# Solutions

This section of the manual describes how to access a solved solution to a problem. It uses the following model as an example:

julia> begin
model = Model(HiGHS.Optimizer)
set_silent(model)
@variable(model, x >= 0)
@variable(model, y[[:a, :b]] <= 1)
@objective(model, Max, -12x - 20y[:a])
@expression(model, my_expr, 6x + 8y[:a])
@constraint(model, my_expr >= 100)
@constraint(model, c1, 7x + 12y[:a] >= 120)
optimize!(model)
print(model)
end
Max -12 x - 20 y[a]
Subject to
6 x + 8 y[a] ≥ 100
c1 : 7 x + 12 y[a] ≥ 120
x ≥ 0
y[a] ≤ 1
y[b] ≤ 1

## Check if an optimal solution exists

Use is_solved_and_feasible to check if the solver found an optimal solution:

julia> is_solved_and_feasible(model)
true

By default, is_solved_and_feasible returns true for both global and local optima. Pass allow_local = false to check if the solver found a globally optimal solution:

julia> is_solved_and_feasible(model; allow_local = false)
true

Pass dual = true to check if the solver found an optimal dual solution in addition to an optimal primal solution:

julia> is_solved_and_feasible(model; dual = true)
true

If this function returns false, use the functions mentioned below like solution_summary, termination_status, primal_status, and dual_status to understand what solution (if any) the solver found.

## Solutions summary

solution_summary can be used for checking the summary of the optimization solutions.

julia> solution_summary(model)
* Solver : HiGHS

* Status
Result count       : 1
Termination status : OPTIMAL
Message from the solver:
"kHighsModelStatusOptimal"

* Candidate solution (result #1)
Primal status      : FEASIBLE_POINT
Dual status        : FEASIBLE_POINT
Objective value    : -2.05143e+02
Objective bound    : -0.00000e+00
Relative gap       : Inf
Dual objective value : -2.05143e+02

* Work counters
Solve time (sec)   : 6.70987e-04
Simplex iterations : 2
Barrier iterations : 0
Node count         : -1

julia> solution_summary(model; verbose = true)
* Solver : HiGHS

* Status
Result count       : 1
Termination status : OPTIMAL
Message from the solver:
"kHighsModelStatusOptimal"

* Candidate solution (result #1)
Primal status      : FEASIBLE_POINT
Dual status        : FEASIBLE_POINT
Objective value    : -2.05143e+02
Objective bound    : -0.00000e+00
Relative gap       : Inf
Dual objective value : -2.05143e+02
Primal solution :
x : 1.54286e+01
y[a] : 1.00000e+00
y[b] : 1.00000e+00
Dual solution :
c1 : 1.71429e+00

* Work counters
Solve time (sec)   : 6.70987e-04
Simplex iterations : 2
Barrier iterations : 0
Node count         : -1

## Why did the solver stop?

Usetermination_status to understand why the solver stopped.

julia> termination_status(model)
OPTIMAL::TerminationStatusCode = 1

The MOI.TerminationStatusCode enum describes the full list of statuses that could be returned.

Common return values include OPTIMAL, LOCALLY_SOLVED, INFEASIBLE, DUAL_INFEASIBLE, and TIME_LIMIT.

Info

A return status of OPTIMAL means the solver found (and proved) a globally optimal solution. A return status of LOCALLY_SOLVED means the solver found a locally optimal solution (which may also be globally optimal, but it could not prove so).

Warning

A return status of DUAL_INFEASIBLE does not guarantee that the primal is unbounded. When the dual is infeasible, the primal is unbounded if there exists a feasible primal solution.

Use raw_status to get a solver-specific string explaining why the optimization stopped:

julia> raw_status(model)
"kHighsModelStatusOptimal"

## Primal solutions

### Primal solution status

Use primal_status to return an MOI.ResultStatusCode enum describing the status of the primal solution.

julia> primal_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

Other common returns are NO_SOLUTION, and INFEASIBILITY_CERTIFICATE. The first means that the solver doesn't have a solution to return, and the second means that the primal solution is a certificate of dual infeasibility (a primal unbounded ray).

