction you
must develop the full analytical model, as described
in detail below. Each subsystem will be a single
section. Within each such section, the individual
team member will write the relevant subsections
for the individual subsystem.
section contains a more detailed qualitative statement
of the subsystem design problem. Building on the
introductory description above, you now describe
in more detail the anticipated trade-offs that motivate
the subsystem optimization study. You also comment
on previous work that has been done by others.
all symbols used and give units for each quantity.
Coordinate with the other subsystem designers so
you all use the same symbols for the same quantities
and avoid nomenclature inconsistencies.
Describe the objective function in words and then
derive its analytical expression in terms of design
variables and parameters.
Describe each constraint in words and then derive
its analytical expression in terms of design variables
and parameters. Try to group the constraints in
two categories: physical constraints that express
natural laws and engineering specifications, e.g.,
conservation of mass, energy, strength and deflection
requirements, etc; practical constraints that may
express limitations of current engineering practice,
rules of thumb, etc. — these will often have
the form of upper/lower bounds on the design variables.
Variables and Parameters
Define and list the design variables and parameters.
Give a set of typical values for the parameters
that you can use for the particular application.
Count and state the number of degrees of freedom.
Find a set of values for the design variables that
satisfy all of the constraints, i.e., show that
there is at least one feasible solution in the model
At the end of the model development summarize the
entire problem in one page, if possible, stating
the objective and all the constraints in standard
the derivation of expressions for the objective
and constraint functions be as explicit as possible.
If you do not have yet an explicit functional form,
state it implicitly, e.g., x1 = f(x2, x3), and explain
how you will calculate the function f. Examples
of that may be curve-fitting from tables or a separate
subroutine (e.g., structural analysis). If you have
performed curve fitting already, give the details
in an Appendix. Throughout the derivations you may
cite the references that you used, so that you do
not have to re-derive everything in the proposal.
may be information that you have not obtained yet,
for example, appropriate values for all of the parameters.
In such cases, state how you expect to get this
models used for design optimization in this class
are created in other courses or in student research
work. Acknowledge any such links and the assistance
of any individuals that helped you in the preparation
of the reported work.
all references in alphabetical order, complete with
author, title, publisher, year, and page number.
In the document text give the citation as e.g.,
Cover Page and Abstract
contains, in a single page, the title of the project,
your name and date, and an abstract of approximately
200 words describing your problem and the results
obtained. Do not write generalities but be specific
about your work.
Table of Contents
list of all sections, subsections, appendices, etc.,
contained in your report.
in the proposal/interim report (following the Optimization
Checklist in Chapter 8 of your textbook), with any
you are designing a system composed of several subsystems,
state the overall system problem and identify the
individual subsystems you will first optimize separately,
and the rationale for selecting these subsystems.
Please note the individual that worked on each subsystem.
all symbols that you use, particularly for the mathematical
model development, as should have been provided in
your proposal/interim report. If you have several
subsystems, you should make sure you use a consistent
nomenclature and set of symbols. It may also be convenient
to divide the symbols list to subsystems.
following sections 5-9 should be done for each subsystem
separately. For some projects, sections 8 and 9 may
be more suitable for inclusion after the system integration
study in section 10.
is the section provided in the proposal/interim report,
with any needed corrections. The last part of this
section must summarize the model, give list of variables,
number of equality and inequality constraints and
number of degrees of freedom. Since it is likely that
the final model evolved from its original statement,
you should describe briefly how the model evolved
and what made you change it. If the changes were a
result of the optimization study itself, you may include
these modeling decisions in the later section on optimization
study (Section 7 below). Relegate to appendices any
lengthy explanations you feel you need to include,
so that the overall report flow is not disrupted.
section describes any possible bounding agreements,
monotonicity properties and tables, constraint activity
identification, model transformations and simplifications,
scaling, case decomposition and anything else you
have done to make the problem easier to solve numerically
is suggested that model analysis may be first described
for a specific set of parameter values and then generalized
to other parameter values to the extent possible.
of the solution and a description of how it was obtained
should be presented. Unsuccessful attempts should
be reported and documented in an appendix.
solution should not be given as just a set of numbers.
Other issues must be examined and described, e.g.,
constraint activity, values of multipliers, interior
vs. boundary solution, global vs. local results, numerical
stability, satisfaction of KKT conditions, different
obtained numerically should motivate attempts for
analytical verification. Examine and explain, for
example, if monotonicity analysis results agree with
solution should be obtained for different sets of
parameter values. Does the optimum change? Can the
results be generalized? Are there ranges of parameter
values that may dictate the type of solution expected?
Discussion of Results
the results of the optimization study are given with
an engineering interpretation. What are the design
implications? Can you identify a "design rule"
for an optimum solution? Do the results make sense?
How does the model limit the solution? Are there "practical"
constraints active and what would this imply? What
would you do next to improve the answers or make the
problem more interesting?
a system design study, you must identify any conflicting
requirements stemming from optimizing the subsystems
separately. Do the subsystems have common variables,
parameters, objectives or constraints? Are some variables
in one subsystem parameters in the other? Is there
an expected sequence of solving one subsystem before
you solve another?
System Integration Study
this section you examine the issues you raised in
Section 9 regarding the linking of the subsystems.
Can the combined subsystem optima give you the overall
system optimum or are there conflicts to be resolved?
In the latter case you must attempt the following:
Select a system objective and combine all variables
and constraints into a single optimization model.
Solve this overall system problem as a single optimization
problem. This is what we call the All-in-One (AiO)
approach. If you can obtain a solution, compare it
with the solutions you obtained from the subsystems.
Discuss your results.
The AiO approach may give you a problem that is too
complicated and you cannot obtain numerical results,
and a decomposition method is applied. Identify the
problem partition into subproblems, each with their
own local variables. These may be just the subsystems
you identified in your earlier individual studies.
Further, define a master problem with an appropriate
objective that has as design variables the linking
variables among the subproblems. Apply a coordination
strategy where the master problem is solved wrt the
linking variables (local variables fixed) and the
subproblems are solved wrt the local variables (linking
variables fixed). Examine how the coordination strategy
Even if you do get results from the AiO strategy,
you should perform the study in (b) and compare the
results you get from the two approaches.
reference list of any sources that you used to complete
may be several appendices containing anything that
would distract the reader if used in the main text,
for example, elaborate algebraic manipulations, proofs
of monotonicities, coding of the models, and selected
Note: Your report should be a high-quality piece of
work similar to technical papers, something you should
be proud of. In fact, several student reports have
resulted in scientific publications in the past. In
any case, you should remember that others must be
able to read, understand and duplicate what you have
done with only the information contained in your report.