# Multi-objective MINLP problem in JuMP

**URL:** <https://discourse.julialang.org/t/multi-objective-minlp-problem-in-jump/103117>\
**Category:** Optimization (Mathematical)\
**Tags:** jump, gurobi\
**Created:** [August 23, 2023, 4:12pm UTC](https://discourse.julialang.org/t/multi-objective-minlp-problem-in-jump/103117 "2023-08-23T16:12:19Z")\
**Posts on this page:** 1\
**Showing post:** 3

<div class="post-metadata">

**Author:** ![ashefa](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ashefa/32/2525_2.png) [@ashefa](https://discourse.julialang.org/u/ashefa)\
**Post date:** [August 24, 2023, 8:47am UTC](https://discourse.julialang.org/t/multi-objective-minlp-problem-in-jump/103117/3 "2023-08-24T08:47:02Z")

</div>

Thanks for the response. Consider the following code:

```julia
A = rand(1,10)
B = rand(1,10)

T=length(A)
M=10^6

c_S = 60
c_R = 100

using JuMP
import Juniper
import HiGHS
import MultiObjectiveAlgorithms as MOA

model = JuMP.Model(() -> MOA.Optimizer(HiGHS.Optimizer))

set_attribute(model, MOA.Algorithm(), MOA.Dichotomy())
set_attribute(model, MOA.SolutionLimit(), 5)

@variable(model, S >= 0);
@variable(model, 1 <= R <= 3);
@variable(model, y1[t = 1:T], Bin);
@variable(model, y2[t = 1:T], Bin);
@variable(model, Y[t = 1:T] >= 0);
@variable(model, O[t = 1:T] >= 0);
@variable(model, V[t = 0:T] >= 0);
@variable(model, Q[t = 1:T] >= 0);

@constraint(model, V[0] == 0);
@constraint(model, [t in 1:T], Q[t] == B[t] * R);

@constraint(model, [t in 1:T], Y[t] <= A[t]);
@constraint(model, [t in 1:T], Y[t] <= V[t-1]);
@constraint(model, [t in 1:T], Y[t] >= A[t] - M * (1 - y1[t]));
@constraint(model, [t in 1:T], Y[t] >= V[t-1] - M * y1[t]);

@constraint(model, [t in 1:T], V[t] <= V[t-1] + Q[t] - O[t] - Y[t]);
@constraint(model, [t in 1:T], V[t] <= S - Y[t]);
@constraint(model, [t in 1:T], V[t] >= V[t-1] + Q[t] - O[t] - Y[t] - M * (1 - y2[t]));
@constraint(model, [t in 1:T], V[t] >= S - Y[t] - M * y2[t]);

@objective(model, Max, [-c_S * S - c_R * R, sum(Y[t] for t in 1:T)])

optimize!(model)
solution_summary(model)

println(objective_value(model))
println("S = ", value.(S), "\n","A = ",value.(R))

```

Now I want to have a solution when I am changing the first objective function to the quadratic term `-c_S * S^2 - c_R * R` (similar to the approach of @Arpit_Bhatia in [NL MO portfolio optimization](https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/portfolio/#Multi-objective-portfolio-optimization)).

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