All of the examples can be found in Jupyter notebook form here.

Paper examples

using Convex, ECOS

Summation.

println("Summation example")
x = Variable();
e = 0;
@time begin
  for i = 1:1000
    global e
    e += x;
  end
  p = minimize(e, x>=1);
end
@time solve!(p, ECOSSolver())

Summation example
  0.027473 seconds (4.64 k allocations: 11.763 MiB, 46.26% gc time)
----------------------------------------------------------------------------
	SCS v2.1.1 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
----------------------------------------------------------------------------
Lin-sys: sparse-indirect, nnz in A = 125, CG tol ~ 1/iter^(2.00)
eps = 1.00e-05, alpha = 1.50, max_iters = 5000, normalize = 1, scale = 1.00
acceleration_lookback = 10, rho_x = 1.00e-03
Variables n = 9, constraints m = 75
Cones:	primal zero / dual free vars: 39
	sd vars: 36, sd blks: 1
Setup time: 2.06e-04s
----------------------------------------------------------------------------
 Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)
----------------------------------------------------------------------------
     0| 4.97e+18  4.10e+19  1.00e+00 -8.89e+18  7.29e+19  3.72e+19  2.79e-04
    20| 3.16e-09  3.34e-09  5.90e-10 -2.40e-09 -1.81e-09  3.79e-17  1.54e-03
----------------------------------------------------------------------------
Status: Solved
Timing: Solve time: 1.54e-03s
	Lin-sys: avg # CG iterations: 1.90, avg solve time: 3.51e-06s
	Cones: avg projection time: 5.22e-05s
	Acceleration: avg step time: 1.35e-05s
----------------------------------------------------------------------------
Error metrics:
dist(s, K) = 8.2379e-10, dist(y, K*) = 3.2732e-09, s'y/|s||y| = -1.6903e-12
primal res: |Ax + s - b|_2 / (1 + |b|_2) = 3.1596e-09
dual res:   |A'y + c|_2 / (1 + |c|_2) = 3.3352e-09
rel gap:    |c'x + b'y| / (1 + |c'x| + |b'y|) = 5.9001e-10
----------------------------------------------------------------------------
c'x = -0.0000, -b'y = -0.0000
============================================================================
----------------------------------------------------------------------------
	SCS v2.1.1 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
----------------------------------------------------------------------------
Lin-sys: sparse-indirect, nnz in A = 27, CG tol ~ 1/iter^(2.00)
eps = 1.00e-05, alpha = 1.50, max_iters = 5000, normalize = 1, scale = 1.00
acceleration_lookback = 10, rho_x = 1.00e-03
Variables n = 6, constraints m = 17
Cones:	primal zero / dual free vars: 2
	linear vars: 8
	soc vars: 7, soc blks: 2
Setup time: 3.79e-05s
----------------------------------------------------------------------------
 Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)
----------------------------------------------------------------------------
     0| 1.65e+19  1.98e+19  1.00e+00 -3.08e+19  5.62e+18  4.89e+19  1.86e-05
    60| 2.67e-09  6.52e-09  2.01e-09  6.32e-03  6.32e-03  1.25e-16  3.61e-04
----------------------------------------------------------------------------
Status: Solved
Timing: Solve time: 3.63e-04s
	Lin-sys: avg # CG iterations: 3.15, avg solve time: 9.30e-07s
	Cones: avg projection time: 1.14e-07s
	Acceleration: avg step time: 4.04e-06s
----------------------------------------------------------------------------
Error metrics:
dist(s, K) = 1.1734e-16, dist(y, K*) = 2.7756e-17, s'y/|s||y| = -5.0362e-17
primal res: |Ax + s - b|_2 / (1 + |b|_2) = 2.6652e-09
dual res:   |A'y + c|_2 / (1 + |c|_2) = 6.5229e-09
rel gap:    |c'x + b'y| / (1 + |c'x| + |b'y|) = 2.0108e-09
----------------------------------------------------------------------------
c'x = 0.0063, -b'y = 0.0063
============================================================================

ECOS 2.0.5 - (C) embotech GmbH, Zurich Switzerland, 2012-15. Web: www.embotech.com/ECOS

It     pcost       dcost      gap   pres   dres    k/t    mu     step   sigma     IR    |   BT
 0  +1.000e+03  +1.000e+03  +7e+01  1e-02  4e-05  1e+00  4e+01    ---    ---    1  1  - |  -  -
 1  +1.000e+03  +1.000e+03  +8e-01  1e-04  4e-07  1e-02  4e-01  0.9890  1e-04   1  1  0 |  0  0
 2  +1.000e+03  +1.000e+03  +9e-03  1e-06  4e-09  2e-04  5e-03  0.9890  1e-04   1  0  0 |  0  0
 3  +1.000e+03  +1.000e+03  +1e-04  1e-08  5e-11  3e-06  5e-05  0.9890  1e-04   1  0  0 |  0  0
 4  +1.000e+03  +1.000e+03  +1e-06  2e-10  5e-13  3e-08  6e-07  0.9890  1e-04  0.040663 seconds (97.66 k allocations: 5.013 MiB)

Indexing.

println("Indexing example")
x = Variable(1000, 1);
e = 0;
@time begin
  for i = 1:1000
    global e
    e += x[i];
  end
  p = minimize(e, x >= ones(1000, 1));
end
@time solve!(p, ECOSSolver())

Indexing example
  0.181705 seconds (125.90 k allocations: 17.667 MiB, 8.98% gc time)
  0.145975 seconds (160.90 k allocations: 79.098 MiB, 53.70% gc time)

Matrix constraints.

println("Matrix constraint example")
n, m, p = 100, 100, 100
X = Variable(m, n);
A = randn(p, m);
b = randn(p, n);
@time begin
  p = minimize(norm(vec(X)), A * X == b);
end
@time solve!(p, ECOSSolver())

Matrix constraint example
  0.159357 seconds (242.37 k allocations: 11.278 MiB)
  1.867681 seconds (211.75 k allocations: 147.876 MiB, 15.79% gc time)

Transpose.

println("Transpose example")
X = Variable(5, 5);
A = randn(5, 5);
@time begin
  p = minimize(norm2(X - A), X' == X);
end
@time solve!(p, ECOSSolver())

n = 3
A = randn(n, n);
#@time begin
  X = Variable(n, n);
  p = minimize(norm(vec(X' - A)), X[1,1] == 1);
  solve!(p, ECOSSolver())
#end

Transpose example
  0.287097 seconds (76.47 k allocations: 3.767 MiB, 79.17% gc time)
  0.025857 seconds (33.33 k allocations: 1.664 MiB)

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