Generalized distance to a simplex and a new geometrical method for portfolio optimization
GGeneralized distance to a simplex and a newgeometrical method for portfolio optimization
Fr´ed´eric BUTIN ∗ September 21, 2020
Abstract
Risk aversion plays a significant and central role in investors’ decisionsin the process of developing a portfolio. In this framework of portfoliooptimization we determine the portfolio that possesses the minimal riskby using a new geometrical method. For this purpose, we elaborate analgorithm that enables us to compute any generalized Euclidean distanceto a standard simplex. With this new approach, we are able to treat thecase of portfolio optimization without short-selling in its entirety, and wealso recover in geometrical terms the well-known results on portfolio op-timization with allowed short-selling.Then, we apply our results in order to determine which convex combina-tion of the CAC 40 stocks possesses the lowest risk: not only we get avery low risk compared to the index, but we also get a return rate that isalmost three times better than the one of the index.
Keywords:
Portfolio optimization without short-selling, generalized dis-tance to a standard simplex, geometrical approach of portfolio optimiza-tion, geometrical algorithm.
JEL Classification:
G11, C61, C63.
The paper [M52] published by Harry Markowitz in 1952 completely changed themethods of portfolio management and gave birth to the so-called “Modern Port-folio Theory”, thanks to which its author earned the Nobel Prize in Economicsin 1990. Since his works and the paper [S63] of Sharpe, this theme centralizesa lot of interest and many developments have been written in this domain. Let ∗ Universit´e de Lyon, Universit´e Lyon 1, CNRS, UMR5208, Institut Camille Jordan, 43blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France, [email protected] a r X i v : . [ q -f i n . P M ] S e p s recall some recent and important works to which our article is linked.In [DJDY08], J´on Dan´ıelsson, Bjørn N. Jorgensen, Casper G. de Vries andXiaoguang Yang study the portfolio allocation under the probabilistic VaR con-straint and obtain remarkable topological results: the set of feasible portfoliosis not always connected nor convex, and the number of local optima increasesin an exponential way with the number of states. They propose a solution toreduce computational complexity due to this exponential increase.In [FS12], Claudio Fontana and Martin Schweizer give a simple approach tomean-variance portfolio problems: they change the problems’ parametrisationfrom trading strategies to final positions. In this way they are able to solve manyquadratic optimisation problems by using orthogonality techniques in Hilbertspaces and providing explicit formulas.In their important article [BCGD18], Hanene Ben Salah, Mohamed Chaouch,Ali Gannoun and Christian De Peretti (see also the thesis [B15]) define a newportfolio optimization model in which the risks are measured thanks to con-ditional variance or semivariance. They use returns prediction obtained bynonparametric univariate methods to make a dynamical portfolio selection andget better performance.In [PR19], Sarah Perrin and Thierry Roncalli show how four algorithms of op-timization (the coordinate descent, the alternating direction method of multi-pliers, the proximal gradient method and the Dykstra’s algorithm) can be usedto solve problems of portfolio allocation.In [BIPS20], Taras Bodnar, Dmytro Ivasiuk, Nestor Parolya and WolfgangSchmid make an interesting work about the portfolio choice problem for powerand logarithmic utilities: they compute the portfolio weights for these utilityfunctions assuming that the portfolio returns follow an approximate log-normaldistribution, as suggested in [B00]. It is also noticeable that their optimal port-folios belong to the set of mean-variance feasible portfolios. The three aims of this article are the following:— give a new geometrical algorithm (Algorithm 1) to compute any generalizeddistance to a simplex,— determine, by making use of this algorithm, a portfolio with minimal vari-ance,— apply this technique to the CAC 40 stocks, and get a portfolio with returnrate that is almost three times better than the one of the index.After having briefly explained the notations in
Section 1 , we expose in
Sec-tion 2 the portfolio optimization problem and prove by compactness and con-2exity arguments that it possesses a unique solution.Then, in
Section 3 , we solve the problem in the case where short-selling isallowed. For this, we recall the classical method, and we give our very simplegeometrical method.
