In recent years, there has been increased interest in stochastic and evolutionary algorithms, which have proven their efficiency in solving hard instances of combinatorial problems in a reasonable time, when classical techniques (branch and bound, branch and cut, and so on) were unable to solve them. These algorithms consist mainly of minimizing (or maximizing) an objective function, also called a fitness function. The optimization is done by moving from a local solution to another in the same neighborhood based on the value of the fitness function, which serves as a guide to reach the optimal solution.

Determining an objective function in real problems is not straightforward; either this function does not exist explicitly, especially in art design and music composition problems, or its evaluation is computationally time consuming. Note that in some combinatorial problems, there may be many objective functions (multicriteria problems) where a technique like divide and conquer is applied. In all cases, we need to construct an approximation model to set up the objective function, and this is the main contribution of this paper.

The author includes a comprehensive survey on the modeling of objective functions. First, the paper discusses the second-order polynomial model, used frequently in approximation modeling, which associates a polynomial model with the fitness function. The author goes on to discuss the kriging model, which is used mainly in the design and analysis of computer experiments, and then introduces neural network models, which are also used frequently in optimization.

The paper also provides pointers to some research works that make a comparison between these different models. I have to mention that, in such approximation techniques, the best way to improve the fitness function is through experimentation. In fact, we can set the desired model, but nothing can ensure that it is the best for the problem being solved. Only through experimentation can we improve it. The paper reviews approximate model management schemes, as well as model construction techniques. The author ends with a discussion of online and off-line data sampling, which are techniques used to improve the quality of the solution for the problem being optimized. A complete list of good references is given at the end of the paper.

This paper will be a good reference source for researchers who are interested in approximation and modeling techniques for fitness functions in optimizations problems, like aerodynamic design optimization, industrial design, and art and music design.