# Homogeneity Detection Incorporating Prior Constraint Information by ADMM simultaneous homogeneity detection and variable selection incorporating prior constraint by ADMM algorithm. The problem turn to solving quadratic programming problems of the form **min(-d^T b + 1/2 b^T D b)** with the constraints **A^T b >= b_0**. The penalty is the pairwise fusion with p(p-1)/2 number of penalties. ```r HSDiC_ADMM(X, Y, A.eq, A.ge, A.lbs, A.ubs, b.eq, b.ge, b.lbs, b.ubs, penalty = c("MCP", "SCAD", "adlasso", "lasso"), lambda2, admmScale1 = 1/nrow(X), admmScale2 = 1, admmAbsTol = 1e-04, admmRelTol = 1e-04, nADMM = 2000, admmVaryScale = FALSE) ``` ## Arguments - `X`: n-by-p design matrix. - `Y`: n-by-1 response matrix. - `A.eq`: equality constraint matrix. - `A.ge`: inequality constraint matrix. - `A.lbs`: low-bounds matrix on variables, see examples. - `A.ubs`: upper-bounds matrix on variables, see examples. - `b.eq`: equality constraint vector. - `b.ge`: inequality constraint vector. - `b.lbs`: low-bounds on variables, see details. - `b.ubs`: upper-bounds on variables, see details. - `penalty`: The penalty to be applied to the model. Either "lasso" (the default), "SCAD", or "MCP". - `lambda2`: penalty tuning parameter for thresholding function. - `admmScale1`: first ADMM scale parameter, 1/nrow(X) is default. - `admmScale2`: second ADMM scale parameter, 1 is default. - `admmAbsTol`: absolute tolerance for ADMM, 1e-04 is default. - `admmRelTol`: relative tolerance for ADMM, 1e-04 is default. - `nADMM`: maximum number of iterations for ADMM, 2000 is default. - `admmVaryScale`: dynamically chance the ADMM scale parameter, FALSE is default ## Returns - **betahat**: solution vector. - **stats.ADMM_inters**: number of iterations. ## References 'Pairwise Fusion Approach Incorporating Prior Constraint Information' by Yaguang Li ## See Also `solve.QP` ## Examples ```r ## data generation set.seed(111) n=100 p=50 r <- 1 #0.5, 0.8, 1 beta <- r*c(sample(rep(1:2, each = 10)), rep(0,10), -sample(rep(1:2, each = 10)) ) X <- matrix(rnorm(n*p),nrow = n) sigma = 1 Y <- X %*% beta + sigma * rnorm(n, 0, 1) # equalities A.eq <- rbind(rep(1,p)) b.eq <- c(0) # inequalities A.ge <- diag( c(rep(1,30), rep(-1,20)) ) b.ge <- rep(0,p) # lower-bounds A.lbs <- diag(1, p) b.lbs <- rep(-2, p) # upper-bounds on variables A.ubs <- diag(-1, p) b.ubs <- rep(-2, p) ptm <- proc.time() fit <- HSDiC_ADMM(X, Y, A.eq, A.ge, A.lbs, A.ubs, b.eq, b.ge, b.lbs, b.ubs, penalty = "adlasso", lambda2 = 0.8, admmScale2 = 1) proc.time() - ptm ## table(round(fit$beta,1)) plot(beta, type="p", pch = 20, cex = 1) points(fit$beta, col = 3) ```

HSDiC_ADMM function