Objective Sensitivity 2.0

Project.toml

@@ -1,7 +1,7 @@
 name = "DiffOpt"
 uuid = "930fe3bc-9c6b-11ea-2d94-6184641e85e7"
 authors = ["Akshay Sharma", "Mathieu Besançon", "Joaquim Dias Garcia", "Benoît Legat", "Oscar Dowson", "Andrew Rosemberg"]
-version = "0.5.1"
+version = "0.6.0"
 
 [deps]
 BlockDiagonals = "0a1fb500-61f7-11e9-3c65-f5ef3456f9f0"

docs/src/usage.md

@@ -117,4 +117,78 @@ DiffOpt.reverse_differentiate!(model)
 @show MOI.get(model, DiffOpt.ReverseConstraintSet(), ParameterRef(p)) == MOI.Parameter(direction_x * 3 / pc_val)
 @show abs(MOI.get(model, DiffOpt.ReverseConstraintSet(), ParameterRef(pc)).value -
     -direction_x * 3 * p_val / pc_val^2) < 1e-5
-```
+```
+
+## Calculating objective sensitivity with respect to parameters (currently only supported for Nonlinear Programs)
+
+Consider a differentiable model with parameters `p` and `pc` as in the previous example:
+
+```julia
+using JuMP, DiffOpt, HiGHS
+
+model = Model(() -> DiffOpt.diff_optimizer(Ipopt.Optimizer))
+set_silent(model)
+
+p_val = 4.0
+pc_val = 2.0
+@variable(model, x)
+@variable(model, p in Parameter(p_val))
+@variable(model, pc in Parameter(pc_val))
+@constraint(model, cons, pc * x >= 3 * p)
+@objective(model, Min, x^4)
+optimize!(model)
+
+direction_p = 3.0
+MOI.set(model, DiffOpt.ForwardConstraintSet(), ParameterRef(p), Parameter(direction_p))
+DiffOpt.forward_differentiate!(model)
+```
+
+Using Lagrangian duality we could already calculate the objective sensitivity with respect to parameters that appear as constants of the constraints (e.g, `cons` in this case for parameter `p`) - i.e. The objective sensitivity w.r.t. a constant parameter change is given by the optimal multiplier.
+
+On the other hand, if the parameter appears as a coefficient of the constraints, one can calculate the objective sensitivity with respect to the parameter using the sensitivities of the variables with respect to the parameter, \( \frac{\partial x}{\partial p} \), and the gradient of the objective with respect to the variables \( \frac{\partial f}{\partial x} \):
+
+```math
+\frac{\partial f}{\partial p} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial p}
+```
+ - A consequence of the chain-rule.
+
+Note that, if the parameter appears as a constant in a constraint, the objective sensitivity calculated through solution sensitivity is equivalent to the optimal multiplier associated with the constraint. 
+
+In order to calculate the objective perturbation with respect to the parameter perturbation vector, we can use the following code:
+
+```julia
+# Always a good practice to clear previously set sensitivities
+DiffOpt.empty_input_sensitivities!(model)
+
+MOI.set(model, DiffOpt.ForwardConstraintSet(), ParameterRef(p), Parameter(3.0))
+MOI.set(model, DiffOpt.ForwardConstraintSet(), ParameterRef(p_c), Parameter(3.0))
+DiffOpt.forward_differentiate!(model)
+
+MOI.get(model, DiffOpt.ForwardObjectiveSensitivity())
+```
+
+In the backward mode, we can calculate the parameter perturbation with respect to the objective perturbation:
+
+```julia
+# Always a good practice to clear previously set sensitivities
+DiffOpt.empty_input_sensitivities!(model)
+
+MOI.set(
+    model,
+    DiffOpt.ReverseObjectiveSensitivity(),
+    0.1,
+)
+
+DiffOpt.reverse_differentiate!(model)
+
+MOI.get(model, DiffOpt.ReverseConstraintSet(), ParameterRef(p))
+```
+
+It is important to note that the (reverse) parameter perturbation given an objective perturbation is somewhat equivalent to the perturbation with respect to solution (since one can be calculated from the other). Therefore, one cannot set both the objective sensitivity (`DiffOpt.ReverseObjectiveSensitivity`) and the solution sensitivity (e.g. `DiffOpt.ReverseVariablePrimal`) at the same time - the code will throw an error if you try to do so.
+
+**Dual Objective Sensitivity**
+
+In addition to the primal objective sensitivity, one could also calculate the dual objective sensitivity with respect to the parameters using the gradients of the dual objective and the sensitivities of the dual variables with respect to the parameters.
+This is currently not implemented for any problem class, but will be available in future releases.
+
+Note that the dual objective sensitivity is equivalent to the primal objective sensitivity problems where strong duality holds.

src/ConicProgram/ConicProgram.jl

@@ -450,4 +450,12 @@ function MOI.get(
     return MOI.get(model.model, attr, ci)
 end
 
+function MOI.get(::Model, ::DiffOpt.ForwardObjectiveSensitivity)
+    return error("Not implemented")
+end
+
+function MOI.set(::Model, ::DiffOpt.ReverseObjectiveSensitivity, val)
+    return error("Not implemented")
+end
+
 end

src/NonLinearProgram/NonLinearProgram.jl

@@ -27,10 +27,11 @@ end
 Base.@kwdef struct ForwCache
     primal_Δs::Dict{MOI.VariableIndex,Float64}  # Sensitivity for primal variables (indexed by VariableIndex)
     dual_Δs::Vector{Float64}           # Sensitivity for constraints and bounds (indexed by ConstraintIndex)
+    dual_p::Float64  # Objective Sensitivity wrt parameters
 end
 
 Base.@kwdef struct ReverseCache
-    Δp::Vector{Float64}  # Sensitivity for parameters
+    Δp::Dict{MOI.ConstraintIndex,Float64}  # Sensitivity for parameters
 end
 
 # Define the form of the NLP
@@ -525,15 +526,19 @@ function DiffOpt.forward_differentiate!(model::Model; tol = 1e-6)
         end
 
         # Compute Jacobian
-        Δs = _compute_sensitivity(model; tol = tol)
+        Δs, df_dp = _compute_sensitivity(model; tol = tol)
 
         # Extract primal and dual sensitivities
         primal_Δs = Δs[1:length(model.cache.primal_vars), :] * Δp # Exclude slacks
         dual_Δs = Δs[cache.index_duals, :] * Δp # Includes constraints and bounds
 
+        # obj sensitivity wrt parameters
+        dual_p = df_dp * Δp
+
         model.forw_grad_cache = ForwCache(;
             primal_Δs = Dict(model.cache.primal_vars .=> primal_Δs),
             dual_Δs = dual_Δs,
+            dual_p = dual_p,
         )
     end
     return nothing
@@ -545,50 +550,53 @@ function DiffOpt.reverse_differentiate!(model::Model; tol = 1e-6)
         form = model.model
 
