Inequality Constraint

An Inequality Constraint is an equation that contains an inequality relationship.

• AKA: Inequality Equation.
  • …
• Example(s):
  • [math]\displaystyle{ x \gt 0 }[/math]
  • [math]\displaystyle{ x \ge 0 }[/math]
  • [math]\displaystyle{ x \lt \infty }[/math]
  • [math]\displaystyle{ x + y \le 50 }[/math]
  • a Linear Inequality Constraint, such as [math]\displaystyle{ 3x - 4 \le 15 }[/math].
  • …
• Counter-Example(s):
  • an Equality Equation.
• See: Linear Program.

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References

2013

• http://en.wikipedia.org/wiki/Linear_inequality
  • In mathematics a linear inequality is an inequality which involves a linear function.

• http://en.wikipedia.org/wiki/Linear_inequality#Definitions
  • When the two expressions are connected by 'greater than' or 'less than' sign, we get an inequality.

    When operating in terms of real numbers, linear inequalities are the ones written in the forms :[math]\displaystyle{ f(x) \lt b \, }[/math] or [math]\displaystyle{ f(x) \leq b }[/math], where [math]\displaystyle{ f(x) }[/math] is a linear functional in real numbers and b is a constant real number. Alternatively, these may be viewed as :[math]\displaystyle{ g(x) \lt 0 \, }[/math] or [math]\displaystyle{ g(x) \leq 0 }[/math], where [math]\displaystyle{ g(x) }[/math] is an affine function.

    The above are commonly written out as :[math]\displaystyle{ a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \lt 0 }[/math] or :[math]\displaystyle{ a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq 0 }[/math]

    Sometimes they may be written out in the forms :[math]\displaystyle{ a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \lt b }[/math] or :[math]\displaystyle{ a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq b }[/math]

    Here [math]\displaystyle{ x_1,\ x_2,...,x_n }[/math] are called the unknowns, [math]\displaystyle{ a_{0},\ a_{1},\ a_{2},...,\ a_{n} }[/math] are called the coefficients, and [math]\displaystyle{ b }[/math] is the constant term.

    A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.

    A system of linear inequalities is a set of linear inequalities in the same variables: :[math]\displaystyle{ \begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; \leq \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; \leq \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; \leq \;&&& b_m \\ \end{alignat} }[/math] Here [math]\displaystyle{ x_1,\ x_2,...,x_n }[/math] are the unknowns, [math]\displaystyle{ a_{11},\ a_{12},...,\ a_{mn} }[/math] are the coefficients of the system, and [math]\displaystyle{ b_1,\ b_2,...,b_m }[/math] are the constant terms.

    This can be concisely written as the matrix inequality:  :[math]\displaystyle{ Ax \leq b }[/math] where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants.

    In the above systems both strict and non-strict inequalities may be used.

    Not all systems of linear inequalities have solutions.

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