The source files can be found in user_guide/.
Constraints and costs
Real mandates are not unconstrained. You cap concentration, hold a sector band, limit how much you trade at each rebalance, and pay transaction costs. In PortfolioOptimisers.jl these are keywords on the JuMPOptimiser — the optimiser carries the constraints and costs, the estimator carries the objective. This page shows the common ones with one minimal call each; for the full treatment see the constraints & costs examples.
We fix one empirical prior and a minimum-risk objective so each keyword's effect is visible against the same baseline.
julia
using PortfolioOptimisers, CSV, TimeSeries, DataFrames, PrettyTables, Clarabel, StatsPlots,
GraphRecipes
resfmt = (v, i, j) -> begin
return if j == 1
v
else
isa(v, AbstractFloat) ? "$(round(v*100, digits=3)) %" : v
end
end;
X = TimeArray(CSV.File(joinpath(@__DIR__, "../examples/SP500.csv.gz")); timestamp = :Date)[(end - 252):end]
rd = prices_to_returns(X)
pr = prior(EmpiricalPrior(), rd)
slv = Solver(; name = :clarabel, solver = Clarabel.Optimizer,
settings = Dict("verbose" => false),
check_sol = (; allow_local = true, allow_almost = true))
res_base = optimise(MeanRisk(; obj = MinimumRisk(),
opt = JuMPOptimiser(; pe = pr, slv = slv)))MeanRiskResult
jr ┼ JuMPOptimisationResult
│ pa ┼ ProcessedJuMPOptimiserAttributes
│ │ pr ┼ LowOrderPrior
│ │ │ X ┼ 252×20 Matrix{Float64}
│ │ │ mu ┼ 20-element Vector{Float64}
│ │ │ sigma ┼ 20×20 Matrix{Float64}
│ │ │ chol ┼ nothing
│ │ │ w ┼ nothing
│ │ │ ens ┼ nothing
│ │ │ kld ┼ nothing
│ │ │ ow ┼ nothing
│ │ │ rr ┼ nothing
│ │ │ f_mu ┼ nothing
│ │ │ f_sigma ┼ nothing
│ │ │ f_w ┴ nothing
│ │ wb ┼ WeightBounds
│ │ │ lb ┼ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ │ ub ┴ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ lt ┼ nothing
│ │ st ┼ nothing
│ │ lcsr ┼ nothing
│ │ ctr ┼ nothing
│ │ gcardr ┼ nothing
│ │ sgcardr ┼ nothing
│ │ smtx ┼ nothing
│ │ sgmtx ┼ nothing
│ │ slt ┼ nothing
│ │ sst ┼ nothing
│ │ sglt ┼ nothing
│ │ sgst ┼ nothing
│ │ tn ┼ nothing
│ │ fees ┼ nothing
│ │ plr ┼ nothing
│ │ ret ┼ ArithmeticReturn
│ │ │ ucs ┼ nothing
│ │ │ lb ┼ nothing
│ │ │ mu ┴ 20-element Vector{Float64}
│ retcode ┼ OptimisationSuccess
│ │ res ┴ Dict{Any, Any}: Dict{Any, Any}()
│ sol ┼ JuMPOptimisationSolution
│ │ w ┴ 20-element Vector{Float64}
│ model ┼ A JuMP Model
│ │ ├ solver: Clarabel
│ │ ├ objective_sense: MIN_SENSE
│ │ │ └ objective_function_type: QuadExpr
│ │ ├ num_variables: 21
│ │ ├ num_constraints: 4
│ │ │ ├ AffExpr in MOI.EqualTo{Float64}: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonnegatives: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonpositives: 1
│ │ │ └ Vector{AffExpr} in MOI.SecondOrderCone: 1
│ │ └ Names registered in the model
│ │ └ :G, :bgt, :dev_1, :dev_1_soc, :k, :lw, :obj_expr, :ret, :risk, :risk_vec, :sc, :so, :variance_flag, :variance_risk_1, :w, :w_lb, :w_ub
fb ┴ nothing1. Weight bounds
WeightBounds sets the per-asset lower and upper bound through the wb keyword. The default is lb = 0, ub = 1 (long-only, fully invested). Capping ub forces diversification — no single name can exceed the bound.