You can also use has_values, which returns true if there is a solution that can be queried, and false otherwise.

julia> has_values(model)
true

### Objective values

The objective value of a solved problem can be obtained via objective_value. The best known bound on the optimal objective value can be obtained via objective_bound. If the solver supports it, the value of the dual objective can be obtained via dual_objective_value.

julia> objective_value(model)
-205.14285714285714

julia> objective_bound(model)  # HiGHS only implements objective bound for MIPs
-0.0

julia> dual_objective_value(model)
-205.1428571428571

### Primal solution values

If the solver has a primal solution to return, use value to access it:

julia> value(x)
15.428571428571429

Broadcast value over containers:

julia> value.(y)
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, [:a, :b]
And data, a 2-element Vector{Float64}:
1.0
1.0

value also works on expressions:

julia> value(my_expr)
100.57142857142857

and constraints:

julia> value(c1)
120.0
Info

Calling value on a constraint returns the constraint function evaluated at the solution.

## Dual solutions

### Dual solution status

Use dual_status to return an MOI.ResultStatusCode enum describing the status of the dual solution.

julia> dual_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

Other common returns are NO_SOLUTION, and INFEASIBILITY_CERTIFICATE. The first means that the solver doesn't have a solution to return, and the second means that the dual solution is a certificate of primal infeasibility (a dual unbounded ray).

You can also use has_duals, which returns true if there is a solution that can be queried, and false otherwise.

julia> has_duals(model)
true

### Dual solution values

If the solver has a dual solution to return, use dual to access it:

julia> dual(c1)
1.7142857142857142

Query the duals of variable bounds using LowerBoundRef, UpperBoundRef, and FixRef:

julia> dual(LowerBoundRef(x))
0.0

julia> dual.(UpperBoundRef.(y))
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, [:a, :b]
And data, a 2-element Vector{Float64}:
-0.5714285714285694
0.0
Warning

JuMP's definition of duality is independent of the objective sense. That is, the sign of feasible duals associated with a constraint depends on the direction of the constraint and not whether the problem is maximization or minimization. This is a different convention from linear programming duality in some common textbooks. If you have a linear program, and you want the textbook definition, you probably want to use shadow_price and reduced_cost instead.

julia> shadow_price(c1)
1.7142857142857142

julia> reduced_cost(x)
-0.0

julia> reduced_cost.(y)
1-dimensional DenseAxisArray{Float64,1,...} with index sets:
Dimension 1, [:a, :b]
And data, a 2-element Vector{Float64}:
0.5714285714285694
-0.0

You should always check whether the solver found a solution before calling solution functions like value or objective_value.

A simple approach for small scripts and notebooks is to use is_solved_and_feasible:

julia> function solve_and_print_solution(model)
optimize!(model)
if !is_solved_and_feasible(model; dual = true)
error(
"""
The model was not solved correctly:
termination_status : $(termination_status(model)) primal_status :$(primal_status(model))
dual_status        : $(dual_status(model)) raw_status :$(raw_status(model))
""",
)
end
println("Solution is optimal")
println("  objective value = ", objective_value(model))
println("  primal solution: x = ", value(x))
println("  dual solution: c1 = ", dual(c1))
return
end
solve_and_print_solution (generic function with 1 method)

julia> solve_and_print_solution(model)
Solution is optimal
objective value = -205.14285714285714
primal solution: x = 15.428571428571429
dual solution: c1 = 1.7142857142857142

For code like libraries that should be more robust to the range of possible termination and result statuses, do some variation of the following:

