Adaptive l1-regularization for short-selling control in portfolio selection
Stefania Corsaro Valentina De Simone







Adaptive l_1-regularization for short-selling control in portfolio selection
    S. CorsaroDipartimento di Studi aziendali e quantitativi, Università di Napoli “Parthenope”,
Via Generale Parisi, 13, I-80133 Napoli, Italy, email: Italy stefania.corsaro@uniparthenope.it
V. De SimoneDipartimento di Matematica e Fisica,
Università della Campania “Luigi Vanvitelli”,
Viale Lincoln, 5, I-81100 Caserta, Italy, email: valentina.desimone@unicampania.it
    
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We consider the l_1-regularized Markowitz model, where a
l_1-penalty term is added to the objective function of the classical mean-variance
one to stabilize the solution process, promoting
sparsity in the solution.
The l_1-penalty term can also be interpreted in terms of short sales,
on which several financial markets have posed restrictions.
The choice of the regularization parameter plays a key role
to obtain optimal portfolios that meet the financial requirements.
We propose an updating rule for the regularization parameter
in Bregman iteration to control both the sparsity and the number
of short positions.
We show that the modified scheme preserves the properties
of the original one.
Numerical tests are reported, which show the effectiveness of the approach.




Keywords:  Portfolio selection. Markowitz model. l_1-regularization. Bregman iteration.






§ INTRODUCTION


In the classical Markowitz mean-variance framework <cit.>,
portfolio selection aims at the construction of an
investment portfolio that exposes investor to minimum risk providing him
a fixed expected return.
This approach was proposed by Markowitz
in his aforementioned seminal paper, where he stated that
portfolio selection strategy should provide an optimal trade-off between
expected return and risk (mean-variance approach).
In a successive work <cit.>,
Markowitz reinforced his theory arguing that, under certain, mild conditions,
a portfolio from a mean-variance efficient frontier
will approximately maximize the investor's expected utility. 

Markowitz model relies on information about future, since expected returns
should actually be computed discounting future flows, that are clearly not
available. A common choice is to use historical data as predictive of the
future behavior of asset returns. This practice has certain drawbacks;
indeed, a limited amount of relevant historical data is often
available. Moreover, correlation between assets returns can lead to
ill-conditioned covariance matrices.

It is well known that errors in estimation of expected values
affect solutions more severely than errors on variances.
For this reason, to overcome this issue
some authors focus on minimum-variance portfolios,
which do not take into account the return constraint. We
recall <cit.> and references therein.
Moreover, different regularization techniques have been suggested;
a review of them can be found in <cit.>. Among these,
penalization techniques have been considered, both for the
minimum- and the mean-variance approach.
In <cit.> l_1 and squared-l_2 norm constraints are proposed
for the minimum-variance criterion.
In <cit.> an algorithm for the optimal minimum-variance portfolio selection
with a weighted l_1 and squared-l_2 norm penalty is presented.

In <cit.> authors regularize the mean-variance objective
function with a weighted elastic net penalty.

In this paper, we consider the l_1 mean-variance regularized model
introduced in <cit.>, where a
l_1-penalty term is added to promote sparsity in the solution.
Since solutions establish the amount of capital to be invested in each
available security,
sparsity means that money are invested in a few securities, the active positions.
This allows investor to reduce both the number of positions to be monitored
and the transaction costs, particularly relevant for small investors,
that are not taken into account
in the theoretical Markowitz model.
Another useful interpretation of l_1 regularization is related to the amount
of shorting in the portfolio;
from the financial point of view negative solutions correspond to
short sales.



In many markets, among which Italy, Germany and Switzerland,
restrictions
on short sales have been established in the last years,
thus short-controlling is desired as well.

Then, the choice of the regularization parameter is crucial
in order to provide sparse solutions, with either a limited or null
number of negative components, preserving fidelity to data.


In this paper we propose an iterative algorithm based on a modified Bregman iteration.
Bregman iteration is a well established method for the solution
of l_1-regularized optimization problems.
It has been successfully applied in different fields,
as image restoration <cit.>
matrix rank minimization <cit.>, compressed sensing <cit.>
and finance <cit.>.


Our modification to the original scheme introduces an adaptive updating
rule for the regularization parameter in the regularized model.
The algorithm selects a value capable to provide solutions
satisfying a fixed financial target, formulated in terms of
limited number of active and/or short positions.

We show that our modified scheme preserves the properties
of the original one and is able to select a good value of the
regularization parameter within a negligible computational time.
Numerical tests confirm the effectiveness of the proposed algorithm.

