In [1]:

    

using JuMP
using PyPlot
using ImplicitEquations


    

    




/usr/lib/python2.7/dist-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')
WARNING: Method definition max(ValidatedNumerics.Interval, ValidatedNumerics.Interval) in module ValidatedNumerics at /home/magno/.julia/v0.4/ValidatedNumerics/src/intervals/arithmetic.jl:230 overwritten in module ImplicitEquations at /home/magno/.julia/v0.4/ImplicitEquations/src/intervals.jl:6.
WARNING: Method definition min(ValidatedNumerics.Interval, ValidatedNumerics.Interval) in module ValidatedNumerics at /home/magno/.julia/v0.4/ValidatedNumerics/src/intervals/arithmetic.jl:225 overwritten in module ImplicitEquations at /home/magno/.julia/v0.4/ImplicitEquations/src/intervals.jl:7.
WARNING: Base.MathConst is deprecated, use Base.Irrational instead.
  likely near /home/magno/.julia/v0.4/ImplicitEquations/src/intervals.jl:221
WARNING: New definition 
    ^(ImplicitEquations.OInterval, Real) at /home/magno/.julia/v0.4/ImplicitEquations/src/intervals.jl:249
is ambiguous with: 
    ^(Real, ForwardDiff.GradientNumber) at /home/magno/.julia/v0.4/ForwardDiff/src/GradientNumber.jl:139.
To fix, define 
    ^(ImplicitEquations.OInterval, ForwardDiff.GradientNumber)
before the new definition.
WARNING: New definition 
    ^(ImplicitEquations.OInterval, Real) at /home/magno/.julia/v0.4/ImplicitEquations/src/intervals.jl:249
is ambiguous with: 
    ^(Real, ForwardDiff.HessianNumber) at /home/magno/.julia/v0.4/ForwardDiff/src/HessianNumber.jl:209.
To fix, define 
    ^(ImplicitEquations.OInterval, ForwardDiff.HessianNumber)
before the new definition.
WARNING: New definition 
    ^(ImplicitEquations.OInterval, Real) at /home/magno/.julia/v0.4/ImplicitEquations/src/intervals.jl:249
is ambiguous with: 
    ^(Real, ForwardDiff.TensorNumber) at /home/magno/.julia/v0.4/ForwardDiff/src/TensorNumber.jl:264.
To fix, define 
    ^(ImplicitEquations.OInterval, ForwardDiff.TensorNumber)
before the new definition.

WARNING: deprecated syntax "[a=>b, ...]" at /home/magno/.julia/v0.4/ImplicitEquations/src/pyplotgraph.jl:39.
Use "Dict(a=>b, ...)" instead.

In [2]:

    

m = Model()

@variable m x
@variable m y

@NLobjective(m, Min, (x+1)^2 + (y-1)^2)

c1(x, y) = 2*y - 1
c2(x, y) = (1-x)*(4-x^2-y^2)
c3(x, y) = 100 - 2*x^2 - y^2
@constraint(m, 2*y - 1 == 0)
@NLconstraint(m, (1-x)*(4-x^2-y^2) <= 0)
@NLconstraint(m, 100 - 2*x^2 - y^2 >= 0)

setvalue(x, -2)
setvalue(y, 0)

m


    

    
Out[2]:





$$ \begin{alignat*}{1}\min\quad & (nonlinear expression)\\
\text{Subject to} \quad & 2 y = 1\\
 & 2 nonlinear constraints\\
 & x free\\
 & y free\\
\end{alignat*}
 $$

In [7]:

    

solve(m)


    

    




This is Ipopt version 3.12.1, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:        1
Number of nonzeros in inequality constraint Jacobian.:        4
Number of nonzeros in Lagrangian Hessian.............:        7

Total number of variables............................:        2
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:        1
Total number of inequality constraints...............:        2
        inequality constraints with only lower bounds:        1
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        1

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  2.0000000e+00 1.00e+00 1.15e-01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.2681427e+00 0.00e+00 1.09e+00  -1.0 5.00e-01    -  9.92e-01 1.00e+00f  1
   2  1.2246371e+00 0.00e+00 1.21e-04  -1.0 4.83e-01    -  1.00e+00 1.00e+00h  1
   3  1.1322171e+00 0.00e+00 6.45e-04  -2.5 5.76e-01    -  1.00e+00 1.00e+00f  1
   4  1.1271754e+00 0.00e+00 3.87e-06  -3.8 1.62e-02    -  1.00e+00 1.00e+00h  1
   5  1.1270185e+00 0.00e+00 2.52e-09  -5.7 9.04e-04    -  1.00e+00 1.00e+00h  1
   6  1.1270167e+00 0.00e+00 3.06e-13  -8.6 1.12e-05    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 6

                                   (scaled)                 (unscaled)
Objective...............:   1.1270166546526288e+00    1.1270166546526288e+00
Dual infeasibility......:   3.0553337637684308e-13    3.0553337637684308e-13
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   2.5058222494813427e-09    2.5058222494813427e-09
Overall NLP error.......:   2.5058222494813427e-09    2.5058222494813427e-09


Number of objective function evaluations             = 7
Number of objective gradient evaluations             = 7
Number of equality constraint evaluations            = 7
Number of inequality constraint evaluations          = 7
Number of equality constraint Jacobian evaluations   = 7
Number of inequality constraint Jacobian evaluations = 7
Number of Lagrangian Hessian evaluations             = 6
Total CPU secs in IPOPT (w/o function evaluations)   =      0.000
Total CPU secs in NLP function evaluations           =      0.000

EXIT: Optimal Solution Found.






    
Out[7]:





:Optimal

In [8]:

    

println("got ", getobjectivevalue(m), " at ", [getvalue(x),getvalue(y)])


    

    




got 1.1270166546526288 at [-1.9364916735628934,0.5]

In [ ]:

    

plot(c2 <= 0)


    

In [ ]: