Weight norm constraints

PortfolioOptimisers.set_weight_norm_2_constraints! Function

julia

set_weight_norm_2_constraints!(model::JuMP.Model, val::Number)
set_weight_norm_2_constraints!(args...)

Constrain the 2-norm of the weights.

val is a direct upper bound on $‖ w ‖_{2}$ , expressed relative to the budget k: the constraint is $‖ w ‖_{2} \leq val \cdot k$ . Smaller val forces the weights to spread more evenly across the assets.

Mathematical definition

\begin{aligned} l 2 c & \geq ‖ w ‖_{2}, \\ l 2 c & \leq val \cdot k . \end{aligned}

Where:

$l 2 c$ : Auxiliary variable upper-bounding $‖ w ‖_{2}$ .
$w$ : Portfolio weights vector $N \times 1$ .
$k$ : Budget scaling / homogenisation variable.
$val$ : Upper bound on the 2-norm of the weights.

Diversification interpretation

The 2-norm and the effective number of assets are reciprocally related: for a fully invested portfolio ( $k = 1$ ), $ENA (w) = 1 / ‖ w ‖_{2}^{2}$ . To require at least m effective assets, set val = 1 / sqrt(m):

\begin{array}{r} ‖ w ‖_{2} \leq \frac{1}{\sqrt{m}} ⟺ ENA (w) = \frac{1}{‖ w ‖_{2}^{2}} \geq m . \end{array}

Arguments

model::JuMP.Model: The JuMP optimisation model.
val::Number: Upper bound on the 2-norm of the weights.

Returns

nothing.

Details

val::Number: Introduces the auxiliary variable l2c, bounds it below by $‖ w ‖_{2}$ with a SecondOrderCone constraint, and adds the linear constraint l2c <= val * k.
args...: No-op, used when no 2-norm weight constraint is configured.

Related

source

PortfolioOptimisers.set_weight_norm_p_constraints! Function

julia

set_weight_norm_p_constraints!(model::JuMP.Model, lps::LpReg_VecLpReg)
set_weight_norm_p_constraints!(args...)

Constrain the p-norm of the weights.

Generalises set_weight_norm_2_constraints! to an arbitrary norm order $p > 1$ . Each term supplies its own norm order and bound, so several may be imposed at once.

Each term is an LpRegularisation, reused here as a constraint rather than a penalty: its p field is the norm order, and its val field is a direct upper bound on $‖ w ‖_{p}$ , expressed relative to the budget k. Smaller val forces a more evenly spread portfolio.

Mathematical definition

\begin{aligned} t_{p, i} & \geq ‖ w ‖_{p_{i}}, \\ t_{p, i} & \leq {val}_{i} \cdot k . \end{aligned}

Where:

$t_{p, i}$ : Auxiliary variable upper-bounding $‖ w ‖_{p_{i}}$ .
$p_{i}$ : Norm order of the $i$ -th term, its p field.
${val}_{i}$ : Upper bound on the $p_{i}$ -norm of the weights, its val field.
$w$ : Portfolio weights vector $N \times 1$ .
$k$ : Budget scaling / homogenisation variable.

Diversification interpretation

For a fully invested portfolio ( $k = 1$ ), the p-norm effective number of assets is ${ENA}_{p} (w) = 1 / ‖ w ‖_{p}^{p}$ . To require at least m p-norm effective assets, set val = m^(-1/p):

\begin{array}{r} ‖ w ‖_{p} \leq m^{- 1 / p} ⟺ {ENA}_{p} (w) = \frac{1}{‖ w ‖_{p}^{p}} \geq m . \end{array}

Arguments

model::JuMP.Model: The JuMP optimisation model.
lps::LpReg_VecLpReg: One or more p-norm weight constraints.

Returns

nothing.

Details

lps::LpReg_VecLpReg: For each term, introduces the auxiliary variables t_lpc_i and r_lpc_i, bounds $‖ w ‖_{p_{i}}$ above by t_lpc_i with a set of MOI.PowerCone constraints, and adds the linear constraint t_lpc_i <= val * k. Variables and constraints are suffixed by the term's index, so terms do not collide, and are named distinctly from those of set_lp_regularisation! so a model may carry both an Lp penalty and a p-norm constraint.
args...: No-op, used when no p-norm weight constraint is configured.

Related

source

PortfolioOptimisers.set_weight_norm_inf_constraints! Function

julia

set_weight_norm_inf_constraints!(model::JuMP.Model, val::Number)
set_weight_norm_inf_constraints!(args...)

Constrain the ∞-norm of the weights, capping the largest absolute weight.

The limiting case of set_weight_norm_p_constraints!. val is a direct upper bound on the largest absolute weight, expressed relative to the budget k: the constraint is $‖ w ‖_{\infty} \leq val \cdot k$ . So a fully invested portfolio ( $k = 1$ ) constrained with val = 0.2 holds no position larger than 20%. Smaller val forces a more evenly spread portfolio.

Mathematical definition

\begin{aligned} t_{\infty} & \geq ‖ w ‖_{\infty}, \\ t_{\infty} & \leq val \cdot k . \end{aligned}

Where:

$t_{\infty}$ : Auxiliary variable upper-bounding $‖ w ‖_{\infty}$ .
$val$ : Upper bound on the largest absolute weight.
$w$ : Portfolio weights vector $N \times 1$ .
$k$ : Budget scaling / homogenisation variable.

Diversification interpretation

Capping the largest weight spreads the portfolio across a minimum number of assets. To spread across at least m assets, set val = 1 / m: no single position can then exceed a $1 / m$ share of a fully invested portfolio.

Arguments

model::JuMP.Model: The JuMP optimisation model.
val::Number: Upper bound on the largest absolute weight.

Returns

nothing.

Details

val::Number: Introduces the auxiliary variable t_linfc, bounds it below by $‖ w ‖_{\infty}$ with a MOI.NormInfinityCone constraint, and adds the linear constraint t_linfc <= val * k.
args...: No-op, used when no ∞-norm weight constraint is configured.

Related

source

Weight norm constraints ​

Weight norm constraints