| # Standard forms Optimization Models > Gauss’ Bet | Functions and Maps | Standard forms | Nomenclature | Problem Classes | Complexity | History Functional form Epigraph form Other standard forms # Functional form An optimization problem is a problem of the form where is the decision variable; is the objective function, or cost; , represent the constraints; is the optimal value. In the above, the term ‘‘subject to’’ is sometimes replaced with the shorthand colon notation. Often the above is referred to as a ‘‘mathematical program’’. The term “programming” (or “program”) does not refer to a computer code. It is used mainly for historical purposes. We will use the more rigorous (but less popular) term “optimization problem”. Example: An optimization problem in two variables. # Epigraph form In optimization, we can always assume that the objective is a linear function of the variables. This can be done via the epigraph representation of the problem, which is based on adding a new scalar variable : At optimum, . In the above, the objective function is , with values . We can picture this as follows. Consider the sub-level sets of the objective function, which are of the form for some . The problem amounts to finding the smallest for which the corresponding sub-level set intersects the set of points that satisfy the constraints. Example: Geometric view of the optimization problem in two variables. # Other standard forms Sometimes we single out equality constraints, if any: where 's are given. Of course, we may reduce the above problem to the standard form above, representing each equality constraint by a pair of inequalities. Sometimes, the constraints are described abstractly via a set condition, of the form for some subset of . The corresponding notation is Some problems come in the form of maximization problems. Such problems are readily cast in standard form via the expression where .