The paper is organized as follows. In section <ref> we
briefly recall Markowitz mean-variance model. In section <ref>
we introduce Bregman iteration for portfolio selection.
Our main results are in section <ref>, where we introduce our
algorithm, based on a modified Bregman iteration, for the l_1-regularized
Markowitz model. In section <ref> we validate our approach
by means of several numerical experiments. Finally, in section <ref>
we give some conclusion and outline future work.



§ PORTFOLIO SELECTION MODEL



We refer to the classical Markowitz mean-variance framework.
Given n traded assets, the core of the problem is to establish the amount
of capital to be invested in each available security.

We assume that one unit of capital is available and define

    𝐰 = (w_1,w_2,…,w_n)^T

the portfolio weight vector, that is, the amount w_i is invested in the i-th security.
Asset returns are assumed to be stationary.
If we denote with

    μ = (μ_1,μ_2,…,μ_n)^T

the expected asset returns, then
the expected portfolio return is their weighted sum:

    ∑_i=1^n w_i μ_i.

We moreover denote with σ_ij is the covariance between returns of securities i and j.
The portfolio risk is measured by means of its variance, given by:

    V = ∑_i=1^n ∑_j=1^n σ_ij w_iw_j.

Let ρ be the fixed expected portfolio return and C
the covariance matrix of returns.
Portfolio selection is formulated as the following quadratic constrained
optimization problem:

    [   min_𝐰𝐰^TC𝐰;         s.t.;    𝐰^T μ = ρ; 𝐰^T 1_n = 1,;              ]

where 1 is the vector
of ones of length n.
The first constraint fixes the expected return, according to (<ref>). The second one
is a budget constraint which establishes that all the available capital is invested.
The non-negativity constraint is often
added to avoid short positions.
We do not consider it here, since we aim at controlling short positions
by tuning the regularization parameter, as it is discussed in the following. 

Let us consider a set of m evenly spaced dates

    𝐭 = (t_1,t_2,…,t_m)

at which asset returns
are estimated and build the matrix R∈ℝ^m× n that contains
observed historical returns of asset i on its i-th column.
It can be shown that problem (<ref>) can be stated in the following form:

    [ min_𝐰1/mρ1_m -R𝐰_2^2;                 s.t.;            𝐰^T μ = ρ;         𝐰^T 1_n = 1.;                      ]

As the asset returns are typically correlated, the matrix R could have some
singular values close to zero; therefore regularization techniques,
that add to objective function some form of a priori knowledge about the
solution, must be considered.
In this paper we consider the following l_1-regularized problem:

    [ min_𝐰ρ1_m -R𝐰_2^2+τ𝐰_1;                   s.t.;              𝐰^T μ = ρ;           𝐰^T 1_n = 1,;                        ]

where the 1/m term has been incorporated into the regularization one.

From the second constraint in (<ref>)
it follows that the objective function
can be equivalently written as:

    || ρ1_m - R 𝐰 ||_2^2 + 2τ∑_i:w_i<0 |w_i| + τ.

This form points out that l_1 penalty is equivalent to a penalty on short positions.
In the limit of very large values
of the regularization parameter, we obtain a portfolio with only positive weights,
as observed also in <cit.>.



§ BREGMAN ITERATION FOR PORTFOLIO SELECTION


Portfolio selection can be formulated as the constrained nonlinear optimization problem:

    [ min_𝐰 E(𝐰);       s.t.;    A𝐰 = 𝐛, ]

where

    E(𝐰)= ρ1 -R𝐰_2^2+τ𝐰_1

is strictly convex and non-smooth due to the presence of
the l_1 penalty term,
 
    A = (
    [ μ^T; 1_n ]) ∈ℝ^2 × n    𝐛 = (ρ,1)^T  ∈ℝ^2.

 One way to solve (<ref>) is to convert it into an
 unconstrained problem,
 for example by using a penalty function/continuation method,
 which approximates it by a sequence:

    [ min_𝐰 E(𝐰) + λ_k/2
        A𝐰 - 𝐛_2^2,  λ_k∈ℝ^+. ]

It is well known that, if the k-th subproblem (<ref>) has solution
w_k and {λ_k} is an increasing sequence tending to
∞ as k ⟶∞, any limit point of {w_k} is a solution
of (<ref>) <cit.>. Therefore, in many problems it is necessary
to choose very large values of λ_k and it makes (<ref>) extremely difficult
to solve numerically.
Alternatively, Bregman iteration can be used to reduce (<ref>) in a
short sequence of unconstrained problems by using the Bregman distance
associated with E  <cit.>,
where, conversely, the value of λ_k in (<ref>) remains constant.

