What is the meaning of multiple objective functions?

I have multiple objective functions in my code, created by a loop around the @objective macro. Here’s the code:

``````using JuMP
using MosekTools
K = 3
N = 2
penalties = [1.0, 3.9, 8.7]
function fac1(r::Number, i::Number, l::Number)
fac1 = 1.0
for m in 0:r-1
fac1 *= (i-m)*(l-m)
end
return fac1
end

function fac2(r::Number, i::Number, l::Number, tau::Float64)
return tau ^ (i + l - 2r + 1)/(i + l - 2r + 1)
end

function Q_r(i::Number, l::Number, r::Number, tau::Float64)
if i >= r && l >= r
return 2 * fac1(r, i, l) * fac2(r, i, l, tau)
else
return 0.0
end
end

function Q(i::Number, l::Number, tau::Number)
elem = 0
for r in 0:N
elem += penalties[r + 1] * Q_r(i, l, r, tau)
end
return elem
end

# discrete segment starting times
mat = Array{Float64, 3}(undef, K, N+1, N+1)
function Q_mat()
for k in 0:K-1
for i in 1:N+1
for j in 1:N+1
mat[k+1, i, j] = Q(i, j, convert(Float64, k))
end
end
return mat
end
end

function A_tau(r::Number, n::Number, tau::Float64)
fac = 1
for m in 1:r
fac *= (n - (m - 1))
end
if n >= r
return fac * tau ^ (n - r)
else
return 0.0
end
end

function A_tau_mat(tau::Float64)
mat = Array{Float64, 2}(undef, N+1, N+1)
for i in 1:N+1
for j in 1:N+1
mat[i, j] = A_tau(i, j, tau)
end
end
return mat
end

function A_0(r::Number, n::Number)
if r == n
fac = 1
for m in 1:r
fac *= r - (m - 1)
end
return fac
else
return 0.0
end
end

m = Model(optimizer_with_attributes(Mosek.Optimizer, "QUIET" => false, "INTPNT_CO_TOL_DFEAS" => 1e-7))

@variable(m, A[i=1:K+1,j=1:K,k=1:N+1,l=1:N+1])
@variable(m, p[i=1:K+1,j=1:N+1])

# constraint difference might be a small fractional difference.
# assuming that time difference is 1 second starting from 0.
for i in 1:K
@constraint(m, -A_tau_mat(convert(Float64, i-1)) * p[i] .+ A_tau_mat(convert(Float64, i-1)) * p[i+1] .== [0.0, 0.0, 0.0])
end

for i in 1:K+1
@constraint(m, A_tau_mat(convert(Float64, i-1)) * p[i] .== [1.0 12.0 13.0])
end
@constraint(m, A_tau_mat(convert(Float64, K+1)) * p[K+1] .== [0.0 0.0 0.0])

for i in 1:K+1
@objective(m, Min, p[i]' * Q_mat()[i] * p[i])
end

optimize!(m)
println("p value is ", value.(p))

``````

The code runs successfully. What does it mean to have multiple objective functions? If there are multiple objective functions, then which one is being minimized? Are they being minimized jointly?

There is only one objective function, `@objective` replaces the last objective function set so only the last `@objective` call is taken into account.

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Is @constraint replaced in the for loop too? If that’s the case, then how to programmatically specify K number of constraints as is the case in the above code?

No, each `@constraint` macro creates a new constraint. There can by arbitrarily many constraints but only one objective.

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