Section 4 is the heart of the article: in this section, we solve the portfoliooptimization problem in the case where short-selling is not allowed. For thispurpose, we give a new geometrical algorithm to compute the distance from apoint to a standard simplex, which can be used for every Euclidean distance.We can eventually apply this algorithm to the example of the CAC 40 stocksand determine the portfolio with the lowest risk. This portfolio also has theproperty of being almost three times more efficient than the underlying index.This is done in
Section 5 . We consider n stocks S , S , . . . , S n and denote by X , X , . . . , X n the randomvariables that represent their return rate (for example, daily, monthly or yearly).For every i ∈ [[1 , n ]], we set m i = E ( X i ) (mean of X i ) and V i = V ( X i ) (varianceof X i ). We set C = ( Cov ( X i , X j )) ( i, j ) ∈ [[1 , n ]] = Cov ( X , X ) · · · Cov ( X , X n )... ... Cov ( X n , X ) · · · Cov ( X n , X n ) ,X = X ... X n , and m = m ... m n . Matrix C is the covariance matrix of X , X , . . . , X n , and X is a random vector. Definition 1.
We call portfolio (with allowed short-selling) every linear com-bination P x = n (cid:88) i =1 x i S i , where x = ( x , . . . , x n ) ∈ R n and n (cid:88) i =1 x i = 1 .The return rate of the portfolio is the linear combination R x = n (cid:88) i =1 x i X i . If we don’t allow short-selling, then every x i must be nonnegative, and in thatcase the linear combination is a convex combination . n − Let us recall some classical results that we will use in the following.3 roposition 1.
We have E ( R x ) = t x m and V ( R x ) = t x C x .Proof. We immediately have E ( R x ) = n (cid:88) i =1 x i E ( X i ) = t x m .Moreover, V ( R x ) = E (cid:32) n (cid:88) i =1 x i X i − E (cid:32) n (cid:88) i =1 x i E ( X i ) (cid:33)(cid:33) = E (cid:32) n (cid:88) i =1 x i ( X i − E ( X i )) (cid:33) , hence V ( R x ) = E n (cid:88) i =1 n (cid:88) j =1 x i x j ( X i − E ( X i ))( X j − E ( X j )) = n (cid:88) i =1 n (cid:88) j =1 x i x j Cov ( X i , X j ) . The following proposition is immediate.
Proposition 2.
The matrix C has the two following properties. (i) It is symmetric positive, (ii)
It is symmetric definite positive if and only if X , . . . , X n are almost surelyaffinely independent. In all the following, we assume that C is symmetric definite positive: this isalways true in practise.Let us denote by H the affine hyperplane of R n with equation n (cid:88) j =1 x j = 1, andby K the standard ( n − − simplex (also called standard simplex of dimension n − K := x = ( x , . . . , x n ) ∈ [0 , n / n (cid:88) j =1 x j = 1 . The simplex K is a Haussdorff compact subset of R n that is contained in thehyperplane H .For example, if n = 2, H is the line of equation x + y = 1 in the usual plane,and the 1 − standard simplex K is the segment [(0 , , (1 , .2 Existence and uniqueness of the solution Minimizing the variance of the portfolio is equivalent to finding the minimumon K of the quadratic form f : x (cid:55)→ V ( R x ) = t x C x . Let us consider the scalar product( x , y ) (cid:55)→ (cid:104) x , y (cid:105) := t x Cy and the Euclidean norm x (cid:55)→ (cid:107) x (cid:107) := √ t x C x . The aim is to determine the point of K that realizes the minimal distance to K from the origin point in the sense of (cid:107) · (cid:107) .As K is a Haussdorff compact subset, and as f is continuous, we know thatthis minimum does exist. We will prove that it is also unique. For this, theconvexity plays a central role (see for example [R17]). Proposition 3.
The map f is strictly convex.Proof. For every x, y ∈ R n and for every λ ∈ [0 , f ( λx + (1 − λ ) y ) = λ t xCx + (1 − λ ) t yCy + λ ( λ − (cid:0) t xCy + t yCx (cid:1) , hence λf ( x ) + (1 − λ ) f ( y ) − f ( λx + (1 − λ ) y ) = λ (1 − λ ) t ( x − y ) C ( x − y ) , which is nonnegative, as C is positive.Moreover, if x (cid:54) = y , this quantity is positive as C is definite positive. Proposition 4.
