Here’s an example of a warning when using set_start_value with GLPK.

┌ Warning: MathOptInterface.VariablePrimalStart() is not supported by MathOptInterface.Bridges.LazyBridgeOptimizer{GLPK.Optimizer}. This information will be discarded.
└ @ MathOptInterface.Utilities C:\Users\plowman\.julia\packages\MathOptInterface\5WwpK\src\Utilities\copy.jl:285

Julia 1.5.3, Julia 1.6.0-rc1
JuMP v0.21.6
GLPK v0.14.7

using JuMP, GLPK
model = Model(GLPK.Optimizer)
x = [1,2,3]
@variable(model, w[1:3] >= 1, Int)
set_start_value.(w, [2,2,6])
start_value.(w)
@constraint(model, sum(w) == 10)
@objective(model, Min, sum(x[i] * w[i] for i in 1:3))
optimize!(model)
println("Objective value: ", objective_value(model))
println("Weights: ", value.(model[:w]))