penoptpersp: penalized optimization of the constrained linearized perspective function

Description

penalized optimization of the constrained linearized perspective function

Usage

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

Arguments

$n \times K$ matrix

length $n$ vector

$n \times J$ matrix

length $J$ vector

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 + \rho \times \mathrm{pen} $ for a fixed $\rho$, we stop when the Newton decrement $f(x) - inf_y \hat{f}(y) \leq f(x)* \mathrm{reltol}$, 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 $\rho$, default to be 1

maxin

maximum number of inner iterations

maxout

maximum number of outer iterations

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

Details

To minimize $\sum_i \frac{(y_i - x_i^T \beta)^2}{z_i^T\alpha}$ over $\alpha$ and $\beta$, subject to $\alpha_j > \epsilon_j$ for $j = 1, \cdots, J$ and $\sum_{j=1}^J \alpha_j < 1$,

Instead we minimize $ \sum_i \frac{(y_i - x_i^T \beta)^2}{z_i^T\alpha} + \rho \times \mathrm{pen}$ for a decreasing sequence of $\rho$

where $ \mathrm{pen} = -( \sum_{j = 1}^J( \log(\alpha_j-\epsilon_j) ) + \log(1-\sum_{j = 1}^J \alpha_j) )$

starting values are $\alpha = a0$ and $\beta = b0$. They can be missing.

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