>>11577353Explaining the concept to yourself is really fucking easy.
Take a sheet of paper, and draw the intersection of two active inequality constraints.
Draw the normal vectors of each of the inequality constraints at the intersection point such that the vectors are pointing in the direction of the feasible region.
Draw a third vector at the intersection point, and the line normal to said vector.
Clearly, whenever the third vector is a positive coefficient linear combination of the normal vectors, then the feasible region lies completely within the half of the space (the halves are split by the line) that contains the third vector. However, if any one of the coefficients in the linear combination is negative, then at least part of the feasible region will lie in the other half of said space.
Clearly, this is relevant to optimization problems whenever the third vector is the gradient of the objective function. If the intersection point satisfies the LICQ, and the coefficients of the active inequality constraints are all positive, then you can say that the intersection point is worth checking for a minima. However, if any of the coefficients of the active inequality constraints are negative, and the gradient of the objective function at the intersection point is non-zero, then you know there is a direction in the feasible region that decreases the objective function, at least locally, which means its pointless to check for a minima at the intersection point.
Similar reasoning can be used for equality constraints.
