# What Stopping Criteria to Use in Projected Gradient Descent

Suppose we want to solve a convex constrained minimization problem. We want to use projected gradient descent. If there was no constraint the stopping condition for a gradient descent algorithm would be that the gradient of function is close to zero.

But for a constrained problem the gradient at the optimal point is not necessarily (close to) zero. Now what stopping condition should we use for a projected gradient descent algorithm?

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@RahulNarain Could you complete the answer? –  user25004 Nov 13 '12 at 0:43
I don't know the answer, or I would have posted it. I just didn't understand your acceptance criteria. Usually when one accepts an answer, it means that one has found out what they wanted to know and are not looking for any more answers. –  Rahul Nov 13 '12 at 17:21

To solve the problem \begin{align} \min f( x) \\ \text{ subject to: } c_i( x) \ge 0 ~\forall i \in I \end{align} We reformulate the problem for the Penalty Method as follows \begin{align} \min F(x)_k = f(x) + {\sigma_k}_i p(c_i(x)) \end{align} With $p(c) = \min(0,c)^2$.
Then inside each of your optimization iteration, you iterate over k, increasing the penalty coefficients ${\sigma_k}_i$.Thus you hope to obtain a solution to your minimization problem $F$ which also minimizes $f$, while fullfilling the constraints.