penoptpersp.alpha.only: penalized optimization of the constrained linearized...

In thq80/Owen_2000_Safe-and-Effective-IS: Optimal Mixture Weights in Multiple Importance Sampling

Description

penalized optimization of the constrained linearized perspective function

Usage

penoptpersp.alpha.only(y, z, a0, eps = NULL, reltol = NULL, relerr = NULL,
  rho0 = NULL, maxin = NULL, maxout = NULL)

Arguments

y	length n vector
z	n \times J matrix
a0	length J 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	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^2}{z_i^Tα} over α 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^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 can be missing.

The optimization stops when within (1+relerr) of minimum variance.

Value

a list of

y

input y

z

input z

alpha

optimized alpha

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

thq80/Owen_2000_Safe-and-Effective-IS documentation built on May 10, 2019, 9:53 a.m.