julia
res_cap = optimise(MeanRisk(; obj = MinimumRisk(),
opt = JuMPOptimiser(; pe = pr, slv = slv,
wb = WeightBounds(; lb = 0.0, ub = 0.10))))MeanRiskResult
jr ┼ JuMPOptimisationResult
│ pa ┼ ProcessedJuMPOptimiserAttributes
│ │ pr ┼ LowOrderPrior
│ │ │ X ┼ 252×20 Matrix{Float64}
│ │ │ mu ┼ 20-element Vector{Float64}
│ │ │ sigma ┼ 20×20 Matrix{Float64}
│ │ │ chol ┼ nothing
│ │ │ w ┼ nothing
│ │ │ ens ┼ nothing
│ │ │ kld ┼ nothing
│ │ │ ow ┼ nothing
│ │ │ rr ┼ nothing
│ │ │ f_mu ┼ nothing
│ │ │ f_sigma ┼ nothing
│ │ │ f_w ┴ nothing
│ │ wb ┼ WeightBounds
│ │ │ lb ┼ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ │ ub ┴ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ lt ┼ nothing
│ │ st ┼ nothing
│ │ lcsr ┼ nothing
│ │ ctr ┼ nothing
│ │ gcardr ┼ nothing
│ │ sgcardr ┼ nothing
│ │ smtx ┼ nothing
│ │ sgmtx ┼ nothing
│ │ slt ┼ nothing
│ │ sst ┼ nothing
│ │ sglt ┼ nothing
│ │ sgst ┼ nothing
│ │ tn ┼ nothing
│ │ fees ┼ nothing
│ │ plr ┼ nothing
│ │ ret ┼ ArithmeticReturn
│ │ │ ucs ┼ nothing
│ │ │ lb ┼ nothing
│ │ │ mu ┴ 20-element Vector{Float64}
│ retcode ┼ OptimisationSuccess
│ │ res ┴ Dict{Any, Any}: Dict{Any, Any}()
│ sol ┼ JuMPOptimisationSolution
│ │ w ┴ 20-element Vector{Float64}
│ model ┼ A JuMP Model
│ │ ├ solver: Clarabel
│ │ ├ objective_sense: MIN_SENSE
│ │ │ └ objective_function_type: QuadExpr
│ │ ├ num_variables: 21
│ │ ├ num_constraints: 4
│ │ │ ├ AffExpr in MOI.EqualTo{Float64}: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonnegatives: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonpositives: 1
│ │ │ └ Vector{AffExpr} in MOI.SecondOrderCone: 1
│ │ └ Names registered in the model
│ │ └ :G, :bgt, :dev_1, :dev_1_soc, :k, :lw, :obj_expr, :ret, :risk, :risk_vec, :sc, :so, :variance_flag, :variance_risk_1, :w, :w_lb, :w_ub
fb ┴ nothingThe budget itself is the bgt keyword (default 1.0); BudgetRange and a separate short budget sbgt let you build long/short and leveraged mandates — see Budget Constraints.
2. Linear and group constraints
Group and linear constraints are written as plain strings over an AssetSets and passed through lcse as a LinearConstraintEstimator — the same syntax used for views. Name a group, then bound it. Here we require the tech group to hold at least 15% (a floor the unconstrained minimum-risk portfolio would not give it).
julia
sets = AssetSets(; dict = Dict("nx" => rd.nx, "tech" => ["AAPL", "AMD", "MSFT"]))
res_grp = optimise(MeanRisk(; obj = MinimumRisk(),
opt = JuMPOptimiser(; pe = pr, slv = slv, sets = sets,
lcse = LinearConstraintEstimator(;
val = ["tech >= 0.15"]))))MeanRiskResult
jr ┼ JuMPOptimisationResult
│ pa ┼ ProcessedJuMPOptimiserAttributes
│ │ pr ┼ LowOrderPrior
│ │ │ X ┼ 252×20 Matrix{Float64}
│ │ │ mu ┼ 20-element Vector{Float64}
│ │ │ sigma ┼ 20×20 Matrix{Float64}
│ │ │ chol ┼ nothing
│ │ │ w ┼ nothing
│ │ │ ens ┼ nothing
│ │ │ kld ┼ nothing
│ │ │ ow ┼ nothing
│ │ │ rr ┼ nothing
│ │ │ f_mu ┼ nothing
│ │ │ f_sigma ┼ nothing
│ │ │ f_w ┴ nothing
│ │ wb ┼ WeightBounds
│ │ │ lb ┼ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ │ ub ┴ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ lt ┼ nothing
│ │ st ┼ nothing
│ │ lcsr ┼ LinearConstraint
│ │ │ ineq ┼ PartialLinearConstraint
│ │ │ │ A ┼ 1×20 LinearAlgebra.Transpose{Float64, Matrix{Float64}}
│ │ │ │ B ┴ Vector{Float64}: [-0.15]
│ │ │ eq ┴ nothing
│ │ ctr ┼ nothing
│ │ gcardr ┼ nothing
│ │ sgcardr ┼ nothing
│ │ smtx ┼ nothing
│ │ sgmtx ┼ nothing
│ │ slt ┼ nothing
│ │ sst ┼ nothing
│ │ sglt ┼ nothing