Let K be a convex domain and f : K → R a convex map.Then, (i) every local minimum of f is global, (ii) if f is strictly convex, then f possesses at most one minimum.Proof. (i) Let x be a local minimum of f in K . Let us assume by contradictionthat it is not global: there exists y ∈ K such that f ( y ) < f ( x ). For every t ∈ ]0 , y t = ty + (1 − t ) x . Then y t belongs to K , and for t smallenough, (cid:107) y t − x (cid:107) = t (cid:107) y − x (cid:107) is close enough to 0, i.e. y t is close enough to x . Thus f ( x ) ≤ f ( y t ). As f is convex, we have f ( y t ) ≤ tf ( y ) + (1 − t ) f ( x ),hence f ( x ) ≤ tf ( y ) + (1 − t ) f ( x ), i.e. f ( x ) ≤ f ( y ), which is a contradiction.(ii) Let us assume by contradiction that f possesses a least two distinct minima x and x with f ( x ) = f ( x ). Then, as f is strictly convex, f (cid:0) x + x (cid:1) < f ( x ) + f ( x ) = f ( x ), which is a contradiction to the fact that x is aminimum.As a consequence of Proposition 4, the quadratic form f possesses exactly oneminimum on f , and his minimum is global.5 Minimisation of f on H In this section, we give two methods to compute the portfolio that possesses thelowest risk: the classical one, and our geometrical approach.Let us denote by E = ( e , . . . , e n ) the canonical basis of R n . A basis of the vec-tor hyperplane that directs H is B = ( e − e , e − e , . . . , e − e n ). Moreover,for every x = n (cid:88) j =1 x j e j ∈ R n , x belongs to H if and only if n (cid:88) j =1 x j = 1.Let is denote by u the vector u = n (cid:88) j =1 e j , which is orthogonal to the hyper-plane H , and let us set h : x = ( x , . . . , x n ) (cid:55)→ t x u − n (cid:88) j =1 x j − . f on H by the classical method Here we briefly recall the classical method to compute the portfolio that pos-sesses the lowest risk. Several sources, such as [N09], [M10] and [PP14], providea clear presentation of these well-known tools.
Proposition 5.
The unique solution x that minimises the map f on H is thevector x = C − u t uC − u .Proof. According to Lagrange’s multipliers theorem, there exists λ ∈ R suchthat ∇ f ( x ) = λ ∇ h ( x ). Here we have ∇ f ( x ) = 2 Cx and ∇ h ( x ) = u , thus2 Cx = λu , i.e. x = λ C − u . As t x u = 1, we get1 = t x u = t ux = λ t uC − u, hence λ = 2 t uC − u and x = C − u t uC − u . Example 1.
For n = 2 , the formula is very simple: by setting ∆ = V ( X ) V ( X ) − Cov ( X , X ) , we have C = (cid:20) V ( X ) Cov ( X , X ) Cov ( X , X ) V ( X ) (cid:21) and C − = 1∆ (cid:20) V ( X ) − Cov ( X , X ) − Cov ( X , X ) V ( X ) (cid:21) , hence x = V ( X ) − Cov ( X , X ) V ( X ) − Cov ( X , X ) + V ( X ) e + V ( X ) − Cov ( X , X ) V ( X ) − Cov ( X , X ) + V ( X ) e . .2 Minimisation of f on H by the geometrical approach Here we recover the classical results on the portfolio with minimal variance withallowed short-selling by making use of an Euclidean interpretation.This portfolio is P x , where x is the orthogonal projection onto H of the originpoint.Let us define the ( n, n ) − matrix A = c , − c , c , − c , · · · c ,n − c n, c , − c , c , − c , · · · c ,n − c n, ... ... ... c , − c ,n c , − c ,n · · · c ,n − c n,n · · · . Proposition 6.
