## Dual Variables¶

Convex.jl also returns the optimal dual variables for a problem. These are stored in the dual field associated with each constraint.

using Convex

x = Variable()
constraint = x >= 0
p = minimize(x, constraint)
solve!(p)

# Get the dual value for the constraint
p.constraints.dual
# or
constraint.dual


## Warmstarting¶

If you’re solving the same problem many times with different values of a parameter, Convex.jl can initialize many solvers with the solution to the previous problem, which sometimes speeds up the solution time. This is called a warm start.

To use this feature, pass the optional argument warmstart=true to the solve! method.

# initialize data
n = 1000
y = rand(n)
x = Variable(n)

# first solve
lambda = 100
problem = minimize(sumsquares(y - x) + lambda * sumsquares(x - 10))
@time solve!(problem)

# now warmstart
# if the solver takes advantage of warmstarts,
# this run will be faster
lambda = 105
@time solve!(problem, warmstart=true)


## Fixing and freeing variables¶

Convex.jl allows you to fix a variable x to a value by calling the fix! method. Fixing the variable essentially turns it into a constant. Fixed variables are sometimes also called parameters.

fix(x, v) fixes the variable x to the value v.

fix(x) fixes x to the value x.value, which might be the value obtained by solving another problem involving the variable x.

To allow the variable x to vary again, call free!(x).

Fixing and freeing variables can be particularly useful as a tool for performing alternating minimization on nonconvex problems. For example, we can find an approximate solution to a nonnegative matrix factorization problem with alternating minimization as follows. We use warmstarts to speed up the solution.

# initialize nonconvex problem
n, k = 10, 1
A = rand(n, k) * rand(k, n)
x = Variable(n, k)
y = Variable(k, n)
problem = minimize(sum_squares(A - x*y), x>=0, y>=0)

# initialize value of y
y.value = rand(k, n)
# we'll do 10 iterations of alternating minimization
for i=1:10
# first solve for x
# with y fixed, the problem is convex
fix!(y)
solve!(problem, warmstart = i > 1 ? true : false)
free!(y)

# now solve for y with x fixed at the previous solution
fix!(x)
solve!(problem, warmstart = true)
free!(x)
end