The Bregman distance <cit.> associated with a proper convex functional
E(𝐰): ℝ^n ⟶ℝ at point
𝐯 is defined as:
 
    D_E^𝐩(𝐰,𝐯)=E(𝐰)-E(𝐯)- < 𝐩, 𝐰-𝐯 >,

where 𝐩∈∂ E(𝐯)
is a subgradient in the subdifferential of E at point 𝐯
and < . , . > denotes the canonical inner product in ℝ^n.
It is not a distance in the usual sense because it is not in general symmetric
but  it does measure closeness between 𝐰 and 𝐯 in the sense that if 𝐮 lies
on the line segment (𝐰,𝐯), then the line segment (𝐰,𝐮) has smaller Bregman
distance than (𝐰,𝐯) does.
At each Bregman iteration E(𝐰) is replaced by the Bregman distance
so a subproblem in the form of (<ref>) is solved
according to the following iterative scheme:

    {[ 𝐰_k+1 = argmin_𝐰 D_E^𝐩_𝐤(𝐰, 𝐰_k) +
       λ/2   A𝐰 - 𝐛_2^2,;        𝐩_k+1 =  𝐩_k-λ A^T(A  𝐰_k+1 -b)
       ∈∂ E( 𝐰_k+1).;                                                         ] .

The updating rule of 𝐩_k+1 is chosen according to the first-order optimality
condition for 𝐰_k+1 and ensures that
D_E^𝐩_𝐤+1(𝐰, 𝐰_k+1) is well defined.
Under suitable hypotheses the convergence of the sequence
{𝐰_k } to the solution of the constrained problem (<ref>)
is guaranteed in a finite number of steps <cit.>;
furthermore, using the equivalence of Bregman iteration with the augmented Lagrangian
one <cit.>, convergence is proved also in <cit.>.
Note that the convergence results guarantee the monotonic decrease of
A𝐰_k- 𝐛_2^2, thus for large k the constraint conditions
are satisfied to an arbitrary high degree of accuracy.
This yields a natural stopping criterion  according to a discrepancy principle.

Since there is generally no explicit expression for the solution of the sub-minimization
problem involved in (<ref>), at each iteration the solution is computed
inexactly using an iterative solver.
So, in the last years there has been a growing interest about inexact solution of the
subproblem involved in Bregman iteration.
In recent papers it is proved that, for many applications, Bregman iterations yield
very accurate solutions even if subproblems are not solved as accurately
<cit.>.
In <cit.> convergence results are obtained for piece-wise linear
convex functionals.
In <cit.> the inexactness in the inner solution is controlled by a
criterion that preserves the convergence of the Bregman iteration and its features in
image restoration.



§ MODIFIED BREGMAN ITERATION



A crucial issue in the solution of (<ref>) is the choice of
a suitable value for the regularization parameter τ, as already pointed out.
The aim is to select τ so to realize a trade-off between sparsity and short-controlling 
(requiring sufficiently large values) and fidelity to data (requiring small values).
While the literature offers
a significative number of methods for Tikhonov regularization <cit.>,
l_1 regularization parameter selection is often based on problem-dependent criteria
and related to iterative empirical estimates, that require a high computational cost.
In <cit.> least-angle regression (LARS) algorithm proceeds by decreasing the value
of τ progressively from very large values,
exploiting the fact that the dependence of the optimal
weights on τ is piecewise linear.

In this section, we present a numerical algorithm, based on a
modified Bregman iteration with adaptive updating rule for τ.
Our basic idea for defining the rule for τ comes from the
well-known properties of the l_1 norm and the
following proposition <cit.>:

Let 𝐰_τ_1 and 𝐰_τ_2  be solution of the l_1-regularized
problem (<ref>) with τ_1 and τ_2 respectively.
If some of (𝐰_τ_2)_i are negative and all the entries in
𝐰_τ_1 are positive or zero,  we have
τ_1 > τ_2.




We then propose an updating rule for τ that generates an increasing sequence of values.
Our aim is to modify Bregman iteration, in order
to produce solutions satisfying a fixed financial target, defined
in terms of sparsity or short-controlling or a combination of them.