The unique solution x that minimises f on H is the vectorwhose coordinates in E are given by the last column of the inverse of matrix A .Proof. For every x in H , x is the orthogonal projection onto H of the originpoint if and only if x is orthogonal to H , that is to say, for every i ∈ [[2 , n ]], (cid:104) x , e − e i (cid:105) = 0.Since (cid:104) x , e − e i (cid:105) = n (cid:88) j =1 x j (cid:104) e j , e − e i (cid:105) = n (cid:88) j =1 x j ( c ,j − c i,j ), we deduce that x isthe solution if and only if A x = (cid:20) n − , (cid:21) , i.e. x = A − (cid:20) n − , (cid:21) , which meansthat the coordinates in E of x are given by the last column of A − . K from a point of R n In this section, we will solve the problem of portfolio optimization without short-selling, by giving an explicit and calculable solution that doesn’t seem to appearin the literature. R m Let us now consider J a subset of [[1 , n ]] and the vector subspace E = (cid:77) j ∈ J R e j of R n , identified with R m , where m = | J | . Let H (cid:48) be the affine hyperplaneof E defined by the equation (cid:88) j ∈ J x j = 0. Let us fix i ∈ J and define J by J = J \ { i } , and let us denote by J the complementary of J in [[1 , n ]]. Then,a basis of the vector hyperplane of E parallel to H (cid:48) is B (cid:48) = ( e i − e i / i ∈ J ).Let a be a point of R n . 7et us set B (cid:48) = ( c i,j − c i ,j ) ( i,j ) ∈ J × J and B = (cid:20) B (cid:48) ,m (cid:21) , then b (cid:48) = n (cid:88) j =1 a j ( c i,j − c i ,j ) i ∈ J and b = (cid:20) b (cid:48) (cid:21) . Proposition 7.
The orthogonal projection of a onto H (cid:48) is the vector whosenonzero coordinates in E are given by B − b , which means that ( x i ) i ∈ J = B − b and ( x i ) i ∈ J = 0 n − m, .Proof. For every x = n (cid:88) j =1 x j e j ∈ R n , x is the orthogonal projection of a onto H (cid:48) if and only if the three following conditions hold(i) for every j ∈ J , x j = 0,(ii) n (cid:88) j =1 x j = 1,(iii) for every i ∈ J , x − a is orthogonal to e i − e i .Since (cid:104) x − a , e i − e j (cid:105) = n (cid:88) j =1 ( x j − a j ) (cid:104) e j , e i − e i (cid:105) , we have (cid:104) x − a , e i − e j (cid:105) = 0 if and only if (cid:88) j ∈ J x j (cid:104) e i − e i , e j (cid:105) = n (cid:88) j =1 a j (cid:104) e i − e i , e j (cid:105) , i.e. (cid:88) j ∈ J x j ( c i,j − c i ,j ) = n (cid:88) j =1 a j ( c i,j − c i ,j ). K from a point of R n We now propose a recursive algorithm to compute the point x realizing thedistance to K from a point a ∈ R n . In his article [C16], L. Condat gave a newand fast algorithm to project a vector onto a simplex. However, his algorithmwas made only for the usual Euclidean distance. Our algorithm can be used forevery Euclidean distance. The reader can also have a look at the paper [CY11]about the projection onto a simplex. 8igure 1: Geometric explanation of Algorithm 1 Algorithm 1.
Entry: ( a , K )Compute x the orthogonal projection of a onto H If x belongs to K Return x ElseIf K is a − simplex (i.e. K has exactly vertices)Return the vertex that is the closest to x ElseDetermine the hyperface K (cid:48) of K that is the closest to x Compute y the orthogonal projection of x onto H (cid:48) (the affinesubspace defined by K (cid:48) )Apply recursively the algorithm to ( y , K (cid:48) ) Proposition 8.
Algorithm 1 ends.Proof.
This is straightforward since at each step of the algorithm the dimensionof the simplex decreases of one unit.
Lemma 1. If x belongs to H \ K , then the distance from x to K is realized ina point of the frontier of K .Proof. Let us proceed by contradiction by assuming that the distance from x to K is realized in a point z in ◦ K . Let us denote by y the intersection of theline ( x , z ) with an hyperface of K crossed by this line. Then, by Minkowski,we get (cid:107) x − z (cid:107) = (cid:107) x − y (cid:107) + (cid:107) y − z (cid:107) > (cid:107) x − y (cid:107) , which is absurd since z realizesthe minimal distance from x to K .As a consequence, if x belongs to H \ K , then the distance from x to K is thedistance from x to the hyperface of K that is the closest to x . Proposition 9.
Algorithm 1 is correct. roof. Let us prove by induction on the dimension of K that the algorithmprovides us x ∈ K such that d ( a , K ) = (cid:107)−−→ ax (cid:107) . • If K has dimension 1, the result is clear. • Now assume that the algorithm is correct for every ( n − − simplex. Let usconsider K a n − simplex (with n ≥ K . Let x be the orthogonal projection of a onto H .— If x belongs to K , then x is the solution, and the algorithm is correct.— If x does not belong to K , as n ≥
2, we consider the simplex K (cid:48) definedabove, the affine subspace H (cid:48) and y the orthogonal projection of x onto H (cid:48) . Byinduction hypothesis applied to y and the ( n − − simplex K (cid:48) , the algorithmprovides us x ∈ K (cid:48) such that d ( y , K (cid:48) ) = (cid:107)−−→ yx (cid:107) . In particular, x belongs to K . Let us now prove that d ( a , K ) = (cid:107)−−→ ax (cid:107) . According to the Pythagoreantheorem, as −→ ax is orthogonal to H , we have d ( a , K ) = (cid:107)−→ ax (cid:107) + d ( x , K ) = (cid:107)−→ ax (cid:107) + d ( x , K (cid:48) ) , thanks to Lemma 1.Moreover, as −→ xy is orthogonal to H (cid:48) , we have d ( x , K (cid:48) ) = (cid:107)−→ xy (cid:107) + d ( y , K (cid:48) ) = (cid:107)−→ xy (cid:107) + (cid:107)−−→ x y (cid:107) = (cid:107)−−→ xx (cid:107) since −→ xy is orthogonal to −−→ x y .Finally, d ( a , K ) = (cid:107)−→ ax (cid:107) + (cid:107)−−→ xx (cid:107) = (cid:107)−−→ ax (cid:107) as −→ ax is orthogonal to −−→ xx , hence d ( a , K ) = (cid:107)−−→ ax (cid:107) and the algorithm is correct for K . Remark 1.
Let x be in H \ K . Then the hyperface of K that is the closestto x is not necessarily the hyperface of K obtained by suppressing the (or one)vertex of K that is the furthest of x .Proof. Let us consider the following example: let K be the 2 − simplex in thehyperplane { x + y + z = 1 } of R . Let us set C = .
012 0 .
004 0 . .
004 0 .
011 0 . .
008 0 .
007 0 . , x = . . − . . Then the distances from x to the vertices e , e , e are respectively d (cid:39) . d (cid:39) . d (cid:39) . x onto the edges defined by { e , e } , { e , e } , { e , e } are respectively p = . . . , p = . . . , p = . . . , and the distances from x to these edges are δ (cid:39) . δ (cid:39) . δ (cid:39) . d < d < d but δ < δ < δ . Here, the vertex of K that is thefurthest of x is e , but the distance from x to K is realized in a point of theedge defined by { e , e } . 10 .3 Minimisation of f on K Now that we have the algorithm to compute the generalized distance to a stan-dard simplex, it is easy to find the solution that minimises f on K : the portfoliothat possesses the lowest risk is P x , where x is the point of K that realizesthe distance from the origin point to K , i.e. d (0 , K ) = (cid:107) x (cid:107) . Here we determine the portfolio with lowest risk: we determine the convexcombination of CAC 40 stocks for which the variance is minimal .We use the mean and the standard deviation of monthly variation. Here we consider the period from 2007-04-23 to 2020-07-21, that is to say westart from the highest point of CAC 40 index. Table 1 gives the mean and thestandard deviation of stocks’ return rates that appear in the results.By using Algorithm 1, we determine the portfolio with allowed short-selling thatpossesses the lowest risk: this linear combination is given by Table 2. The meanof its monthly variation is 0 .
