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.

I have a simple (few variables), continuous, twice differentiable convex function that I wish to minimize over the unit simplex. In other words, $\min. f(\mathbf{x})$, $\text{s.t. } \mathbf{0} \preceq x$ and $\mathbf{1}^\top \mathbf{x} = {1}$. This will have to be performed multiple times.

What is a good optimization method to use? Preferably fairly simple to describe and implement?

There are some really fancy methods (such as “Mirror descent and nonlinear projected subgradient methods for convex optimization”, Beck and Teboulle) that specifically have methods for minimizing over the unit simplex. But these methods only use the gradient and not the Hessian.

share|improve this question
    
If the function is convex, why wouldn't a simple line search method work? Since the objective function is twice-differentiable, I'd think Newton's method would work fine -- if only once-differentiable, gradient descent should work. Both are extremely well documented online with pseudocode and application-specific code abound. –  jedwards Dec 6 '11 at 3:34
    
How about the "Karush–Kuhn–Tucker conditions"? en.wikipedia.org/wiki/… –  matt Dec 6 '11 at 12:35
    
Thanks for the kind reply. I was wondering if there is a specific algorithm for minimization over the unit simplex that uses the Hessian (since it is easily calculated and the problem size is small). (I am still new to convex optimization) In the end I implemented the Interior Point method and it works well. –  dcm29 Dec 12 '11 at 3:05
1  
This answer probably comes to late, but have you already looked into the paper "Projected newton methods for optimization problems with simple constraints" by Bertsekas (SIAM J. Control and Optimization, Vol. 20, No. 2, March 1982)? As the title says, a projected Newton algorithm is introduced together with an Armijo-like linesearch. Also, superlinear convergence is proved for simple constraints. The above problem is listed as an example. –  user103342 Oct 26 '13 at 11:12
    
See the paper... Efficient Projections onto the l1-Ball for Learning in High Dimensions This paper can perfectly solve your objective. –  Leon Nov 22 '13 at 13:16

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.