Wrong objective definition/transcription

[DarioSlaifsteinSk]

Hi,
I'm working on a tracking problem and can't define the objective correctly for the transcription/solver to recognize properly.
My objective is defined as:

@expression(model, ΔSoC[n ∈ 1:nEV], SoCev[n] .- SoCdep[n])
@objective(model, Min, ∫(cgrid,t)/Dt[end] +
        WSoCDep * ∫(sum(γf .* ΔSoC[n].^2 for n ∈ 1:nEV),t)/Dt[end] +
        Wlims*∫(sTin .* 𝒪,t)/Dt[end])

the first and third parts are linear objectives and are transcribed correctly but the second part $$||SoC_{EV} - SoC_{EV}^*||^2_2$$ should be trascribed as a quadratic obj.
From my solver I get:

=======================================
           Academic License
       (NOT FOR COMMERCIAL USE)
         Artelys Knitro 14.2.0
=======================================

Knitro presolve eliminated 443 variables and 61 constraints.

algorithm                1
datacheck                0
hessian_no_f             1
maxtime_real             900
mip_maxtime_real         900
mip_numthreads           1
ms_enable                1
ms_maxsolves             20
ms_numthreads            1
numthreads               1
outlev                   3
The problem is identified as an MPEC.

Problem Characteristics                                 (   Presolved)
-----------------------
Objective goal:  Minimize
Objective type:  general <----- WHY GENERAL?
Number of variables:                               2784 (        2341)

what am I doing wrong?

[pulsipher]

Can you provide a minimum working example? I don't have a license for Knitro, but it is possible this is just a quirk of the Knitro.jl wrapper. The wrapper might just pass any quadratic objective as a general one.

[DarioSlaifsteinSk]

here's the MWE:

model = InfiniteModel(KNITRO.Optimizer);
@infinite_parameter(model, t ∈ [0., 24.], supports = collect(0.:1.:24.),
                        derivative_method = FiniteDifference(Backward()))
@variable(model, 0.2 .≤ SoCev[n ∈ 1:2] .≤ 1, Infinite(t))
@expression(model, ΔSoC[n ∈ 1:2], SoCev[n] .- 0.85)
γ = ones(2,length(0.:1.:24.)); γ[:,10:15] .= 0.;
@parameter_function(model, γf[n ∈ 1:2] == (t) -> γ[n, round(Int,t+1)])
@objective(model, Min, ∫(sum((γf[n].* ΔSoC[n]).^2 for n ∈ 1:2), t))
optimize!(model)
=======================================
           Academic License
       (NOT FOR COMMERCIAL USE)
         Artelys Knitro 14.2.0
=======================================

No start point provided -- Knitro computing one.

Knitro presolve eliminated 0 variables and 0 constraints.

datacheck                0
hessian_no_f             1
mip_numthreads           1
ms_numthreads            1
numthreads               1
The problem is identified as bound constrained only.

Problem Characteristics                                 (   Presolved)
-----------------------
Objective goal:  Minimize
Objective type:  general
Number of variables:                                 50 (          50)
    bounded below only:                               0 (           0)
    bounded above only:                               0 (           0)
    bounded below and above:                         50 (          50)
    fixed:                                            0 (           0)
    free:                                             0 (           0)
Number of constraints:                                0 (           0)
    linear equalities:                                0 (           0)
    quadratic equalities:                             0 (           0)
    gen. nonlinear equalities:                        0 (           0)
    linear one-sided inequalities:                    0 (           0)
    quadratic one-sided inequalities:                 0 (           0)
    gen. nonlinear one-sided inequalities:            0 (           0)
    linear two-sided inequalities:                    0 (           0)
    quadratic two-sided inequalities:                 0 (           0)
    gen. nonlinear two-sided inequalities:            0 (           0)
Number of nonzeros in Jacobian:                       0 (           0)
Number of nonzeros in Hessian:                       38 (          38)

Knitro using the Interior-Point/Barrier Direct algorithm.

  Iter      Objective      FeasError   OptError    ||Step||    CGits
--------  --------------  ----------  ----------  ----------  -------
       0    3.522340e+00   0.000e+00
       5    4.584285e-20   0.000e+00   1.086e-11   3.385e-05        0

EXIT: Locally optimal solution found.

