# Polynomial optimization problem can't deal with redundant constraints

**URL:** <https://discourse.julialang.org/t/polynomial-optimization-problem-cant-deal-with-redundant-constraints/41839>\
**Category:** Optimization (Mathematical)\
**Tags:** jump, optimization\
**Created:** [June 22, 2020, 5:58am UTC](https://discourse.julialang.org/t/polynomial-optimization-problem-cant-deal-with-redundant-constraints/41839 "2020-06-22T05:58:30Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![aliceee](https://avatars.discourse-cdn.com/v4/letter/a/8baadc/32.png) [@aliceee](https://discourse.julialang.org/u/aliceee)\
**Post date:** [June 22, 2020, 5:58am UTC](https://discourse.julialang.org/t/polynomial-optimization-problem-cant-deal-with-redundant-constraints/41839/1 "2020-06-22T05:58:30Z")

</div>

I’m exploring the following polynomial optimization problem:

```julia
using SumOfSquares
using DynamicPolynomials
using MosekTools

# Create symbolic variables (not JuMP decision variables)
@polyvar x1 x2

# Create a Sum of Squares JuMP model with the Mosek solver
model = SOSModel(with_optimizer(Mosek.Optimizer))

# Create a JuMP decision variable for the lower bound
@variable(model, γ)
@variable(model, x)

# f(x) is the Goldstein-Price function
f1 = x1+x2+1
f2 = 19-14*x1+3*x1^2-14*x2+6*x1*x2+3*x2^2
f3 = 2*x1-3*x2
f4 = 18-32*x1+12*x1^2+48*x2-36*x1*x2+27*x2^2

f = (1+f1^2*f2)*(30+f3^2*f4)

```

when the constraint takes the form of

```julia
@constraint(model, f >= γ)

```

this problem can be solved to optimal:

```julia
@objective(model, Max, γ)
optimize!(model)
# The lower bound found is 3
println(objective_value(model))

```

However, when I added a redundant constraint

```julia
@constraint(model, f1>= x)

```

the output indicates that this problem has not been solved:

```julia
Problem
  Name :                 
  Objective sense : max             
  Type : CONIC (conic optimization problem)
  Constraints : 49              
  Cones : 0               
  Scalar variables : 3               
  Matrix variables : 1               
  Integer variables : 0               

Optimizer started.
Presolve started.
Eliminator - tries : 0 time : 0.00            
Lin. dep. - tries : 0 time : 0.00            
Lin. dep. - number : 0               
Presolve terminated. Time: 0.00    
Optimizer terminated. Time: 0.00    

0.0

```

I’m pretty confused about this result since the additional constraint is obviously redundant. Why adding a redundant ruins everything? Thanks for your help in advance!

---

<div class="post-metadata">

**Author:** ![blegat](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/blegat/32/217090_2.png) [@blegat](https://discourse.julialang.org/u/blegat)\
**Post date:** [June 25, 2020, 7:12am UTC](https://discourse.julialang.org/t/polynomial-optimization-problem-cant-deal-with-redundant-constraints/41839/2 "2020-06-25T07:12:44Z")

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In the first problem, you are trying to find a value of `γ` such that `f >= γ` **for all** `x1` and `x2`. Therefore `γ` gives you a lower bound for the function `f` but you still don’t know what are the values of `x1` and `x2` that minimize `f` (although you may be able to get it from the dual, see [here](https://github.com/jump-dev/SumOfSquares.jl/blob/master/examples/Polynomial_Optimization.ipynb)).  
In the second example, you want to find a value of `x` such that `x1 + x2 + 1 >= x` **for all** `x1` and `x2`. As the linear expression `x1 + x2 + 1` is not bounded below, e.g. take `x1 -> -Inf` or `x2 -> -Inf`, there is no value of `x` satisfying this so the problem is infeasible.