Problem:
Given n points (x_1, x_2, ...x_n) as independent variables, where x_i can be a value or a vector, and the dependent variables (y_1, y_2, ... y_i), design a regression that satifies:
1. f(x) = t(w) * x (assume the the attribute of x corresponding w_0 is 1)
2. sigma(w) = 1
3. w_i >= 0
3. minimize sigma(y - f(x))^2
Solution:
The solution is based on KKT condition which is a generized from langrange muptiply method.
KKT conditions of this question:
1. delta(siagma(y - f(x))^2 + a * delta(w) + sigma(b_i * delta(w_i)) = 0
2. sigma(w) = 1
solve about equotions, we get:
b_i = 0
rbind(
cbind(2 * t(X) *X, c(1, ....),
c(w, 1)
) * c(w, a) = 2 * t(X) * Y
R code: ( assume x is a vector, whose first element is 1 correponding w_0)
regress <- function(x, y) {
A <- 2 * t(x) %*% x
B <- rep(1, nrow(A))
C <- 2 * t(x) %*% y
D <- c(rep(1, nrow(A)), 0)
left <- rbind(cbind(A, B), D)
coef <- solve(left, c(C, 1))
coef[1:length(coef) - 1]
}