Convex.jl norm of multiple variables

Specific Domains Optimization (Mathematical)
1

I would like to use Convex.jl to solve a problem along the lines of

using Convex
x = Variable(1)
y = Variable(1)
z = Variable(1)
problem = minimize(x+y+z,norm([x,y,z])<=1)

However, when I run this code, I get the error

MethodError: no method matching AbstractFloat(::Convex.MaxAtom)

Closest candidates are:
  (::Type{T})(::AbstractChar) where T<:Union{AbstractChar, Number}
   @ Base char.jl:50
  (::Type{T})(::Base.TwicePrecision) where T<:Number
   @ Base twiceprecision.jl:266
  (::Type{T})(::Complex) where T<:Real
   @ Base complex.jl:44
  ...


Stacktrace:
 [1] float(x::Convex.MaxAtom)
   @ Base ./float.jl:294
 [2] generic_normInf(x::Vector{Variable})
   @ LinearAlgebra ~/.julia/juliaup/julia-1.9.0-rc2+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:453
 [3] normInf
   @ ~/.julia/juliaup/julia-1.9.0-rc2+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:527 [inlined]
 [4] generic_norm2(x::Vector{Variable})
   @ LinearAlgebra ~/.julia/juliaup/julia-1.9.0-rc2+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:463
 [5] norm2
   @ ~/.julia/juliaup/julia-1.9.0-rc2+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:529 [inlined]
 [6] norm(itr::Vector{Variable}, p::Int64)
   @ LinearAlgebra ~/.julia/juliaup/julia-1.9.0-rc2+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:598
 [7] norm(itr::Vector{Variable})
   @ LinearAlgebra ~/.julia/juliaup/julia-1.9.0-rc2+0.x64.linux.gnu/share/julia/stdlib/v1.9/LinearAlgebra/src/generic.jl:596
 [8] top-level scope
   @ In[9]:5

From this I understand that the norm function can only be used on multi-dimensional variables, not multiple variables. Is there a way to combine multiple one-dimensional variables into a multi-dimensional variable so that this works?

2

Can you just formulate your problem with a single vector variable from the start?

3

I suppose in principle this is always possible, but it would improve readability etc to be able to use multiple names for conceptually different variables.

4

I think you might have better luck by defining the vector variable first and then giving individual names to new variables which you constrain to be equal to the vector components?

5

You can combine scalar variables into a vector-valued variable using hcat or vcat.

julia> using Convex

julia> import SCS

julia> x = Variable(1)
Variable
size: (1, 1)
sign: real
vexity: affine
id: 351…156

julia> y = Variable(1)
Variable
size: (1, 1)
sign: real
vexity: affine
id: 464…161

julia> z = Variable(1)
Variable
size: (1, 1)
sign: real
vexity: affine
id: 749…425

julia> problem = minimize(x + y + z, norm(hcat(x, y, z)) <= 1)
minimize
└─ + (affine; real)
   ├─ real variable (id: 351…156)
   ├─ real variable (id: 464…161)
   └─ real variable (id: 749…425)
subject to
└─ <= constraint (convex)
   ├─ norm2 (convex; positive)
   │  └─ hcat (affine; real)
   │     ├─ …
   │     ├─ …
   │     └─ …
   └─ 1

status: `solve!` not called yet

julia> solve!(problem, SCS.Optimizer)
------------------------------------------------------------------
	       SCS v3.2.3 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 5, constraints m: 6
cones: 	  z: primal zero / dual free vars: 1
	  l: linear vars: 1
	  q: soc vars: 4, qsize: 1
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): 9, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 1.68e+01  1.68e+00  6.00e+01 -3.00e+01  1.00e-01  8.72e-05 
    50| 3.30e-06  1.39e-09  2.86e-06 -1.73e+00  1.00e-01  1.17e-04 
------------------------------------------------------------------
status:  solved
timings: total: 1.19e-04s = setup: 6.81e-05s + solve: 5.12e-05s
	 lin-sys: 9.65e-06s, cones: 5.73e-06s, accel: 2.43e-06s
------------------------------------------------------------------
objective = -1.732052
------------------------------------------------------------------

julia> evaluate(x), evaluate(y), evaluate(z)
(-0.5773512251810479, -0.5773512251810479, -0.5773512251810479)
3 Likes
6

Thank you! Of course this perfectly answers my question. Thanks also @gdalle ; what you suggest would work as well I think but it is very nice to be able to write things using multiple variables when this is convenient.