Topics in Applied Math: Methods of Optimization

Professor Andrej Cherkaev,
Department of Mathematics 
Office: JWB 225 
Tel : +1 801 - 581 6822 

 Search for the Perfection: 
An image from 
Bridgeman Art Library

The course is designed for grad students in Math, Science, and Engineering. It covers algorithms and methods for search of extremum of functions of one or several variables  such search methods as gradient, conjugate gradient, quasi-Newton; basics of linear, quadratic, and convex programming. We also discuss modeling, dealing with uncertainty in data, and optimization of large poorly defined objects. The students are encouraged to apply the methods to their research, course projects will be assigned.

Prerequisite: Calculus, Linear Algegra/ODE, elementary programming.


Remarks, Introduction: About the algorithms, Optimization and modeling, Basic rules, Classification

Go to the top

Introduction to Optimization

Everyone 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?

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 the top

Optimization 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 short-term and long-term 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 the top

    Basic rules for optimization algorithms

  • There is no smart algorithm for choosing the oldest person from an alphabetical telephone directory.
  • 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.
  • My maximum is higher than your maximum!
  • 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.
  • Several classical optimization problems serve as testing grounds for optimization algorithms. Those are: maximum of an one-dimensional unimodal function, the mean square approximation, linear and quadratic programming.
  • Go to the top


    To explain how knotty the optimization problems are, one may try to classify them. I recommend to look at the optimization tree by NEOS.

    Below, there are some comments and examples of optimization problems.

    In any practical problem, the researcher meets a unique combination of mentioned factors and has to decide what numerical tools to use or modify to reach the goal. Therefore, the optimization always includes creativity and intuition. It is said that optimization belongs to both science and art.

    Go to the top

    Related Links

    Below, there are several links to the interesting popular and tutorial material in Optimization Theory at the Internet. There are hundreds more.
    Please send me more links: (
  • Personal websites
  • Institutions and Societies

  • Go to Contents
    Go to the top
    Go to Teaching Page
    Go to my Homepage


    NSF support is acknowledged.