julia> function solve_and_print_solution(model)
status = termination_status(model)
if status in (OPTIMAL, LOCALLY_SOLVED)
println("Solution is optimal")
elseif status in (ALMOST_OPTIMAL, ALMOST_LOCALLY_SOLVED)
println("Solution is optimal to a relaxed tolerance")
elseif status == TIME_LIMIT
println(
"Solver stopped due to a time limit. If a solution is available, " *
"it may be suboptimal."
)
elseif status in (
ITERATION_LIMIT, NODE_LIMIT, SOLUTION_LIMIT, MEMORY_LIMIT,
OBJECTIVE_LIMIT, NORM_LIMIT, OTHER_LIMIT,
)
println(
"Solver stopped due to a limit. If a solution is available, it " *
"may be suboptimal."
)
elseif status in (INFEASIBLE, LOCALLY_INFEASIBLE)
println("The problem is primal infeasible")
elseif status == DUAL_INFEASIBLE
println(
"The problem is dual infeasible. If a primal feasible solution " *
"exists, the problem is unbounded. To check, set the objective " *
"to @objective(model, Min, 0) and re-solve. If the problem is " *
"feasible, the primal is unbounded. If the problem is " *
"infeasible, both the primal and dual are infeasible.",
)
elseif status == INFEASIBLE_OR_UNBOUNDED
println(
"The model is either infeasible or unbounded. Set the objective " *
"to @objective(model, Min, 0) and re-solve to disambiguate. If " *
"the problem was infeasible, it will still be infeasible. If the " *
"problem was unbounded, it will now have a finite optimal solution.",
)
else
println(
"The model was not solved correctly. The termination status is $status", ) end if primal_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT) println(" objective value = ", objective_value(model)) println(" primal solution: x = ", value(x)) elseif primal_status(model) == INFEASIBILITY_CERTIFICATE println(" primal certificate: x = ", value(x)) end if dual_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT) println(" dual solution: c1 = ", dual(c1)) elseif dual_status(model) == INFEASIBILITY_CERTIFICATE println(" dual certificate: c1 = ", dual(c1)) end return end solve_and_print_solution (generic function with 1 method) julia> solve_and_print_solution(model) Solution is optimal objective value = -205.14285714285714 primal solution: x = 15.428571428571429 dual solution: c1 = 1.7142857142857142 ## OptimizeNotCalled errors Due to differences in how solvers cache solutions internally, modifying a model after calling optimize! will reset the model into the OPTIMIZE_NOT_CALLED state. If you then attempt to query solution information, an OptimizeNotCalled error will be thrown. If you are iteratively querying solution information and modifying a model, query all the results first, then modify the problem. For example, instead of: julia> model = Model(HiGHS.Optimizer); julia> set_silent(model) julia> @variable(model, x >= 0); julia> optimize!(model) julia> termination_status(model) OPTIMAL::TerminationStatusCode = 1 julia> set_upper_bound(x, 1) julia> x_val = value(x) ┌ Warning: The model has been modified since the last call to optimize! (or optimize! has not been called yet). If you are iteratively querying solution information and modifying a model, query all the results first, then modify the model. └ @ JuMP ~/work/JuMP.jl/JuMP.jl/src/optimizer_interface.jl:712 ERROR: OptimizeNotCalled() Stacktrace: [...] julia> termination_status(model) OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0 do julia> model = Model(HiGHS.Optimizer); julia> set_silent(model) julia> @variable(model, x >= 0); julia> optimize!(model); julia> x_val = value(x) 0.0 julia> termination_status(model) OPTIMAL::TerminationStatusCode = 1 julia> set_upper_bound(x, 1) julia> set_lower_bound(x, x_val) julia> termination_status(model) OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0 If you know that your particular solver supports querying solution information after modifications, you can use direct_model to bypass the OPTIMIZE_NOT_CALLED state: julia> model = direct_model(HiGHS.Optimizer()); julia> set_silent(model) julia> @variable(model, x >= 0); julia> optimize!(model) julia> termination_status(model) OPTIMAL::TerminationStatusCode = 1 julia> set_upper_bound(x, 1) julia> x_val = value(x) 0.0 julia> set_lower_bound(x, x_val) julia> termination_status(model) OPTIMAL::TerminationStatusCode = 1 Warning Be careful doing this. If your particular solver does not support querying solution information after modification, it may silently return incorrect solutions or throw an error. ## Accessing attributes MathOptInterface defines many model attributes that can be queried. Some attributes can be directly accessed by getter functions. These include: ## Sensitivity analysis for LP Given an LP problem and an optimal solution corresponding to a basis, we can question how much an objective coefficient or standard form right-hand side coefficient (c.f., normalized_rhs) can change without violating primal or dual feasibility of the basic solution. Note that not all solvers compute the basis, and for sensitivity analysis, the solver interface must implement MOI.ConstraintBasisStatus. Tip Read the Sensitivity analysis of a linear program for more information on sensitivity analysis. To give a simple example, we could analyze the sensitivity of the optimal solution to the following (non-degenerate) LP problem: julia> begin model = Model(HiGHS.Optimizer) set_silent(model) @variable(model, x[1:2]) set_lower_bound(x[2], -0.5) set_upper_bound(x[2], 0.5) @constraint(model, c1, x[1] + x[2] <= 1) @constraint(model, c2, x[1] - x[2] <= 1) @objective(model, Max, x[1]) print(model) end Max x[1] Subject to c1 : x[1] + x[2] ≤ 1 c2 : x[1] - x[2] ≤ 1 x[2] ≥ -0.5 x[2] ≤ 0.5 To analyze the sensitivity of the problem we could check the allowed perturbation ranges of, for example, the cost coefficients and the right-hand side coefficient of the constraint c1 as follows: julia> optimize!