         # Compute Jacobian
-        Δs = _compute_sensitivity(model; tol = tol)
-        num_primal = length(cache.primal_vars)
-        # Fetch primal sensitivities
-        Δx = zeros(num_primal)
-        for (i, var_idx) in enumerate(cache.primal_vars)
-            if haskey(model.input_cache.dx, var_idx)
-                Δx[i] = model.input_cache.dx[var_idx]
+        Δs, df_dp = _compute_sensitivity(model; tol = tol)
+        Δp = if !iszero(model.input_cache.dobj)
+            model.input_cache.dobj * df_dp
+        else
+            num_primal = length(cache.primal_vars)
+            # Fetch primal sensitivities
+            Δx = zeros(num_primal)
+            for (i, var_idx) in enumerate(cache.primal_vars)
+                if haskey(model.input_cache.dx, var_idx)
+                    Δx[i] = model.input_cache.dx[var_idx]
+                end
             end
-        end
-        # Fetch dual sensitivities
-        num_constraints = length(cache.cons)
-        num_up = length(cache.has_up)
-        num_low = length(cache.has_low)
-        Δdual = zeros(num_constraints + num_up + num_low)
-        for (i, ci) in enumerate(cache.cons)
-            idx = form.nlp_index_2_constraint[ci]
-            if haskey(model.input_cache.dy, idx)
-                Δdual[i] = model.input_cache.dy[idx]
+            # Fetch dual sensitivities
+            num_constraints = length(cache.cons)
+            num_up = length(cache.has_up)
+            num_low = length(cache.has_low)
+            Δdual = zeros(num_constraints + num_up + num_low)
+            for (i, ci) in enumerate(cache.cons)
+                idx = form.nlp_index_2_constraint[ci]
+                if haskey(model.input_cache.dy, idx)
+                    Δdual[i] = model.input_cache.dy[idx]
+                end
             end
-        end
-        for (i, var_idx) in enumerate(cache.primal_vars[cache.has_low])
-            idx = form.constraint_lower_bounds[var_idx.value].value
-            if haskey(model.input_cache.dy, idx)
-                Δdual[num_constraints+i] = model.input_cache.dy[idx]
+            for (i, var_idx) in enumerate(cache.primal_vars[cache.has_low])
+                idx = form.constraint_lower_bounds[var_idx.value].value
+                if haskey(model.input_cache.dy, idx)
+                    Δdual[num_constraints+i] = model.input_cache.dy[idx]
+                end
             end
-        end
-        for (i, var_idx) in enumerate(cache.primal_vars[cache.has_up])
-            idx = form.constraint_upper_bounds[var_idx.value].value
-            if haskey(model.input_cache.dy, idx)
-                Δdual[num_constraints+num_low+i] = model.input_cache.dy[idx]
+            for (i, var_idx) in enumerate(cache.primal_vars[cache.has_up])
+                idx = form.constraint_upper_bounds[var_idx.value].value
+                if haskey(model.input_cache.dy, idx)
+                    Δdual[num_constraints+num_low+i] = model.input_cache.dy[idx]
+                end
             end
+            # Extract Parameter sensitivities
+            Δw = zeros(size(Δs, 1))
+            Δw[1:num_primal] = Δx
+            Δw[cache.index_duals] = Δdual
+            Δp = Δs' * Δw
         end
-        # Extract Parameter sensitivities
-        Δw = zeros(size(Δs, 1))
-        Δw[1:num_primal] = Δx
-        Δw[cache.index_duals] = Δdual
-        Δp = Δs' * Δw
-
-        # Order by ConstraintIndex
-        varorder =
-            sort(collect(keys(form.var2ci)); by = x -> form.var2ci[x].value)
-        Δp = [Δp[form.var2param[var_idx].value] for var_idx in varorder]
-
-        model.back_grad_cache = ReverseCache(; Δp = Δp)
+
+        Δp_dict = Dict{MOI.ConstraintIndex,Float64}(
+            form.var2ci[var_idx] => Δp[form.var2param[var_idx].value]
+            for var_idx in keys(form.var2ci)
+        )
+        model.back_grad_cache = ReverseCache(; Δp = Δp_dict)
     end
     return nothing
 end
@@ -620,10 +628,16 @@ function MOI.get(
     ::DiffOpt.ReverseConstraintSet,
     ci::MOI.ConstraintIndex{MOI.VariableIndex,MOI.Parameter{T}},
 ) where {T}
-    form = model.model
-    var_idx = MOI.VariableIndex(ci.value)
-    p_idx = form.var2param[var_idx].value
-    return MOI.Parameter{T}(model.back_grad_cache.Δp[p_idx])
+    return MOI.Parameter{T}(model.back_grad_cache.Δp[ci])
+end
+
+function MOI.get(model::Model, ::DiffOpt.ForwardObjectiveSensitivity)
+    return model.forw_grad_cache.dual_p
+end
+
+function MOI.set(model::Model, ::DiffOpt.ReverseObjectiveSensitivity, val)
+    model.input_cache.dobj = val
+    return
 end
 
 end # module NonLinearProgram

src/NonLinearProgram/nlp_utilities.jl

@@ -27,6 +27,13 @@ function _fill_off_diagonal(H::SparseMatrixCSC)
     return ret
 end
 
+function _compute_gradient(model::Model)
+    evaluator = model.cache.evaluator
+    grad = zeros(length(model.x))
+    MOI.eval_objective_gradient(evaluator, grad, model.x)
+    return grad
+end
+
 """
     _compute_optimal_hessian(evaluator::MOI.Nonlinear.Evaluator, rows::Vector{JuMP.ConstraintRef}, x::Vector{JuMP.VariableRef})
 
