Constraints and feasible regions

A factor or response that is not variable from infinitely negative values to infinitely positive values is said to be constrained. 

Many factors and responses are naturally constrained. Temperature, for example, is usually considered to have a lower limit of -273.16°C temperatures at or below that value or theoretically impossible. There is, however, no natural upper limit to temperature temperature is said to have a natural lower bound, but it has no natural upper bound. 

The voltage from an autotransformer has both a lower bound (0 V a.c.) and an upper bound (usually about 130 V a.c.) and is an example of a naturally constrained discrete factor. The upper constraint could be changed if the autotransformer were 

The natural constraint on temperature, the natural constraints on autotransformer voltage, and the artificial constraints on temperature are all examples of inequality constraints. If T is used to represent temperature and E represents voltage, these inequality constraints can be expressed, in order, as 

Here the less than or equal to symbol is used to indicate that the boundary values themselves are included in the set of possible values these variables may assume. If the boundary values are not included, less than symbols ( ) are used. The presence of the additional equal to symbol in Equations 2.6 and 2.7 does not prevent the constraints from being inequalities. 

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