Contents

 

Intro: Optimization and modeling Intro: Assumptions  Intro: Classification  Links

Prelude

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.

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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.)

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Assumptions

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.

Classification

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

• Numerous optimization methods are designed for various number of independent controls (dimensionality). These ranges corresponds to one variable, several (two to ten), dozens, hundreds, or tens of thousands.
• An important special class -- discrete optimization -- corresponds to a large number of controls that take only integer or Boolean values. It requires combinatorial methods.
• Another special case -- calculus of variations -- corresponds to infinitely many controls.
• A significant parameter is the cost of evaluation of the function: It ranges from milliseconds of the computer time to lengthy and costly measurements in natural experiments. Accordingly, the realistic goals and the methods are different.

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.

• The presence of constraints and their character is very important for the choice of the strategy. The constrained problem typically possess the sharp maximum in a corner of the permitted domain while the smooth unconstrained function reaches its maximum in a stationary point.
•  Stochastic optimization deals  with noise in measurements or uncertainty of some factors in the model. The optimization process should account for possibly non accurate data (click here to see an example). Here, the goal may be to maximize the expected gain or to maximize the probability of the high profit.
• Another approach asks to play the game: Design versus Uncertainty, considering for the worst possible scenario. Find here an example of such game.

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.