sparse_project: Successive projections with sparsely defined restrictions

Description

Compute a vector, closest to \(x\) satisfying a set of linear (in)equality restrictions.

Usage

sparse_project(
  x,
  A,
  b,
  neq = length(b),
  w = rep(1, length(x)),
  eps = 0.01,
  maxiter = 1000L,
  ...
)

Value

A list with the following entries:

x: the adjusted vector
status: Exit status:
- 0: success
- 1: could not allocate enough memory (space for approximately \(2(m+n)\) doubles is necessary).
- 2: divergence detected (set of restrictions may be contradictory)
- 3: maximum number of iterations reached
eps: The tolerance achieved after optimizing (see Details).
iterations: The number of iterations performed.
duration: the time it took to compute the adjusted vector
objective: The (weighted) Euclidean distance between the initial and the adjusted vector

Arguments

x: [numeric] Vector to optimize, starting point.
A: [data.frame] Coeffiencient matrix in [row,column,coefficient] format.
b: [numeric] Constant vector of the system \(Ax\leq b\)
neq: [integer] Number of equalities
w: [numeric] weight vector of same length of x
eps: maximally allowed tolerance
maxiter: maximally allowed number of iterations.
...: extra parameters passed to sparse_constraints

Details

The tolerance eps is defined as the maximum absolute value of the difference vector \(\boldsymbol{Ax}-\boldsymbol{b}\) for equalities. For inequalities, the difference vector is set to zero when it's value is lesser than zero (i.e. when the restriction is satisfied). The algorithm iterates until either the tolerance is met, the number of allowed iterations is exceeded or divergence is detected.

Examples

Run this code


# the system 
# x + y = 10
# -x <= 0   # ==> x > 0
# -y <= 0   # ==> y > 0
# Defined in the row-column-coefficient form:

A <- data.frame(
    row = c(1,1,2,3)
  , col = c(1,2,1,2)
  , coef= c(1,1,-1,-1)
)
b <- c(10,0,0)

sparse_project(x=c(4,5),A=A,b=b)