(model) julia> value.(x) 2-element Vector{Float64}: 1.0 -0.0 julia> report = lp_sensitivity_report(model); julia> x1_lo, x1_hi = report[x[1]] (-1.0, Inf) julia> println("The objective coefficient of x[1] could decrease by$(x1_lo) or increase by $(x1_hi).") The objective coefficient of x[1] could decrease by -1.0 or increase by Inf. julia> x2_lo, x2_hi = report[x[2]] (-1.0, 1.0) julia> println("The objective coefficient of x[2] could decrease by$(x2_lo) or increase by $(x2_hi).") The objective coefficient of x[2] could decrease by -1.0 or increase by 1.0. julia> c_lo, c_hi = report[c1] (-1.0, 1.0) julia> println("The RHS of c1 could decrease by$(c_lo) or increase by $(c_hi).") The RHS of c1 could decrease by -1.0 or increase by 1.0. The range associated with a variable is the range of the allowed perturbation of the corresponding objective coefficient. Note that the current primal solution remains optimal within this range; however the corresponding dual solution might change since a cost coefficient is perturbed. Similarly, the range associated with a constraint is the range of the allowed perturbation of the corresponding right-hand side coefficient. In this range the current dual solution remains optimal, but the optimal primal solution might change. If the problem is degenerate, there are multiple optimal bases and hence these ranges might not be as intuitive and seem too narrow, for example, a larger cost coefficient perturbation might not invalidate the optimality of the current primal solution. Moreover, if a problem is degenerate, due to finite precision, it can happen that, for example, a perturbation seems to invalidate a basis even though it doesn't (again providing too narrow ranges). To prevent this, increase the atol keyword argument to lp_sensitivity_report. Note that this might make the ranges too wide for numerically challenging instances. Thus, do not blindly trust these ranges, especially not for highly degenerate or numerically unstable instances. ## Conflicts When the model you input is infeasible, some solvers can help you find the cause of this infeasibility by offering a conflict, that is, a subset of the constraints that create this infeasibility. Depending on the solver, this can also be called an IIS (irreducible inconsistent subsystem). If supported by the solver, use compute_conflict! to trigger the computation of a conflict. Once this process is finished, query the MOI.ConflictStatus attribute to check if a conflict was found. If found, copy the IIS to a new model using copy_conflict, which you can then print or write to a file for easier debugging: julia> using JuMP julia> import Gurobi julia> model = Model(Gurobi.Optimizer) A JuMP Model Feasibility problem with: Variables: 0 Model mode: AUTOMATIC CachingOptimizer state: EMPTY_OPTIMIZER Solver name: Gurobi julia> set_silent(model) julia> @variable(model, x >= 0) x julia> @constraint(model, c1, x >= 2) c1 : x ≥ 2.0 julia> @constraint(model, c2, x <= 1) c2 : x ≤ 1.0 julia> optimize!(model) julia> compute_conflict!(model) julia> if get_attribute(model, MOI.ConflictStatus()) == MOI.CONFLICT_FOUND iis_model, _ = copy_conflict(model) print(iis_model) end Feasibility Subject to c1 : x ≥ 2.0 c2 : x ≤ 1.0 If you need more control over the list of constraints that appear in the conflict, iterate over the list of constraints and query the MOI.ConstraintConflictStatus attribute: julia> list_of_conflicting_constraints = ConstraintRef[] ConstraintRef[] julia> for (F, S) in list_of_constraint_types(model) for con in all_constraints(model, F, S) if get_attribute(con, MOI.ConstraintConflictStatus()) == MOI.IN_CONFLICT push!(list_of_conflicting_constraints, con) end end end julia> list_of_conflicting_constraints 2-element Vector{ConstraintRef}: c1 : x ≥ 2.0 c2 : x ≤ 1.0 ## Multiple solutions Some solvers support returning multiple solutions. You can check how many solutions are available to query using result_count. Functions for querying the solutions, for example, primal_status, dual_status, value, dual, and solution_summary all take an additional keyword argument result which can be used to specify which result to return. Warning Even if termination_status is OPTIMAL, some of the returned solutions may be suboptimal. However, if the solver found at least one optimal solution, then result = 1 will always return an optimal solution. Use objective_value to assess the quality of the remaining solutions. julia> using JuMP julia> import MultiObjectiveAlgorithms as MOA julia> import HiGHS julia> model = Model(() -> MOA.Optimizer(HiGHS.Optimizer)); julia> set_attribute(model, MOA.Algorithm(), MOA.Dichotomy()) julia> set_silent(model) julia> @variable(model, x1 >= 0) x1 julia> @variable(model, 0 <= x2 <= 3) x2 julia> @objective(model, Min, [3x1 + x2, -x1 - 2x2]) 2-element Vector{AffExpr}: 3 x1 + x2 -x1 - 2 x2 julia> @constraint(model, 3x1 - x2 <= 6) 3 x1 - x2 ≤ 6 julia> optimize!(model) julia> solution_summary(model; result = 1) * Solver : MOA[algorithm=MultiObjectiveAlgorithms.Dichotomy, optimizer=HiGHS] * Status Result count : 3 Termination status : OPTIMAL Message from the solver: "Solve complete. Found 3 solution(s)" * Candidate solution (result #1) Primal status : FEASIBLE_POINT Dual status : NO_SOLUTION Objective value : [0.00000e+00,0.00000e+00] Objective bound : [0.00000e+00,-9.00000e+00] Relative gap : Inf Dual objective value : -9.00000e+00 * Work counters Solve time (sec) : 1.82720e-03 Simplex iterations : 0 Barrier iterations : 0 Node count : -1 julia> for i in 1:result_count(model) println("Solution$i")
println("   x = ", value.([x1, x2]; result = i))
println(" obj = ", objective_value(model; result = i))
end
Solution 1
x = [0.0, 0.0]
obj = [0.0, 0.0]
Solution 2
x = [0.0, 3.0]
obj = [3.0, -6.0]
Solution 3
x = [3.0, 3.0]
obj = [12.0, -9.0]
Tip

