Math 5750, 3 credit hours. Spring 2001. T, Th
2:003:25
in JWB 333
Topics in Applied Math: Methods of Optimization
Instructor :
Professor Andrej Cherkaev, Department of Mathematics Office: JWB 225 Email: cherk@math.utah.edu Tel : +1 801  581 6822 Fax: +1 801  581 4148 
Search for the Perfection: An image from Bridgeman Art Library 
The course is addressed to senior undergraduate and graduate
students in
Applied Mathematics, Science, and Engineering.
The desire for optimality (perfection) is inherent for humans. The search for extremes inspires mountaineers, scientists, mathematicians, and the rest of the human race. A beautiful and practical mathematical theory of optimization (i.e. searchforoptimum strategies) is developed since the sixties when computers become available. Every new generation of computers allows for attacking new types of problems and calls for new methods. The goal of the theory is the creation of reliable methods to catch the extremum of a function by an intelligent arrangement of its evaluations (measurements). This theory is vitally important for modern engineering and planning that incorporate optimization at every step of the complicated decision making process.
This course discusses classical direct searchforoptimum methods, such as Golden Mean, Conjugate Gradients, Modified Newton Method, methods for constrained optimization, including Linear and Quadratic Programming, and others. We will also briefly review genetic algorithms that mimic evolution and stochastic algorithms that account for uncertainties of mathematical models. The course work includes several homework assignments that ask to implement the studied methods and a final project, that will also be orally presented in the class.
The textbook Practical optimization by Philip Gill, Walter Murray, and Margaret H. Wright is interesting and readable (a British colleague of my recommended it as "a bloody good book"); the authors are among the top experts in the field. The book discusses pros and cons of various methods, often in the context of specific applications. The review part of the course will also use the instructor's notes.
Prerequisite: Calculus, ODE, elementary programming.
To learn more about the course, please visit instructor's website
www.math.utah.edu/~cherk; then follow the online instructions.
The final grade is composed as follows: 60%  the homework, 40%  the project.
Preliminary Plan
Some Frivolous Remarks Introduction 
Intro: Optimization and modeling
Intro: Assumptions Intro: Classification Links 
The inherent human desire to optimize is cerebrated in the
famous Dante
quotation:
All that is superfluous displeases God and Nature All that displeases God and Nature is evil. In engineering, optimal projects are considered beautiful and rational, and the farfromoptimal ones are called ugly and meaningless. Obviously, every engineer tries to create the best project and he/she relies on optimization methods to achieve the goal. 
The general principle by Maupertuis
proclaims:
If there occur some changes in nature, the amount of action necessary for this change must be as small as possible. This principle proclaims that the nature always finds the "best" way to reach a goal. It leads to an interesting inverse optimization problem: Find the essence of optimality of a natural "project." 
Optimization theory is developed by ingenious and creative people,
who
regularly appeal to vivid common sense associations, formulating them
in
a general mathematical form. For instance, the theory steers numerical
searches through
canyons
and passes (saddles), towards
the
peaks;
it fights the curse of dimensionality,
models
evolution,
gambling,
and other human passions. The optimizing algorithms themselves are
mathematical
models of intellectual and intuitive decision making.
The essence of an optimization problem is: Catching
a black cat in a dark room in minimal time.
(A constrained optimization
problem corresponds to a room full of furniture.)
A light, even dim, is needed: Hence optimization methods explore assumptions about the character of response of the goal function to varying parameters and suggest the best way to change them. The variety of a priori assumptions corresponds to the variety of optimization methods. This variety explains why there is no silver bullet in optimization theory. 

Go to ContentsEveryone who studied calculus knows that an extremum of a smooth function is reached at a stationary point where its gradient vanishes. Some may also remember the Weierstrass theorem which proclaims that the minimum and the maximum of a function in a closed finite domain do exist. Does this mean that the problem is solved? Introduction to Optimization
A small thing remains: To actually find that maximum. This problem is the subject of the optimization theory that deals with algorithms for search of the extremum. More precisely, we are looking for an algorithm to approach a proximity of this maximum and we are allowed to evaluate the function (to measure it) in a finite number of points. Below, some links to mathematical societies and group in optimization are placed that testify how popular the optimization theory is today: Many hundreds of groups are intensively working on it.
Go to Contents
Go to the topOptimization and Modeling
The modeling of the optimizing process is conducted along with the optimization. Inaccuracy of the model is emphasized in optimization problem, since optimization usually brings the control parameters to the edge, where a model may fail to accurately describe the prototype. For example, when a linearized model is optimized, the optimum often corresponds to infinite value of the linearized control. (Click here to see an example) On the other hand, the roughness of the model should not be viewed as a negative factor, since the simplicity of a model is as important as the accuracy. Recall the joke about the most accurate geographical map: It is done in the 1:1 scale. Unlike the models of a physical phenomena, an optimization models critically depend on designer's will. Firstly, different aspects of the process are emphasized or neglected depending on the optimization goal. Secondly, it is not easy to set the goal and the specific constrains for optimization. Naturally, one wants to produce more goods, with lowest cost and highest quality. To optimize the production, one either may constrain by some level the cost and the quality and maximize the quantity, or constrain the quantity and quality and minimize the cost, or constrain the quantity and the cost and maximize the quality. There is no way to avoid the difficult choice of the values of constraints. The mathematical tricks go not farther than: "Better be healthy and wealthy than poor and ill". True, still not too exciting.The maximization of the monetary profit solves the problem to some extent by applying an universal criterion. Still, the shortterm and longterm profits require very different strategies; and it is necessary to assign the level of insurance, to account for possible market variations, etc. These variables must be a priori assigned by the researcher. Sometimes, a solution of an optimization problem shows unexpected features: for example, an optimal trajectory zigzags infinitely often. Such behavior points to an unexpected, but optimal behavior of the solution. It should not be rejected as a mathematical extravaganza, but thought through! (Click here for some discussion.)
Go to Contents
Go to the top
Assumptions
This says that some properties of the maximized function be a priori assumed. Without assumptions, no rational algorithms can be suggested. The search methods approximate  directly or indirectly  the behavior of the function in the neighborhood of measurements. The approximation is based on the assumed smoothness or sometimes the convexity Various methods assume different types of the approximation. There is no smart algorithm for choosing the oldest person from an alphabetical telephone directory.
Generally, there are no ways to predict the behavior of the function everywhere in the permitted domain. An optimized function may have more than one local maximum. Most of the methods pilot the search to a local maximum without a guarantee that this maximum is also a global one. Those methods that guarantee the global character of the maximum, require additional assumptions as the convexity. My maximum is higher than your maximum!
Several classical optimization problems serve as testing grounds for optimization algorithms. Those are: maximum of an onedimensional unimodal function, the mean square approximation, linear and quadratic programming.
Below, there are some comments and examples of optimization problems.
Sharp Maximum is achieved! The photo by and courtesy of Dr. Robert (Bob) Palais 
The sharpness of the maximum is of a special interest. It is very difficult to find a maximum of a flat function especially in the presence of numerical errors. For instance, a point of the maximum elevation in a flat region is much harder to determine than to locate a peak in the mountains. A sharp maximum (left figure) also requires some special methods since the gradient is discontinuous; it does not exist in the optimal point. 
Structural
Optimization is an interesting and tricky class of optimization
problem.
It is characterized by a large number of variables that represent the
shape
of the design and and stiffness of the available materials.
The control is the layout of the materials in the volume of the structure. 
Go to Contents
Go to the top
Go to
Teaching Page
Go to my Homepage
NSF support is acknowledged.