## 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, SCS

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

# Get the dual value for the constraint
p.constraints[1].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, SCSSolver())

# now warmstart
# if the solver takes advantage of warmstarts,
# this run will be faster
lambda = 105
@time solve!(problem, SCSSolver(), 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, SCSSolver(), warmstart = i > 1 ? true : false)
free!(y)

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

## Printing and the tree structure

A Convex problem is structured as a tree, with the root being the problem object, with branches to the objective and the set of constraints. The objective is an AbstractExpr which itself is a tree, with each atom being a node and having children which are other atoms, variables, or constants. Convex provides children methods from AbstractTrees.jl so that the tree-traversal functions of that package can be used with Convex.jl problems and structures. This is what allows powers the printing of problems, expressions, and constraints. The depth to which the tree corresponding to a problem, expression, or constraint is printed is controlled by the global variable Convex.MAXDEPTH, which defaults to 3. This can be changed by e.g. setting

Convex.MAXDEPTH[] = 5

Likewise, Convex.MAXWIDTH, which defaults to 15, controls the "width" of the printed tree. For example, when printing a problem with 20 constraints, only the first MAXWIDTH of the constraints will be printed. Vertical dots, "⋮", will be printed indicating that some constraints were omitted in the printing.

The AbstractTrees methods can also be used to analyze the structure of a Convex.jl problem. For example,

julia> using Convex, AbstractTrees

julia> x = Variable()
Variable
size: (1, 1)
sign: real
vexity: affine
id: 111…426

julia> p = maximize( log(x), x >= 1, x <= 3 )
maximize
└─ log (concave; real)
└─ real variable (id: 111…426)
subject to
├─ >= constraint (affine)
│  ├─ real variable (id: 111…426)
│  └─ 1
└─ <= constraint (affine)
├─ real variable (id: 111…426)
└─ 3

current status: not yet solved

julia> for leaf in AbstractTrees.Leaves(p)
println("Here's a leaf: $(summary(leaf))") end Here's a leaf: real variable (id: 111…426) Here's a leaf: real variable (id: 111…426) Here's a leaf: constant (constant; positive) Here's a leaf: real variable (id: 111…426) Here's a leaf: constant (constant; positive) We can also iterate over the problem in various orders. The following descriptions are taken from the AbstractTrees.jl docstrings, which have more information. ### PostOrderDFS Iterator to visit the nodes of a tree, guaranteeing that children will be visited before their parents. julia> for (i, node) in enumerate(AbstractTrees.PostOrderDFS(p)) println("Here's node$i via PostOrderDFS: $(summary(node))") end Here's node 1 via PostOrderDFS: real variable (id: 111…426) Here's node 2 via PostOrderDFS: log (concave; real) Here's node 3 via PostOrderDFS: real variable (id: 111…426) Here's node 4 via PostOrderDFS: constant (constant; positive) Here's node 5 via PostOrderDFS: >= constraint (affine) Here's node 6 via PostOrderDFS: real variable (id: 111…426) Here's node 7 via PostOrderDFS: constant (constant; positive) Here's node 8 via PostOrderDFS: <= constraint (affine) Here's node 9 via PostOrderDFS: 2-element Array{Constraint,1} Here's node 10 via PostOrderDFS: Problem ### PreOrderDFS Iterator to visit the nodes of a tree, guaranteeing that parents will be visited before their children. julia> for (i, node) in enumerate(AbstractTrees.PreOrderDFS(p)) println("Here's node$i via PreOrderDFS: $(summary(node))") end Here's node 1 via PreOrderDFS: Problem Here's node 2 via PreOrderDFS: log (concave; real) Here's node 3 via PreOrderDFS: real variable (id: 111…426) Here's node 4 via PreOrderDFS: 2-element Array{Constraint,1} Here's node 5 via PreOrderDFS: >= constraint (affine) Here's node 6 via PreOrderDFS: real variable (id: 111…426) Here's node 7 via PreOrderDFS: constant (constant; positive) Here's node 8 via PreOrderDFS: <= constraint (affine) Here's node 9 via PreOrderDFS: real variable (id: 111…426) Here's node 10 via PreOrderDFS: constant (constant; positive) ### StatelessBFS Iterator to visit the nodes of a tree, guaranteeing that all nodes of a level will be visited before their children. julia> for (i, node) in enumerate(AbstractTrees.StatelessBFS(p)) println("Here's node$i via StatelessBFS: \$(summary(node))")
end
Here's node 1 via StatelessBFS: Problem
Here's node 2 via StatelessBFS: log (concave; real)
Here's node 3 via StatelessBFS: 2-element Array{Constraint,1}
Here's node 4 via StatelessBFS: real variable (id: 111…426)
Here's node 5 via StatelessBFS: >= constraint (affine)
Here's node 6 via StatelessBFS: <= constraint (affine)
Here's node 7 via StatelessBFS: real variable (id: 111…426)
Here's node 8 via StatelessBFS: constant (constant; positive)
Here's node 9 via StatelessBFS: real variable (id: 111…426)
Here's node 10 via StatelessBFS: constant (constant; positive)