│ │ sgst ┼ nothing
│ │ tn ┼ nothing
│ │ fees ┼ nothing
│ │ plr ┼ nothing
│ │ ret ┼ ArithmeticReturn
│ │ │ ucs ┼ nothing
│ │ │ lb ┼ nothing
│ │ │ mu ┴ 20-element Vector{Float64}
│ retcode ┼ OptimisationSuccess
│ │ res ┴ Dict{Any, Any}: Dict{Any, Any}()
│ sol ┼ JuMPOptimisationSolution
│ │ w ┴ 20-element Vector{Float64}
│ model ┼ A JuMP Model
│ │ ├ solver: Clarabel
│ │ ├ objective_sense: MIN_SENSE
│ │ │ └ objective_function_type: QuadExpr
│ │ ├ num_variables: 21
│ │ ├ num_constraints: 5
│ │ │ ├ AffExpr in MOI.EqualTo{Float64}: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonnegatives: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonpositives: 2
│ │ │ └ Vector{AffExpr} in MOI.SecondOrderCone: 1
│ │ └ Names registered in the model
│ │ └ :G, :bgt, :dev_1, :dev_1_soc, :k, :lcs_ineq_1, :lw, :obj_expr, :ret, :risk, :risk_vec, :sc, :so, :variance_flag, :variance_risk_1, :w, :w_lb, :w_ub
fb ┴ nothingThe same lcse handles absolute and relative bounds ("AAPL <= 0.1", "MSFT >= AMD"). For constraints built from the asset hierarchy — phylogeny and centrality — see Phylogeny & Centrality.
3. Turnover
Costs enter the same way. Turnover (tn) limits how far the new weights may drift from a reference portfolio w — your current holdings — so a rebalance stays cheap. Here we anchor at the current minimum-risk portfolio and re-solve under a turnover budget.
julia
res_tn = optimise(MeanRisk(; obj = MinimumRisk(),
opt = JuMPOptimiser(; pe = pr, slv = slv,
tn = Turnover(; w = res_base.w, val = 0.02))))MeanRiskResult
jr ┼ JuMPOptimisationResult
│ pa ┼ ProcessedJuMPOptimiserAttributes
│ │ pr ┼ LowOrderPrior
│ │ │ X ┼ 252×20 Matrix{Float64}
│ │ │ mu ┼ 20-element Vector{Float64}
│ │ │ sigma ┼ 20×20 Matrix{Float64}
│ │ │ chol ┼ nothing
│ │ │ w ┼ nothing
│ │ │ ens ┼ nothing
│ │ │ kld ┼ nothing
│ │ │ ow ┼ nothing
│ │ │ rr ┼ nothing
│ │ │ f_mu ┼ nothing
│ │ │ f_sigma ┼ nothing
│ │ │ f_w ┴ nothing
│ │ wb ┼ WeightBounds
│ │ │ lb ┼ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ │ ub ┴ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ lt ┼ nothing
│ │ st ┼ nothing
│ │ lcsr ┼ nothing
│ │ ctr ┼ nothing
│ │ gcardr ┼ nothing
│ │ sgcardr ┼ nothing
│ │ smtx ┼ nothing
│ │ sgmtx ┼ nothing
│ │ slt ┼ nothing
│ │ sst ┼ nothing
│ │ sglt ┼ nothing
│ │ sgst ┼ nothing
│ │ tn ┼ Turnover
│ │ │ w ┼ 20-element Vector{Float64}
│ │ │ val ┼ Float64: 0.02
│ │ │ fixed ┴ Bool: false
│ │ fees ┼ nothing
│ │ plr ┼ nothing
│ │ ret ┼ ArithmeticReturn
│ │ │ ucs ┼ nothing
│ │ │ lb ┼ nothing
│ │ │ mu ┴ 20-element Vector{Float64}
│ retcode ┼ OptimisationSuccess
│ │ res ┴ Dict{Any, Any}: Dict{Any, Any}()
│ sol ┼ JuMPOptimisationSolution
│ │ w ┴ 20-element Vector{Float64}
│ model ┼ A JuMP Model
│ │ ├ solver: Clarabel
│ │ ├ objective_sense: MIN_SENSE
│ │ │ └ objective_function_type: QuadExpr
│ │ ├ num_variables: 41
│ │ ├ num_constraints: 25
│ │ │ ├ AffExpr in MOI.EqualTo{Float64}: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonnegatives: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonpositives: 2
│ │ │ ├ Vector{AffExpr} in MOI.NormOneCone: 20
│ │ │ └ Vector{AffExpr} in MOI.SecondOrderCone: 1
│ │ └ Names registered in the model
│ │ └ :G, :bgt, :ctn_1, :ctn_noc_1, :dev_1, :dev_1_soc, :k, :lw, :obj_expr, :ret, :risk, :risk_vec, :sc, :so, :t_tn_1, :tn_1, :variance_flag, :variance_risk_1, :w, :w_lb, :w_ub
fb ┴ nothingA tighter val keeps the result closer to the reference holdings; a looser one frees the optimiser to move toward the unconstrained solution.
