fix: initial objective for termination conditions

Author: avik-pal

lib/NonlinearSolveFirstOrder/test/least_squares_tests.jl

@@ -54,3 +54,24 @@ end
         @test norm(sol.resid, 2) < 1e-6
     end
 end
+
+@testitem "Correct Best Solution: #565" setup=[CoreNLLSTesting] tags=[:core] begin
+    using StableRNGs
+
+    x = collect(0:0.1:10)
+
+    line_fct(x, p) = p[1] .+ p[2] .* x
+
+    y_line = line_fct(x, [1, 3])
+    y_line_n = line_fct(x, [1, 3]) + randn(StableRNG(0), length(x))
+
+    res(β, (x, y)) = line_fct(x, β) .- y
+
+    prob = NonlinearLeastSquaresProblem(res, [1, 3], p = (x, y_line_n))
+    sol1 = solve(prob; maxiters = 1000)
+
+    prob = NonlinearLeastSquaresProblem(res, [1, 5], p = (x, y_line_n))
+    sol2 = solve(prob; maxiters = 1000)
+
+    @test sol1.u ≈ sol2.u
+end