Weight norm constraints

PortfolioOptimisers.set_weight_norm_2_constraints! Function

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

Constrain the 2-norm of the weights, bounding the effective number of assets from below.

Diversification is expressed as a bound on a norm of the weight vector: the smaller the norm, the more evenly the weights spread across the assets. For the 2-norm the reciprocal of the squared norm is the effective number of assets, so bounding the norm above by $k/\sqrt{\mathrm{val}}$ is the same as requiring at least val effective assets.

Mathematical definition

$$\begin{array}{rl}\mathrm{wn}2& \ge \Vert \mathit{w}{\Vert }_2,\\ \mathrm{wn}2\cdot \sqrt{\mathrm{val}}& \le k.\end{array}$$

Which, for a fully invested portfolio ($k=1$), is equivalent to:

$$\begin{array}{rl}\mathrm{ENA}(\mathit{w})& =\frac{1}{\Vert \mathit{w}{\Vert }_2^2}\ge \mathrm{val}.\end{array}$$

Where:

• $\mathrm{wn}2$: Auxiliary variable upper-bounding $\Vert \mathit{w}{\Vert }_2$.

• $\mathit{w}$: Portfolio weights vector $N\times 1$.

• $k$: Budget scaling / homogenisation variable.

• $\mathrm{ENA}(\mathit{w})$: Effective number of assets.

• $\mathrm{val}$: Minimum required effective number of assets.

Arguments

• model::JuMP.Model: The JuMP optimisation model.

• val::Number: Minimum required effective number of assets.

Returns

• nothing.

Details

• val::Number: Introduces the auxiliary variable wn2, bounds it below by $\Vert \mathit{w}{\Vert }_2$ with a SecondOrderCone constraint, and adds the linear constraint wn2 * sqrt(val) <= k.

• args...: No-op, used when no 2-norm weight constraint is configured.

Related

• set_weight_norm_p_constraints!

• set_weight_norm_inf_constraints!

• number_effective_assets

• JuMPOptimiser

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PortfolioOptimisers.set_weight_norm_p_constraints! Function

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

Constrain the p-norm of the weights, bounding the p-norm effective number of assets from below.

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 the minimum p-norm effective number of assets, not the maximum norm. Larger val forces a more evenly spread portfolio.

Mathematical definition

$$\begin{array}{rl}{\mathrm{t}}_{p,i}& \ge \Vert \mathit{w}{\Vert }_{p_i},\\ {\mathrm{t}}_{p,i}\cdot {\mathrm{val}}_i^{1/p_i}& \le k.\end{array}$$

Which, for a fully invested portfolio ($k=1$), is equivalent to:

$$\begin{array}{rl}{\mathrm{ENA}}_{p_i}(\mathit{w})& =\frac{1}{\Vert \mathit{w}{\Vert }_{p_i}^{p_i}}\ge {\mathrm{val}}_i.\end{array}$$

Where:

• ${\mathrm{t}}_{p,i}$: Auxiliary variable upper-bounding $\Vert \mathit{w}{\Vert }_{p_i}$.

• $p_i$: Norm order of the $i$-th term, its p field.

• ${\mathrm{val}}_i$: Minimum required p-norm effective number of assets of the $i$-th term, its val field.

• $\mathit{w}$: Portfolio weights vector $N\times 1$.

• $k$: Budget scaling / homogenisation variable.

• ${\mathrm{ENA}}_p(\mathit{w})$: Effective number of assets under the $p$-norm.

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_wnp_i and r_wnp_i, bounds $\Vert \mathit{w}{\Vert }_{p_i}$ above by t_wnp_i with a set of MOI.PowerCone constraints, and adds the linear constraint t_wnp_i * val^(1 / p) <= 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

• LpRegularisation

• LpReg_VecLpReg

• set_weight_norm_2_constraints!

• set_weight_norm_inf_constraints!

• set_lp_regularisation!

• JuMPOptimiser

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PortfolioOptimisers.set_weight_norm_inf_constraints! Function

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

Constrain the ∞-norm of the weights, bounding the largest weight from above.

The limiting case of set_weight_norm_p_constraints!. linf is the reciprocal of the largest permitted weight, not the weight itself: a fully invested portfolio constrained with linf = 5 holds no position larger than 1 / 5, and so is spread across at least five assets. Larger linf forces a more evenly spread portfolio, consistent with the other members of the family.

Mathematical definition

$$\begin{array}{rl}{\mathrm{t}}_{\mathrm{\infty }}& \ge \Vert \mathit{w}{\Vert }_{\mathrm{\infty }},\\ {\mathrm{t}}_{\mathrm{\infty }}\cdot \mathrm{linf}& \le k.\end{array}$$

Which, for a fully invested portfolio ($k=1$), is equivalent to:

$$\begin{array}{rl}\underset{i}{max}|w_i|& \le \frac{1}{\mathrm{linf}}.\end{array}$$

Where:

• ${\mathrm{t}}_{\mathrm{\infty }}$: Auxiliary variable upper-bounding $\Vert \mathit{w}{\Vert }_{\mathrm{\infty }}$.

• $\mathrm{linf}$: Reciprocal of the largest permitted weight.

• $w_i$: Portfolio weight for asset $i$.

• $\mathit{w}$: Portfolio weights vector $N\times 1$.

• $k$: Budget scaling / homogenisation variable.

Arguments

• model::JuMP.Model: The JuMP optimisation model.

• linf::Number: Reciprocal of the largest permitted weight.

Returns

• nothing.

Details

• linf::Number: Introduces the auxiliary variable t_wninf, bounds it below by $\Vert \mathit{w}{\Vert }_{\mathrm{\infty }}$ with a MOI.NormInfinityCone constraint, and adds the linear constraint t_wninf * linf <= k.

• args...: No-op, used when no ∞-norm weight constraint is configured.

Related

• set_weight_norm_2_constraints!

• set_weight_norm_p_constraints!

• set_linf_regularisation!

• JuMPOptimiser

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