# Extra:Mathematical programming

### From Glossary

# Mathematical Programming

A *mathematical program* is an optimization problem of the form:

where is a subset of and is in the domain of , and , which map into real spaces. The relations, , , and are called *constraints*, and is called the *objective function*.

There are forms that deviate from this paradigm, and it is typically a modeling issue to find an equivalent standard form. Important examples are as follows:

A point is *feasible* if it is in and satisfies the constraints:
, and . A point is *optimal* if it is feasible and if the value of the objective function is not less than that of any other feasible solution: for all feasible . The *sense of optimization* is presented here as *maximization*, but it could just as well be *minimization*, with the appropriate change in the meaning of optimal
solution: for all feasible .

A mathematical program is often extended to indicate a *family* of mathematical programs over a *parameter space*, say . This merely involves extending the domain of the functions, , and , and we use the semi-colon to separate the decision variables from the parameters.

We could also have depend on , but the above form generally suffices.

*Mathematical programming* is the study or use of the mathematical program. It includes any or all of the following:

- Theorems about the form of a solution, including whether one exists;
- Algorithms to seek a solution or ascertain that none exists;
- Formulation of problems into mathematical programs, including understanding the quality of one formulation in comparison with another;
- Analysis of results, including debugging situations, such as infeasible or anomalous values;
- Theorems about the model structure, including properties pertaining to feasibility, redundancy and/or implied relations (such theorems could be to support analysis of results or design of algorithms);
- Theorems about approximation arising from imperfections of model forms, levels of aggregation, computational error, and other deviations;
- Developments in connection with other disciplines, such as a computing environment.

One taxonomy for mathematical programming is by its defining ingredients: