penalized optimization of the constrained linearized perspective function

Description

penalized optimization of the constrained linearized perspective function

Usage

1
2
penoptpersp(x, y, z, a0 = NULL, b0 = NULL, eps = NULL, reltol = NULL,
  relerr = NULL, rho0 = NULL, maxin = NULL, maxout = NULL)

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