# An example of the Sequential Quadratic Programming (SQP)

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

$\min f(x,y)=(x+8.5)^2+(y-14)^2$

$s.t~~-4x+2.5y-15\leq0$

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?

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## migrated from mathoverflow.netJul 24 '13 at 17:42

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I see the optimization problem as trivial. Without applying any algorithm, it's obvious that the optimal solution is $(-6, 10)$ You simply want both expressions in the brackets to be equal to zero. – Waldemar Jul 24 '13 at 10:16
That's true, but in this problem, what I want to know is, why the intermediate point lies out of the feasible region by useing the SQP algorithm. I've just given this simple example to illustrate this case. – ClePIR Jul 25 '13 at 3:11