Advanced Features

DCP Errors

When a problem is solved that involves an expression which is not of DCP form, an error is emitted. For example,

julia> using Convex, SCS

julia> x = Variable();

julia> y = Variable();

julia> p = minimize(log(x) + square(y), [x >= 0, y >= 0]);

julia> solve!(p, SCS.Optimizer; silent = true)
┌ Warning: Problem not DCP compliant: objective is not DCP
└ @ Convex ~/.julia/dev/Convex/src/problems.jl:73
ERROR: DCPViolationError: Expression not DCP compliant. This either means that your problem is not convex, or that we could not prove it was convex using the rules of disciplined convex programming. For a list of supported operations, see https://jump.dev/Convex.jl/stable/operations/. For help writing your problem as a disciplined convex program, please post a reproducible example on https://jump.dev/forum.
Stacktrace:
[...]

See Extended formulations and the DCP ruleset for more discussion on why these errors occur.

Dual Variables

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

julia> using Convex, SCS
julia> x = Variable();
julia> constraint = x >= 0;
julia> p = minimize(x, [constraint]);
julia> solve!(p, SCS.Optimizer; silent = true)Problem statistics
  problem is DCP         : true
  number of variables    : 1 (1 scalar elements)
  number of constraints  : 1 (1 scalar elements)
  number of coefficients : 1
  number of atoms        : 1

Solution summary
  termination status : OPTIMAL
  primal status      : FEASIBLE_POINT
  dual status        : FEASIBLE_POINT
  objective value    : 0.0

Expression graph
  minimize
   └─ real variable (id: 118…164)
  subject to
   └─ ≥ constraint (affine)
      └─ + (affine; real)
         ├─ real variable (id: 118…164)
         └─ [0;;]
julia> constraint.dual0.999999976532818

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.

using Convex, SCS
n = 1_000
y = rand(n);
x = Variable(n)
lambda = Variable(Positive())
fix!(lambda, 100)
problem = minimize(sumsquares(y - x) + lambda * sumsquares(x - 10))
@time solve!(problem, SCS.Optimizer)
# Now warmstart. If the solver takes advantage of warmstarts, this run will be
# faster
fix!(lambda, 105)
@time solve!(problem, SCS.Optimizer; 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 its current 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.

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
set_value!(y, rand(k, n))
# we'll do 10 iterations of alternating minimization
for i in 1:10
    # first solve for x. With y fixed, the problem is convex.
    fix!(y)
    solve!(problem, SCS.Optimizer; warmstart = i > 1 ? true : false)
    # Now solve for y with x fixed at the previous solution.
    free!(y)
    fix!(x)
    solve!(problem, SCS.Optimizer; warmstart = true)
    free!(x)
end

Custom Variable Types

By making subtypes of Convex.AbstractVariable that conform to the appropriate interface (see the Convex.AbstractVariable docstring for details), one can easily provide custom variable types for specific constructions. These aren't always necessary though; for example, one can define the following function probabilityvector:

using Convex
function probability_vector(d::Int)
    x = Variable(d, Positive())
    add_constraint!(x, sum(x) == 1)
    return x
end

probability_vector (generic function with 1 method)

and then use, say, p = probabilityvector(3) in any Convex.jl problem. The constraints that the entries of p are non-negative and sum to 1 will be automatically added to any problem p is used in.

Custom types are necessary when one wants to dispatch on custom variables, use them as callable types, or provide a different implementation. Continuing with the probability vector example, let's say we often use probability vectors variables in taking expectation values, and we want to use function notation for this. To do so, we define:

julia> using Convex
julia> mutable struct ProbabilityVector <: Convex.AbstractVariable
           head::Symbol
           size::Tuple{Int,Int}
           value::Union{Convex.Value,Nothing}
           vexity::Convex.Vexity
           function ProbabilityVector(d)
               return new(:ProbabilityVector, (d, 1), nothing, Convex.AffineVexity())
           end
       end
julia> Convex.get_constraints(p::ProbabilityVector) = [ sum(p) == 1 ]
julia> Convex.sign(::ProbabilityVector) = Convex.Positive()
julia> Convex.vartype(::ProbabilityVector) = Convex.ContVar
julia> (p::ProbabilityVector)(x) = dot(p, x)

Warning

Custom variable types must be mutable, otherwise variables with the same size and value would be treated as the same object.

Then one can call p = ProbabilityVector(3) to construct a our custom variable which can be used in Convex, which already encodes the appropriate constraints (non-negative and sums to 1), and which can act on constants via p(x). For example,

julia> using SCS
julia> p = ProbabilityVector(3)Variable
size: (3, 1)
sign: positive
vexity: affine
id: 517…233
julia> prob = minimize(p([1.0, 2.0, 3.0]))Problem statistics
  problem is DCP         : true
  number of variables    : 1 (3 scalar elements)
  number of constraints  : 1 (1 scalar elements)
  number of coefficients : 4
  number of atoms        : 4

Solution summary
  termination status : OPTIMIZE_NOT_CALLED
  primal status      : NO_SOLUTION
  dual status        : NO_SOLUTION

