# penoptpersp: penalized optimization of the constrained linearized... In optismixture: Optimal Mixture Weights in Multiple Importance Sampling

## 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

optismixture documentation built on May 1, 2019, 10:13 p.m.