The Multi-objective knapsack tutorial provides more examples of querying multiple solutions.

## Checking feasibility of solutions

To check the feasibility of a primal solution, use primal_feasibility_report, which takes a model, a dictionary mapping each variable to a primal solution value (defaults to the last solved solution), and a tolerance atol (defaults to 0.0).

The function returns a dictionary which maps the infeasible constraint references to the distance between the primal value of the constraint and the nearest point in the corresponding set. A point is classed as infeasible if the distance is greater than the supplied tolerance atol.

julia> model = Model(HiGHS.Optimizer);

julia> set_silent(model)

julia> @variable(model, x >= 1, Int);

julia> @variable(model, y);

julia> @constraint(model, c1, x + y <= 1.95);

julia> point = Dict(x => 1.9, y => 0.06);

julia> primal_feasibility_report(model, point)
Dict{Any, Float64} with 2 entries:
x integer         => 0.1
c1 : x + y ≤ 1.95 => 0.01

julia> primal_feasibility_report(model, point; atol = 0.02)
Dict{Any, Float64} with 1 entry:
x integer => 0.1

If the point is feasible, an empty dictionary is returned:

julia> primal_feasibility_report(model, Dict(x => 1.0, y => 0.0))
Dict{Any, Float64}()

To use the primal solution from a solve, omit the point argument:

julia> optimize!(model)

julia> primal_feasibility_report(model; atol = 0.0)
Dict{Any, Float64}()

Calling primal_feasibility_report without the point argument is useful when primal_status is FEASIBLE_POINT or NEARLY_FEASIBLE_POINT, and you want to assess the solution quality.

Warning

To apply primal_feasibility_report to infeasible models, you must also provide a candidate point (solvers generally do not provide one). To diagnose the source of infeasibility, see Conflicts.

Pass skip_mising = true to skip constraints which contain variables that are not in point:

julia> primal_feasibility_report(model, Dict(x => 2.1); skip_missing = true)
Dict{Any, Float64} with 1 entry:
x integer => 0.1

You can also use the functional form, where the first argument is a function that maps variables to their primal values:

julia> optimize!(model)

julia> primal_feasibility_report(v -> value(v), model)
Dict{Any, Float64}()