Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
I thought your question was about projected gradient descent. But the answer you accepted says nothing about projected gradient descent. –  Rahul Nov 13 '12 at 0:20
    
@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
add comment

2 Answers 2

up vote 1 down vote accepted

Just like in the gradient descent method, you want to stop when the norm of the gradient is sufficiently small, in the projected gradient method, you want to stop when the norm of the projected gradient is sufficiently small. Suppose the projected gradient is zero. Geometrically, that means that the negative gradient lies in the normal cone to the feasible set. If you had linear equality constraints only, it would mean that the gradient vector is orthogonal to the feasible set. In other words, it's locally impossible to find a descent direction, and you have first-order optimality.

share|improve this answer
    
Thanks. Do you have a formal proof for this? (Do you know where I can find it?) –  user25004 May 9 at 2:22
    
Look for example in Numerical Optimization (Nocedal & Wright, Springer), Section 12.3 "First-Order Optimality Conditions". For any optimization problem, the first-order conditions are that the negative gradient lie in the normal cone. –  Dominique May 9 at 8:16
    
Thank you Dominique. –  user25004 May 9 at 18:43
add comment

there are actually several methods for dealing with constraint optimization problems. See Penalty Method or Augmented Lagrangian Method as two examples.

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.

Edit:

Note that also the constraints have to be smooth as you evaluate the gradient.

share|improve this answer
    
How is this related to the question??? –  Dominique May 8 at 22:35
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.