Let

    E_k(𝐰) = R𝐰-ρ_2^2+τ_k 𝐰_1,   k=0,1,…

We now prove the main result of this paper:


Given (𝐰_k+1,𝐩_k+1) provided by (<ref>) applied to E_k, it holds

    𝐩̃_k+1= τ_k+1/τ_k𝐩_k+1+ 2(1-τ_k+1/τ_k) R^T(R𝐰_k+1-ρ) ∈∂ E_k+1(𝐰_k+1).


Proof. It holds 𝐩_k+1∈∂ E_k(𝐰_k+1) =
∂ (τ_k 𝐰_k+1_1) + ∇( R𝐰_k+1-ρ_2^2),
thus a vector 𝐪_k+1∈∂ (τ_k 𝐰_k+1_1) exists such that

    𝐩_k+1 = 𝐪_k+1 + 2 R^T(R𝐰_k+1-ρ).

It follows that 𝐪_k+1 = 𝐩_k+1 - 2  R^T(R𝐰_k+1-ρ) ∈∂ (τ_k 𝐰_k+1_1).
It is easy to verify that:

    τ_k+1/τ_k𝐪_k+1∈∂ (τ_k+1𝐰_k+1_1).

Then τ_k+1/τ_k𝐪_k+1  + 2 R^T(R𝐰_k+1-ρ) ∈∂ E_k+1(𝐰_k+1),
which completes the proof.













We propose the following modified Bregman iteration:

    {[      𝐩_k =  τ_k/τ_k-1𝐩_k + 2(1-τ_k/τ_k-1) R^T(R𝐰_k-ρ) ,; 𝐰_k+1 = argmin_𝐰 D_E_k^𝐩_𝐤(𝐰, 𝐰_k) +
       λ/2 A𝐰 - 𝐛_2^2,;                         𝐩_k+1 = 𝐩_k-λ A^T(A  𝐰_k+1 -b),;                                          τ_k+1 = h(τ_k);                                                         ] .

where h: ^+ ⟶^+
is an increasing, bounded function.

Note that relation (<ref>) in Theorem <ref> guarantees that the iterative scheme
(<ref>) is well defined, thus preserves the properties of the original one.

In this paper we choose a multiplicative form for the function h.
We set τ_k+1 = η_k+1τ_k,
 where η_k+1 depends on 𝐰_k+1 according
 to the financial target, as shown in Algorithm <ref>.
Note that we are not ensured that a finite value of τ exists
that satisfies the financial target, thus we force h to be bounded
by setting a maximum value τ_max.
If the financial target is met at a certain step, then η_k=1 for all successive iterations.
Conversely, τ is set to τ_max. In any case, there exists
an iteration k̅ such that τ_k remain fixed at a value τ̅
for k≥k̅.



Let τ̅ be the regularization parameter value produced by the Algorithm <ref>
at step k̅.
Suppose that at a certain step k≥k̅
the iterate
𝐰_k satisfies A𝐰_k=b.
Then 𝐰_k is a solution to the constrained problem

    [ min_𝐰 E_k̅(𝐰);          s.t.;       A𝐰 = 𝐛. ]



Proof.
We note that τ_k = τ̅ ∀  k≥k̅,
thus the objective function is fixed for k≥k̅.
Therefore, the proof follows the proof of Theorem 2.2 in <cit.>.


This result shows that if the sequence provided by
Algorithm <ref> converges in the sense of lim_k ⟶∞ ||A𝐰_k-b||_2=0,
then the iterates 𝐰_k will get arbitrarily close to a solution
to the original constrained problem with τ = τ.



§ EXPERIMENTAL RESULTS



In this section, we discuss some computational issues and
show the effectiveness of Algorithm <ref> for
solving the regularized portfolio optimization problem (<ref>).

In Algorithm <ref> we set λ=1, τ_0= 2^-5,  τ_max = 1 and
θ = 2.
Iterations are stopped as soon as A𝐰_k-b_2≤ Tol with Tol=10^-4
that, from the financial point of the view, guarantees constraints at a sufficient accuracy.
We implement the Fast Proximal Gradient method with backtracking stepsize rule (FISTA) <cit.>
to solve the unconstrained subproblem at each modified Bregman iteration in Algorithm <ref>.
FISTA is an accelerated variant of Forward Backward (FB) algorithm,
built upon the ideas of Güler <cit.> and Nesterov <cit.>.
Note that FB is a first-order method for minimizing objective functions
F(x) ≡ f(x) + g(x),  where g:ℝ^n→ℝ is a proper,
convex, lower semicontinuous function with 𝖽𝗈𝗆(g) closed,
f:ℝ^n →ℝ is convex and ∇ f is L-Lipschitz continuous.
It generates a sequence (x_n)_n ∈ N in two separate stages;
the former performs a forward (explicit) step which involves only f,
while the latter performs a backward (implicit) step involving a proximal
map associated to g <cit.>.
In our case we set
f=ρ1 -R𝐰_2^2 - <𝐩, 𝐰> + λ/2 A𝐰 - 𝐛_2^2
and g=τ𝐰_1,
then the proximal map of g is the simple and explicit Soft threshold operator:

