Logistic Regression With Weights Constraints(non-nagetive,desceding Order)

I was asked in an interview if logistic regression with weights constraints such as 1.weights are all non-negative and 2.weights are keep descending order can get the global optima

Solution 1:

The constraint on the order of the weights just limits the search to a region of the weight space, and the maximum of the likelihood is either within the region or on its boundary. This is the same situation as for nonnegative weights, just with a different region. In two dimensions, w1 < w2 is the region below the line w1 = w2. I'm having trouble generalizing that picture to higher dimensions but maybe you can do it.

Be careful about constrained optimization. Bringing the constraint into play only gives the maximum subject to the constraint, i.e. a solution on the boundary of the allowed region. If the unconstrained global maximum is within the region, then the constraint is satisfied and you don't need to bring the constraints into play. So the way to solve such a problem is to first find the unconstrained maximum. If it's within the region, then you're done. Otherwise, you need to solve the constrained problem.

This question is probably more suitable for stats.stackexchange.com.