Expression graph
  minimize
   └─ sum (affine; positive)
      └─ .* (affine; positive)
         ├─ 3-element positive variable (id: 517…233)
         └─ [1.0; 2.0; 3.0;;]
  subject to
   └─ == constraint (affine)
      └─ + (affine; real)
         ├─ sum (affine; positive)
         │  └─ …
         └─ [-1;;]
julia> solve!(prob, SCS.Optimizer; silent = false);[ Info: [Convex.jl] Compilation finished: 0.16 seconds, 5.898 MiB of memory allocated
------------------------------------------------------------------
	       SCS v3.2.3 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 3, constraints m: 4
cones: 	  z: primal zero / dual free vars: 1
	  l: linear vars: 3
settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
	  acceleration_lookback: 10, acceleration_interval: 10
lin-sys:  sparse-direct-amd-qdldl
	  nnz(A): 6, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 1.94e+01  3.06e+00  4.02e+01 -1.60e+01  1.00e-01  4.98e-05 
    50| 1.09e-04  7.07e-05  1.94e-04  1.00e+00  1.00e-01  7.36e-05 
------------------------------------------------------------------
status:  solved
timings: total: 7.43e-05s = setup: 3.83e-05s + solve: 3.59e-05s
	 lin-sys: 7.33e-06s, cones: 4.24e-06s, accel: 2.19e-06s
------------------------------------------------------------------
objective = 1.000168
------------------------------------------------------------------
julia> evaluate(p)3-element Vector{Float64}:
 0.9998127159848165
 0.00010875388968836632
 7.823604474042554e-5

Subtypes of AbstractVariable must have the fields head and size. Then they must also

either have a field value, or implement Convex._value and Convex.set_value!
either have a field vexity, or implement Convex.vexity and Convex.vexity! (though the latter is only necessary if you wish to support Convex.fix! and Convex.free!)
have a field constraints or implement Convex.get_constraints (optionally implement Convex.add_constraint! to be able to add constraints to your variable after its creation),
either have a field sign or implement Convex.sign, and
either have a field vartype, or implement Convex.vartype (optionally, implement Convex.vartype! to be able to change a variable's vartype after construction).

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 for example, 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.

A related setting is Convex.MAXDIGITS, which controls printing the internal IDs of atoms: if the string representation of an ID is longer than double the value of MAXDIGITS, then it is shortened by printing only the first and last MAXDIGITS characters.

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

using Convex, AbstractTrees
x = Variable()
p = maximize( log(x), x >= 1, x <= 3 )
for leaf in AbstractTrees.Leaves(p)
    println("Here's a leaf: $(summary(leaf))")
end

Here's a leaf: real variable (id: 181…279)
Here's a leaf: real variable (id: 181…279)
Here's a leaf: negative constant
Here's a leaf: real variable (id: 181…279)
Here's a leaf: negative constant

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.

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: 181…279)
Here's node 2 via PostOrderDFS: log (concave; real)
Here's node 3 via PostOrderDFS: real variable (id: 181…279)
Here's node 4 via PostOrderDFS: negative constant
Here's node 5 via PostOrderDFS: + (affine; real)
Here's node 6 via PostOrderDFS: ≥ constraint (affine)
Here's node 7 via PostOrderDFS: real variable (id: 181…279)
Here's node 8 via PostOrderDFS: negative constant
Here's node 9 via PostOrderDFS: + (affine; real)
Here's node 10 via PostOrderDFS: ≤ constraint (affine)
Here's node 11 via PostOrderDFS: 2-element Vector{Constraint}
Here's node 12 via PostOrderDFS: Problem{Float64} (concave; real)

PreOrderDFS

Iterator to visit the nodes of a tree, guaranteeing that parents will be visited before their children.

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{Float64} (concave; real)
Here's node 2 via PreOrderDFS: log (concave; real)
Here's node 3 via PreOrderDFS: real variable (id: 181…279)
Here's node 4 via PreOrderDFS: 2-element Vector{Constraint}
Here's node 5 via PreOrderDFS: ≥ constraint (affine)
Here's node 6 via PreOrderDFS: + (affine; real)
Here's node 7 via PreOrderDFS: real variable (id: 181…279)
Here's node 8 via PreOrderDFS: negative constant
Here's node 9 via PreOrderDFS: ≤ constraint (affine)
Here's node 10 via PreOrderDFS: + (affine; real)
Here's node 11 via PreOrderDFS: real variable (id: 181…279)
Here's node 12 via PreOrderDFS: negative constant

StatelessBFS

Iterator to visit the nodes of a tree, guaranteeing that all nodes of a level will be visited before their children.

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{Float64} (concave; real)
Here's node 2 via StatelessBFS: log (concave; real)
Here's node 3 via StatelessBFS: 2-element Vector{Constraint}
Here's node 4 via StatelessBFS: real variable (id: 181…279)
Here's node 5 via StatelessBFS: ≥ constraint (affine)
Here's node 6 via StatelessBFS: ≤ constraint (affine)
Here's node 7 via StatelessBFS: + (affine; real)
Here's node 8 via StatelessBFS: + (affine; real)
Here's node 9 via StatelessBFS: real variable (id: 181…279)
Here's node 10 via StatelessBFS: negative constant
Here's node 11 via StatelessBFS: real variable (id: 181…279)
Here's node 12 via StatelessBFS: negative constant