Constraint to hold on some region

Specific Domains Optimization (Mathematical)
1

Hi,

I am solving an optimization problem of the following interface:

find max A such that
x^A + y < = 100 for all 0 <= x <= y <= 1

How can I write the following type of constraint in JuMP? And how it is better to define that for non-integer parameter A we set 0^A == 0?

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2

Isn’t the answer A = \infty? 1^\infty + 1 = 2 \le 100.

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3

Thank you for your reply,

yes , it’s better to change the 1 in constraint condition by 2, for example :).

4

In which case the answer is the solution to 2^A + 2 = 100 \implies A = \log_2 98.

5

I’ve wrote in the beginning that this problem is just similar to mine in interface. Of course, no need to use computers to solve directly the problem I’ve mentioned. I want to know the sintaxis of writing constraint in JuMP of the mentioned type.

1 Like
6

You generally can’t write “for all” constraints in JuMP (that’s just not the way that solvers work).

You could write the inverse of the problem though. Find the minimum A such that the constraint is violated.

julia> using JuMP

julia> import Ipopt

julia> model = Model(Ipopt.Optimizer)
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: EMPTY_OPTIMIZER
Solver name: Ipopt

julia> @variable(model, A >= 0.0)
A

julia> @variable(model, 0 <= x <= 2, start = 2)
x

julia> @variable(model, 0 <= y <= 2, start = 2)
y

julia> @NLconstraint(model, x^A + y >= 100)
(x ^ A + y) - 100.0 ≥ 0

julia> @objective(model, Min, A)
A

julia> optimize!(model)
This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:        3
Number of nonzeros in Lagrangian Hessian.............:        3

Total number of variables............................:        3
                     variables with only lower bounds:        1
                variables with lower and upper bounds:        2
                     variables with only upper bounds:        0
Total number of equality constraints.................:        0
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        1
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        0

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  9.9999900e-03 9.70e+01 5.96e-01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.6878344e-02 9.70e+01 1.04e+00  -1.0 5.56e+01    -  5.11e-04 3.53e-04h  1
   2  1.4884628e+00 9.52e+01 2.37e+00  -1.0 1.40e+02    -  1.28e-04 1.05e-02h  1
   3  7.6262848e+00 0.00e+00 2.76e+03  -1.0 4.91e+01    -  5.89e-02 1.25e-01h  4
   4  7.6104472e+00 0.00e+00 1.59e+03  -1.0 1.23e-01   4.0 8.93e-02 1.00e+00h  1
   5  7.6082893e+00 0.00e+00 2.89e+02  -1.0 3.65e-02   3.5 1.00e+00 1.00e+00h  1
   6  7.6072385e+00 0.00e+00 5.54e+00  -1.0 4.09e-03   3.0 1.00e+00 1.00e+00h  1
   7  7.6048327e+00 0.00e+00 3.52e+00  -1.0 9.51e-03   2.6 1.00e+00 1.00e+00f  1
   8  6.4266817e+00 1.25e+01 3.79e+00  -1.0 8.60e+00    -  9.94e-01 1.37e-01f  1
   9  6.7946110e+00 0.00e+00 1.03e+00  -1.0 2.82e+00    -  1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  6.8204764e+00 0.00e+00 2.01e-02  -1.0 3.81e+00    -  1.00e+00 1.00e+00h  1
  11  6.6382699e+00 0.00e+00 1.87e-02  -2.5 6.93e+00    -  9.64e-01 9.59e-01f  1
  12  6.6153426e+00 0.00e+00 4.30e-03  -3.8 8.16e-01    -  8.22e-01 1.00e+00h  1
  13  6.6151594e+00 0.00e+00 7.80e-08  -3.8 8.22e-03    -  1.00e+00 1.00e+00h  1
  14  6.6147160e+00 0.00e+00 1.30e-07  -5.7 1.01e-02    -  1.00e+00 1.00e+00h  1
  15  6.6147098e+00 0.00e+00 2.80e-11  -8.6 1.74e-04    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 15

                                   (scaled)                 (unscaled)
Objective...............:   6.6147097558501411e+00    6.6147097558501411e+00
Dual infeasibility......:   2.8009594643663149e-11    2.8009594643663149e-11
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Variable bound violation:   1.9474300838595582e-08    1.9474300838595582e-08
Complementarity.........:   2.5931984733449229e-09    2.5931984733449229e-09
Overall NLP error.......:   2.5931984733449229e-09    2.5931984733449229e-09


Number of objective function evaluations             = 18
Number of objective gradient evaluations             = 16
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 21
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 16
Number of Lagrangian Hessian evaluations             = 15
Total seconds in IPOPT                               = 0.009

EXIT: Optimal Solution Found.

julia> value(A)
6.614709755850141

julia> log2(98)
6.614709844115208
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