penalized optimization of the constrained linearized perspective function
Description
penalized optimization of the constrained linearized perspective function
Usage
1 2 |
Arguments
x |
n \times K matrix |
y |
length n vector |
z |
n \times J matrix |
a0 |
length J vector |
b0 |
length K vector |
eps |
length J vector, default to be rep(0.1/J, J) |
reltol |
relative tolerence for Newton step, between 0 to 1, default to be 10^{-3}. For each inner loop, we optimize f_0 + ρ \times \mathrm{pen} for a fixed ρ, we stop when the Newton decrement f(x) - inf_y \hat{f}(y) ≤q f(x)* \mathrm{reltol}, where \hat{f} is the second-order approximation of f at x |
relerr |
stop when within (1+relerr) of minimum variance, default to be 10^{-3}, between 0 to 1. |
rho0 |
initial value for ρ, default to be 1 |
maxin |
maximum number of inner iterations |
maxout |
maximum number of outer iterations |
Details
To minimize ∑_i \frac{(y_i - x_i^T β)^2}{z_i^Tα} over α and β, subject to α_j > ε_j for j = 1, \cdots, J and ∑_{j=1}^J α_j < 1,
Instead we minimize ∑_i \frac{(y_i - x_i^T β)^2}{z_i^Tα} + ρ \times \mathrm{pen} for a decreasing sequence of ρ
where \mathrm{pen} = -( ∑_{j = 1}^J( \log(α_j-ε_j) ) + \log(1-∑_{j = 1}^J α_j) )
starting values are α = a0 and β = b0. They can be missing.
The optimization stops when within (1+relerr) of minimum variance.
Value
a list of
- x
input x
- y
input y
- z
input z
- alpha
optimized alpha
- beta
optimized beta
- rho
value of rho
- f
value of the objective function
- rhopen
value of rho*pen when returned
- outer
number of outer loops
- relerr
relative error
- alphasum
sum of optimized alpha