Constraints and slack variables

The upper and lower bounds on the m components of F and Gx are said to define the general constraints of the problem.  SNOPT converts the general constraints to equalities by introducing a set of slack variables s, where s = (s₁, s₂ , ... , s_m)T. For example, the linear constraint

5 2x₁ + 3x₂ +;

is replaced by

2x₁ + 3x₂  - s₁ = 0

together with the bounded slack

5 s₁ +; . 

The problem defined by (1.1) can therefore be rewritten in the following equivalent form

The linear and nonlinear general constraints become equalities of the form

F(x) - s_N = 0 and Gx - s_L = 0,

where s_L and s_N are known as the linear and nonlinear slacks.