    Prox_g(w_i)=sgn(w_i)  ( |w_i|-min {|w_i|,τ} ).

Inner iterations are stopped when the relative difference in Euclidean norm between
two successive iterates is less than Tol_Inn=10^-4.
All our experiments, some of which are reported in the following, show that it is
not worth to require a great accuracy to the inner solver.


The tests have been performed in Matlab R2015a (v. 8.5, 64-bit) environment,
on a six-core Xeon processor with
24 GB of RAM and 12 MB of cache memory, running
Ubuntu/Linux 12.04.5.
We compare our optimal portfolios with the evenly weighted one
(the naive portfolio), usually taken as benchmark in literature <cit.>.
This essentially for three reasons: it is easy to implement, many
investors still use such simple rule to allocate their wealth across assets
and it allows one to diversify the risk.

We evaluate our approach observing the out-of-sample performances of
optimal portfolios as in <cit.>.
This means that for each T-years period of asset
returns, we use historical series to solve (<ref>); the target return ρ
is fixed to the average return provided by the naive portfolio in those years.
The optimal solution obtained in this way is used to build a portfolio that is retained
for one year. We continue this process by moving one year ahead until we reach the end of the
period, ending with a series of out-of-sample portfolios.
We then compare the so obtained average return ρ̂
and standard deviation values σ̂ with the
corresponding ones of the naive portfolio.
We moreover compute the Sharpe ratio SR=ρ̂/σ̂:
since one would desire great return and small variance values,
the Sharpe ratio can be taken as reference value for the comparison.
We present the results on three test problems; the first and the second
one come from Fama and French database[data available at 

],
used in <cit.>. We obtain comparable results in terms of optimal
portfolio Sharpe ratio.
The last test problem is built on data from Italian market.



 §.§ Test 1: FF48

We consider
the first database - FF48 - which contains monthly returns
of 48 industry sector portfolios from July 1926 to December 2015.

Using data from 1970 to 2015, we construct optimal portfolios and analyze
their out-of-sample performance. Starting from July 1970, we use the T=5-years
so 40 optimal portfolios are built, until June 2015. Portfolios in FF48 exhibit
moderate correlation, indeed the condition number of C is O(10^4) for all simulations.
We tested difference values of Tol_Inn; all our experiments,
show that lower values of Tol_Inn do not improve results, thus we show results
obtained for Tol_Inn=10^-4. 

In table <ref>, for both optimal and naive
portfolio, expected return, standard deviation and Sharpe ratio are
reported, all expressed on annual basis. The optimal portfolios are no-short ones,
(n_short = 0, n_act = 48),
that is, the target is to obtain positive solutions.
Values refer to average values computed over
8 years, grouped as described in the first column of the table. The first row
contains average values computed over the all 40-years period of simulation.


In all cases, optimal portfolio exhibits greater values of Sharpe ratio
than the naive one.


In figure <ref> we report the number of active positions
in optimal no-short portfolios (top) and the number of modified Bregman iterations (bottom)
for each year of simulation.
We note the fast convergence of the Algorithm <ref>, with an average number
of iterations equal to 8.
The values of τ range between 2^-5 and 2^-2,

promoting sparsity (the percentage of
sparsity varies from 6% to 21%) and positivity.


In table <ref>, for the same financial target,
we report results on optimal portfolios containing at most
ten active positions (n_short = 48, n_act = 10).
Values are interpreted as in table <ref>.
In all cases, optimal portfolio exhibits again greater values of Sharpe ratio
than the naive one.

In figure <ref> we report the number of active and short positions
in optimal portfolios (top) and the number of modified Bregman iterations (bottom)
for each year of simulation.
In this case the average number of Bregman iterations is equal to 6.
The values of τ range between 2^-5 and 2^-3.