4. Fees
Fees (fees) charges proportional (and optionally fixed) transaction costs on long and short positions, which the objective then trades off against return. The minimal form sets a per-unit long fee.
julia
res_fee = optimise(MeanRisk(; obj = MaximumRatio(; rf = 4.2 / 100 / 252),
opt = JuMPOptimiser(; pe = pr, slv = slv,
fees = Fees(; l = 0.001))))MeanRiskResult
jr ┼ JuMPOptimisationResult
│ pa ┼ ProcessedJuMPOptimiserAttributes
│ │ pr ┼ LowOrderPrior
│ │ │ X ┼ 252×20 Matrix{Float64}
│ │ │ mu ┼ 20-element Vector{Float64}
│ │ │ sigma ┼ 20×20 Matrix{Float64}
│ │ │ chol ┼ nothing
│ │ │ w ┼ nothing
│ │ │ ens ┼ nothing
│ │ │ kld ┼ nothing
│ │ │ ow ┼ nothing
│ │ │ rr ┼ nothing
│ │ │ f_mu ┼ nothing
│ │ │ f_sigma ┼ nothing
│ │ │ f_w ┴ nothing
│ │ wb ┼ WeightBounds
│ │ │ lb ┼ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ │ ub ┴ 20-element StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}
│ │ lt ┼ nothing
│ │ st ┼ nothing
│ │ lcsr ┼ nothing
│ │ ctr ┼ nothing
│ │ gcardr ┼ nothing
│ │ sgcardr ┼ nothing
│ │ smtx ┼ nothing
│ │ sgmtx ┼ nothing
│ │ slt ┼ nothing
│ │ sst ┼ nothing
│ │ sglt ┼ nothing
│ │ sgst ┼ nothing
│ │ tn ┼ nothing
│ │ fees ┼ Fees
│ │ │ tn ┼ nothing
│ │ │ l ┼ Float64: 0.001
│ │ │ s ┼ nothing
│ │ │ fl ┼ nothing
│ │ │ fs ┼ nothing
│ │ │ kwargs ┴ @NamedTuple{atol::Float64}: (atol = 1.0e-8,)
│ │ plr ┼ nothing
│ │ ret ┼ ArithmeticReturn
│ │ │ ucs ┼ nothing
│ │ │ lb ┼ nothing
│ │ │ mu ┴ 20-element Vector{Float64}
│ retcode ┼ OptimisationSuccess
│ │ res ┴ Dict{Any, Any}: Dict{Any, Any}()
│ sol ┼ JuMPOptimisationSolution
│ │ w ┴ 20-element Vector{Float64}
│ model ┼ A JuMP Model
│ │ ├ solver: Clarabel
│ │ ├ objective_sense: MIN_SENSE
│ │ │ └ objective_function_type: QuadExpr
│ │ ├ num_variables: 22
│ │ ├ num_constraints: 6
│ │ │ ├ AffExpr in MOI.EqualTo{Float64}: 2
│ │ │ ├ Vector{AffExpr} in MOI.Nonnegatives: 1
│ │ │ ├ Vector{AffExpr} in MOI.Nonpositives: 1
│ │ │ ├ Vector{AffExpr} in MOI.SecondOrderCone: 1
│ │ │ └ VariableRef in MOI.GreaterThan{Float64}: 1
│ │ └ Names registered in the model
│ │ └ :G, :bgt, :dev_1, :dev_1_soc, :fees, :fl, :k, :lw, :obj_expr, :ohf, :ret, :risk, :risk_vec, :sc, :so, :sr_ret, :variance_flag, :variance_risk_1, :w, :w_lb, :w_ub
fb ┴ nothingSoft alternatives to hard turnover/position limits — L1/L2 weight regularisation and an effective-number-of-assets floor (l1, l2, wn2) — are covered in Regularisation; benchmark Tracking (the tr keyword) in Turnover & Tracking.
5. Comparing the effect
Same prior, same objective — only the constraint or cost changes the allocation.
julia
results = [res_base, res_cap, res_grp, res_tn, res_fee]
labels = ["Base", "Cap 10%", "Tech ≥ 15%", "Turnover", "Fees"]
pretty_table(DataFrame(["Asset" => rd.nx,
[labels[i] => results[i].w for i in eachindex(results)]...]);
formatters = [resfmt], title = "Weights under each constraint / cost")
plot_stacked_bar_composition(results, rd; xticks = (1:length(labels), labels))
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