# project: spg Projection Functions In BB: Solving and Optimizing Large-Scale Nonlinear Systems

## Description

Projection function implementing contraints for spg parameters.

## Usage

 `1` ``` projectLinear(par, A, b, meq) ```

## Arguments

 `par` A real vector argument (as for `fn`), indicating the parameter values to which the constraint should be applied. `A` A matrix. See details. `b` A vector. See details. `meq` See details.

## Details

The function `projectLinear` can be used by `spg` to define the constraints of the problem. It projects a point in R^n onto a region that defines the constraints. It takes a real vector `par` as argument and returns a real vector of the same length.

The function `projectLinear` incorporates linear equalities and inequalities in nonlinear optimization using a projection method, where an infeasible point is projected onto the feasible region using a quadratic programming solver. The inequalities are defined such that: `A %*% x - b > 0 `. The first ‘meq’ rows of A and the first ‘meq’ elements of b correspond to equality constraints.

## Value

A vector of the constrained parameter values.

`spg`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89``` ```# Example fn <- function(x) (x[1] - 3/2)^2 + (x[2] - 1/8)^4 gr <- function(x) c(2 * (x[1] - 3/2) , 4 * (x[2] - 1/8)^3) # This is the set of inequalities # x[1] - x[2] >= -1 # x[1] + x[2] >= -1 # x[1] - x[2] <= 1 # x[1] + x[2] <= 1 # The inequalities are written in R such that: Amat %*% x >= b Amat <- matrix(c(1, -1, 1, 1, -1, 1, -1, -1), 4, 2, byrow=TRUE) b <- c(-1, -1, -1, -1) meq <- 0 # all 4 conditions are inequalities p0 <- rnorm(2) spg(par=p0, fn=fn, gr=gr, project="projectLinear", projectArgs=list(A=Amat, b=b, meq=meq)) meq <- 1 # first condition is now an equality spg(par=p0, fn=fn, gr=gr, project="projectLinear", projectArgs=list(A=Amat, b=b, meq=meq)) # box-constraints can be incorporated as follows: # x[1] >= 0 # x[2] >= 0 # x[1] <= 0.5 # x[2] <= 0.5 Amat <- matrix(c(1, 0, 0, 1, -1, 0, 0, -1), 4, 2, byrow=TRUE) b <- c(0, 0, -0.5, -0.5) meq <- 0 spg(par=p0, fn=fn, gr=gr, project="projectLinear", projectArgs=list(A=Amat, b=b, meq=meq)) # Note that the above is the same as the following: spg(par=p0, fn=fn, gr=gr, lower=0, upper=0.5) # An example showing how to impose other constraints in spg() fr <- function(x) { ## Rosenbrock Banana function x1 <- x[1] x2 <- x[2] 100 * (x2 - x1 * x1)^2 + (1 - x1)^2 } # Impose a constraint that sum(x) = 1 proj <- function(x){ x / sum(x) } spg(par=runif(2), fn=fr, project="proj") # Illustration of the importance of `projecting' the constraints, rather # than simply finding a feasible point: fr <- function(x) { ## Rosenbrock Banana function x1 <- x[1] x2 <- x[2] 100 * (x2 - x1 * x1)^2 + (1 - x1)^2 } # Impose a constraint that sum(x) = 1 proj <- function(x){ # Although this function does give a feasible point it is # not a "projection" in the sense of the nearest feasible point to `x' x / sum(x) } p0 <- c(0.93, 0.94) # Note, the starting value is infeasible so the next # result is "Maximum function evals exceeded" spg(par=p0, fn=fr, project="proj") # Correct approach to doing the projection using the `projectLinear' function spg(par=p0, fn=fr, project="projectLinear", projectArgs=list(A=matrix(1, 1, 2), b=1, meq=1)) # Impose additional box constraint on first parameter p0 <- c(0.4, 0.94) # need feasible starting point spg(par=p0, fn=fr, lower=c(-0.5, -Inf), upper=c(0.5, Inf), project="projectLinear", projectArgs=list(A=matrix(1, 1, 2), b=1, meq=1)) ```

BB documentation built on Sept. 23, 2019, 3:01 a.m.