Final Statistics
----------------
Final objective value               =   4.58428486647783e-20
Final feasibility error (abs / rel) =   0.00e+00 / 0.00e+00
Final optimality error  (abs / rel) =   1.09e-11 / 1.09e-11
# of iterations                     =          5
# of CG iterations                  =          0
# of function evaluations           =          7
# of gradient evaluations           =          7
# of Hessian evaluations            =          5
Total program time (secs)           =       0.015 (     0.000 CPU time)
Time spent in evaluations (secs)    =       0.002

===============================================================================

when removing the $\gamma$ from the objective it does become quadratic:

model = InfiniteModel(KNITRO.Optimizer);
@infinite_parameter(model, t ∈ [0., 24.], supports = collect(0.:1.:24.),
                        derivative_method = FiniteDifference(Backward()))
@variable(model, 0.2 .≤ SoCev[n ∈ 1:2] .≤ 1, Infinite(t))
@expression(model, ΔSoC[n ∈ 1:2], SoCev[n] .- 0.85)
γ = ones(2,length(0.:1.:24.)); γ[:,10:15] .= 0.;
@parameter_function(model, γf[n ∈ 1:2] == (t) -> γ[n, round(Int,t+1)])
@objective(model, Min, ∫(sum(ΔSoC[n].^2 for n ∈ 1:2), t))
optimize!(model)
=======================================
           Academic License
       (NOT FOR COMMERCIAL USE)
         Artelys Knitro 14.2.0
=======================================

No start point provided -- Knitro computing one.

Knitro presolve eliminated 0 variables and 0 constraints.

datacheck                0
hessian_no_f             1
The problem is identified as a convex QP.
The problem is identified as bound constrained only.

Problem Characteristics                                 (   Presolved)
-----------------------
Objective goal:  Minimize
Objective type:  quadratic <---  HERE it's fine
Number of variables:                                 50 (          50)
    bounded below only:                               0 (           0)
    bounded above only:                               0 (           0)
    bounded below and above:                         50 (          50)
    fixed:                                            0 (           0)
    free:                                             0 (           0)
Number of constraints:                                0 (           0)
    linear equalities:                                0 (           0)
    quadratic equalities:                             0 (           0)
    gen. nonlinear equalities:                        0 (           0)
    linear one-sided inequalities:                    0 (           0)
    quadratic one-sided inequalities:                 0 (           0)
    gen. nonlinear one-sided inequalities:            0 (           0)
    linear two-sided inequalities:                    0 (           0)
    quadratic two-sided inequalities:                 0 (           0)
    gen. nonlinear two-sided inequalities:            0 (           0)
Number of nonzeros in Jacobian:                       0 (           0)
Number of nonzeros in Hessian:                       50 (          50)

Knitro using the Interior-Point/Barrier Direct algorithm.

  Iter      Objective      FeasError   OptError    ||Step||    CGits
--------  --------------  ----------  ----------  ----------  -------
       0    4.629723e+00   0.000e+00
       7    3.663736e-15   0.000e+00   8.161e-10   4.356e-04        0

EXIT: Optimal solution found.

Final Statistics
----------------
Final objective value               =   3.66373598126302e-15
Final feasibility error (abs / rel) =   0.00e+00 / 0.00e+00
Final optimality error  (abs / rel) =   8.16e-10 / 8.16e-10
# of iterations                     =          7
# of CG iterations                  =          0
# of function evaluations           =          0
# of gradient evaluations           =          0
# of Hessian evaluations            =          0
Total program time (secs)           =       0.009 (     0.000 CPU time)
Time spent in evaluations (secs)    =       0.000

===============================================================================

The thing is that I don't think in my application I can remove the $\gamma$ because I want the SoC to go to the setpoint when the EV is connected ($\gamma=1$) otherwise I'm afraid the opt won't work.

[pulsipher]

Thanks for the MWE, this explains the issue. The problem is that InfiniteOpt stores the objective as a nonlinear expression since you have parameter function multiplying a quadratic term. Naturally, this amounts to a quadratic program after discretization, but it is nontrivial to detect this in general and convert the nonlinear expression back to a quadratic one. Addressing this is on the todo list, but I am not sure when that will happen.

The work around is to introduce an auxiliary variable. So instead of:

using InfiniteOpt
model = InfiniteModel()
@infinite_parameter(model, t in [0, 1])
@variable(model, y, Infinite(t))
@parameter_function(model, pf == sin(t))

@objective(model, Min, integral(pf * y^2, t)) # nonlinear expression that will not become recognized as quadratic during transcription

do

using InfiniteOpt
model = InfiniteModel()
@infinite_parameter(model, t in [0, 1])
@variable(model, y, Infinite(t))
@parameter_function(model, pf == sin(t))

@variable(model, aux, Infinite(t))
@constraint(model, aux == pf * y) # quadratic expression that will become linear during transcription 
@objective(model, Min, integral(aux * y, t)) # quadratic objective

[DarioSlaifsteinSk]

thanks! that solved the issue! the idea was for the solver to recognize the problem as quadratic and enable parallelism. In KNITRO internal parallelism is desabled for general NLP because it is not thread-safe