Given an objective function $f(x,y)=(x+6)^2+(y-10)^2$, we want to minimize the function $f$ under a constrained condition $xy-5\leq0$. Obviously it is a constrained optimization, a good algorithm to deal with it is the Sequential Quadratic Programming (SQP), and in every iteration, we solve a QP subproblem.
Let the initial iteration point be $x_0=(2.5,-4)$, which satisfying the constrained condition $xy-5\leq 0$, then the QP subproblem is
we can solve it easily, the optimal point is $(1.2079,7.9326)$,then the first iteration point is $x_1=x_0+(1.2079,7.9326)=(3.7079,3.9326)$,and then we can construct another QP subproblem, then $x_2=(-6,10)$, which is the optimal point of the SQP algorithm.
I find that the intermediate iteration point $x_1=(3.7079,3.9326)$ does not satisfy the constrained condition, but I don't know why this happend. I know that some intermediate iteration points do not satisfy the constrained condition, but I want to know when and why can this happen.
And if I don't want the intermediate points violate the constraints, how to solve it?