@@ -104,7 +111,7 @@ function _create_evaluator(form::Form)
         backend,
         MOI.VariableIndex.(1:form.num_variables),
     )
-    MOI.initialize(evaluator, [:Hess, :Jac])
+    MOI.initialize(evaluator, [:Hess, :Jac, :Grad])
     return evaluator
 end
 
@@ -480,6 +487,11 @@ function _compute_sensitivity(model::Model; tol = 1e-6)
     # Dual bounds lower
     ∂s[(num_w+num_cons+1):(num_w+num_cons+num_lower), :] *= _sense_multiplier
     # Dual bounds upper
-    ∂s[(num_w+num_cons+num_lower+1):end, :] *= -_sense_multiplier
-    return ∂s
+    ∂s[((num_w+num_cons+num_lower+1):end), :] *= -_sense_multiplier
+
+    # dual wrt parameter
+    primal_idx = [i.value for i in model.cache.primal_vars]
+    df_dx = _compute_gradient(model)[primal_idx]
+    df_dp = df_dx'∂s[1:num_vars, :]
+    return ∂s, df_dp
 end

src/QuadraticProgram/QuadraticProgram.jl

@@ -501,4 +501,12 @@ function MOI.set(model::Model, ::LinearAlgebraSolver, linear_solver)
     return model.linear_solver = linear_solver
 end
 
+function MOI.get(::Model, ::DiffOpt.ForwardObjectiveSensitivity)
+    return error("Not implemented")
+end
+
+function MOI.set(::Model, ::DiffOpt.ReverseObjectiveSensitivity, val)
+    return error("Not implemented")
+end
+
 end

src/diff_opt.jl

@@ -15,6 +15,7 @@ Base.@kwdef mutable struct InputCache
     dx::Dict{MOI.VariableIndex,Float64} = Dict{MOI.VariableIndex,Float64}()# dz for QP
     dy::Dict{MOI.ConstraintIndex,Float64} = Dict{MOI.ConstraintIndex,Float64}()
     # Dual sensitivity currently only works for NonLinearProgram
+    dobj::Float64 = 0.0 # Objective input sensitivity for reverse differentiation
     # ds
     # dy #= [d\lambda, d\nu] for QP
     # FIXME Would it be possible to have a DoubleDict where the value depends
@@ -35,6 +36,7 @@ end
 function Base.empty!(cache::InputCache)
     empty!(cache.dx)
     empty!(cache.dy)
+    cache.dobj = 0.0
     empty!(cache.parameter_constraints)
     empty!(cache.scalar_constraints)
     empty!(cache.vector_constraints)
@@ -191,6 +193,20 @@ MOI.set(model, DiffOpt.ReverseConstraintDual(), ci, value)
 """
 struct ReverseConstraintDual <: MOI.AbstractConstraintAttribute end
 
+"""
+    ReverseObjectiveSensitivity <: MOI.AbstractModelAttribute
+
+A `MOI.AbstractModelAttribute` to set input data for reverse differentiation.
+
+For instance, to set the sensitivity of the parameter perturbation with respect to the
+objective function perturbation, do the following:
+
+```julia
+MOI.set(model, DiffOpt.ReverseObjectiveSensitivity(), value)
+```
+"""
+struct ReverseObjectiveSensitivity <: MOI.AbstractModelAttribute end
+
 """
     ForwardConstraintDual <: MOI.AbstractConstraintAttribute
 
@@ -206,6 +222,21 @@ struct ForwardConstraintDual <: MOI.AbstractConstraintAttribute end
 
 MOI.is_set_by_optimize(::ForwardConstraintDual) = true
 
+"""
+    ForwardObjectiveSensitivity <: MOI.AbstractModelAttribute
+
+A `MOI.AbstractModelAttribute` to get output objective sensitivity data from forward differentiation.
+
+For instance, to get the sensitivity of the objective function with respect to the parameter perturbation, do the following:
+
+```julia
+MOI.get(model, DiffOpt.ForwardObjectiveSensitivity())
+```
+"""
+struct ForwardObjectiveSensitivity <: MOI.AbstractModelAttribute end
+
+MOI.is_set_by_optimize(::ForwardObjectiveSensitivity) = true
+
 """
     ReverseObjectiveFunction <: MOI.AbstractModelAttribute