Finally in table <ref> we report a comparison
with results exhibited in Table 1 of paper <cit.>.
We denote with Algorithm 1 the results produced by our optimization
procedure and with LARS the results provided in <cit.>.
We refer to the same 30-years simulation period, with the average
taken over 5-years for each break-out period.
 We note that our procedure of regularization parameter selection produces higher values of Sharpe ratio; since the expected return is fixed by the constraint, this means that we obtain less risky portfolios.







 §.§ Test 2: FF100

We here show results on the second database by Fama and French
- FF100 - containing data of 100
portfolios which are the intersections of 10 portfolios formed on size
and 10 portfolios formed on the ratio of book equity to market equity.
Also FF100 contains monthly returns from
from July 1926 to December 2015.



We apply the same strategy as in FF48 (T=5-years, 40 optimal portfolios
constructed).
Correlation values observed in FF100 are higher than in the previous
test, the conditioning of C is O(10^18).

In table <ref>, we show optimal no-short portfolios.
We report the
expected return, the standard deviation and the Sharpe ratio expressed on annual
 basis.

On the overall period, optimal portfolio outperforms the naive one.
The values of τ range between 2^-4 and 2^-2, the percentage of
sparsity varies from 4% to 17% (Fig. <ref>).
Note that, looking at details on each year of simulation, we observe
negative returns for both optimal and naive portfolio. For instance,
in the 8^th year of simulation, optimal portfolio produces a loss of 4%, the
naive one of 12%. In the 10^th year the losses are of 1% and
10% respectively. This happens because almost all components in
portfolios show decreased returns.
Finally, we note that in the period 07/1975-06/1983 naive portfolio
outperforms the optimal one. For instance, in the 3^rd year of simulation
the optimal portfolio, which contains 5 assets (56, 90, 91, 93, 95),
produces a gain of 8%, versus a gain of 18%
of the naive one.
This behavior is essentially due to
a drastic change in asset returns with respect to historical data.








This situation could be controlled by a dynamic asset allocation strategies, for which
at the beginning of each period during
the investment horizon, the investor can freely rearrange the portfolio,  but it isn't the aim of this paper.










We finally report also in this case a comparison with results exhibited in <cit.>,
Table 3. We note that we obtain
higher values of the Sharpe ratio.











 §.§ Test 3: IT72

We here consider a portfolio constructed on
real data from Italian market. It considers the monthly returns of 72 equities,
from September 2009 to August 2016. Assets are reported in table <ref>;
25 assets are included in the FTSE MIB index computation.
The FTSE MIB is the primary benchmark Index for the Italian equity markets.
The Index is comprised of highly liquid, leading companies across different
sectors, indeed
it captures about the 80% of the domestic market capitalization.
The FTSE MIB is computed on 40 Italian equities and seeks to replicate the
broad sector weights of the Italian stock market. 


Starting from September 2009, we use the T=6-years data
to build the optimal portfolio from September 2015 until August 2016.
The conditioning of R^TR is O(10^9).
In figure <ref> we graphically show
the composition of the optimal portfolio we constructed. The optimization strategy
allocates the investor wealth on 14 equities, with weights represented as percentage
in the figure, among which 5 belong to the FTSE MIB set.
The result is obtained in 10 Bregman iterations,
with τ =2^-3.
We note that the optimal portfolio has return and standard deviation,
on annual basis, given by 11% and 34% respectively. The same values for the naive
portfolio are -14% and 60%, thus the latter provides a loss to the investor.    





§ CONCLUSIONS



We have proposed an algorithm, which exploits the Bregman iteration method,
for the portfolio selection problem formulated
as an l1-regularized mean-variance model.
The choice of the regularization parameter is the key point in order to provide
solutions with either a limited or null number of negative components and/or
a limited number of active positions.
Our main contribution is the modification of the Bregman iteration,
which adaptively sets the value of the regularization parameter depending on the financial target.
It is observed that both sparsity and short-controlling are obtained for sufficiently
large values of the regularization parameter. The basic idea is then to generate an
increasing sequence of values and fix it when requirements are met.
We show that our modification to the Bregman iteration preserves the convergence of the original scheme.
Numerical experiments confirm the effectiveness of the proposed algorithm.

We saw in our experiments that sometimes the effectiveness of the optimization strategy
can be affected by changes in market conditions. Future work could consider
dynamic asset allocation, which involves frequent portfolio adjustments.



§ ACKNOWLEDGMENTS

This work was partially supported by FFABR grant, annuity 2017, and INdAM-GNCS project.

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