# Regular constraints

I am going through some exercises in a presentation I found treating the basics of math for machine learning, and they talk about regular constraints.

For example, this set $$K = \{(x,y) \in R^2 / x+y = - \frac{1}{2}, x^2-2y =0\}$$, they talk about these constraints not regular on $$K$$, what do they mean by that?

A point $$(x,y)$$ is a regular point for the constraints $$h_{1}(x,y)=0$$, $$h_{2}(x,y)=0$$ if the constraints are satisfied and the gradients of $$h_{1}$$ and $$h_{2}$$ are linearly independent.

The only feasible point in your set $$K$$ is $$(x,y)=(-1,1/2)$$. For this point,

$$\nabla h_{1}(x,y)=\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$$

and

$$\nabla h_{2}(x,y)=\left[ \begin{array}{c} -2 \\ -2 \end{array} \right]$$.

Since the gradients are linearly dependent, this is not a regular point of the constraints.

The Lagrange multiplier necessary conditions are not guaranteed to work at irregular points. For example, if your problem is

$$\min x_{1}$$

subject to

$$h_{1}(x,y)=0$$

$$h_{2}(x,y)=0$$,

then the minimum is at $$(-1,1/2)$$ because it is the only feasible solution. However, this point doesn't satisfy the Lagrange multiplier conditions.

• The term to mention in this context is "constraint qualification", which will lead to many search hits if you look for it. Commented Jul 30 at 23:02
• Yes, this Linear Independence Constraint Qualification (LICQ) is one of many hypotheses that work. There are whole books on the subject of constraint qualifications that can be used instead of LICQ. Commented Jul 31 at 3:25
• @BrianBorchers can you share some insight on this scicomp.stackexchange.com/questions/44432/… ?
– Papa
Commented Jul 31 at 19:46