44% and its standard-deviation 4 . H already belongs to thesimplex K . We use the following abbreviations.AC : Accor SA ACA : Credit Agricole S.A.AI : L’Air Liquide S.A. AIR : Airbus SEATO : Atos SE BN : Danone S.A.BNP : BNP Paribas SA CA : Carrefour SACAP : Capgemini SE CS : AXA SADG : VINCI SA DSY : Dassault Systemes SEEL : EssilorLuxottica Societe anonyme EN : Bouygues SAENGI : ENGIE SA FP : TOTAL S.A.GLE : Societe Generale Societe anonyme HO : Thales S.A.KER : Kering SA LR : Legrand SAMC : LVMH Moet Hennessy - Louis Vuitton ML : Cie G le des Et. MichelinMT : ArcelorMittal OR : L’Oreal S.A.ORA : Orange S.A. PUB : Publicis Groupe S.A.RI : Pernod Ricard SA RMS : Hermes InternationalRNO : Renault SA SAF : Safran SASAN : Sanofi SGO : Compagnie de Saint-Gobain S.A.STM : STMicroelectronics N.V. SU : Schneider ElectricSW : Sodexo S.A. UG : Peugeot S.A.URW : Unibail-Rodamco-Westfield VIE : Veolia Environnement S.A.VIV : Vivendi WLN : Worldline For this computation, we do not consider EL, GLE and WLN, for which we don’t haveenough data. The French stock market month (that ends the third Friday in the month) is used. x x .
41% and its standard deviation 11 . .
12% and the standard deviation 19 . Year 2007 2008 2008 2010 2011 2012 2013Porfolio’s r. r. 1.85% -23.04% 5.09% 12.84% -0.80% 14.67% 7.81%CAC 40’s r. r. -5.12% -43.87% 22.87% 0.34% -23.45% 23.54% 14.72%Year 2014 2015 2016 2017 2018 2019 2020Porfolio’s r. r. 12.89% 16.16% 3.95% 14.38% 1.47% 33.12% 3.37%CAC 40’s r. r. 0.93% 7.30% 5.64% 12.40% -14.65% 30.33% -15.34%
Figure 3: Portfolio optimization from 2007-04-23 to 2020-07-21
Here we consider the period from 2009-01-19 to 2020-07-21, that is to say westart from the lowest point of CAC 40 index. As in previous section, Table 5gives the mean and the standard deviation of stocks’ return rates that appearin the results.The portfolio with allowed short-selling that possesses the lowest risk is given bythe following linear combination in Table 6. The mean of its monthly variationis 0 .
76% and its standard-deviation 4 . .
87% and its standard-deviation 4 . .
27% and its standard deviation 11 . .
94% and the standard deviation 16 . x x -20.66% 19.75% -3.07% 17.26% 18.36% 3.54%Table 7: Portfolio without allowed short-selling that possesses the lowest riskStock AI BN CA DSY HO x x Appendix — Python programs
Here we give a possible way to program Algorithm 1 in Python as well as thesubroutine used to compute an orthogonal projection.The function orth_proj(c,a,J) computes an orthogonal projection, where— c is the covariance matrix,— a is the point of which we want to compute the orthogonal projection,— J is the list of indices of p vectors of E that define the affine subspace ontowhich we want to project a . def orth_proj(c,a,J):p=len(J); n=len(c); i0=J[0]; L=list(set(range(n))-set(J)) The function mini_dist_fct(c,a) finds the point that realizes the minimaldistance from a to the standard ( n − − simplex and also returns the square ofthis distance: it is a possible version of Algorithm 1. def mini_dist_fct(c,a): s=J[0]if str(set(J)-{s})+str(x) in dico:delta=dico[str(set(J)-{s})+str(x)]else:delta=mini_dist(c,x,list(set(J)-{s}))dico[str(set(J)-{s})+str(x)]=deltad=delta[1]for j in J[1:]:if str(set(J)-{j})+str(x) in dico:delta0=dico[str(set(J)-{j})+str(x)]else:delta0=mini_dist(c,x,list(set(J)-{j}))dico[str(set(J)-{j})+str(x)]=delta0d0=delta0[1]if d0 Annals of Operations Research (1), DOI: 10.1007/s10479-016-2235-z (2018).[BIPS20] Taras Bodnar, Dmytro Ivasiuk, Nestor Parolya, Wolfgang Schmid,Mean-variance efficiency of optimal power and logarithmic utility portfolios, Mathematics and Financial Economics , 675–698 (2020).[C16] Laurent Condat, Fast projection onto the simplex and the l ball, Math-ematical Programming , 575–585 (2016).[CY11] Yunmei Chen, Xiaojing Ye, Projection Onto A Simplex,arXiv:1101.6081v2 [math.OC] (2011).[DJDY08] J´on Dan´ıelsson, Bjørn N. Jorgensen, Casper G. de Vries, XiaoguangYang, Optimal portfolio allocation under the probabilistic VaR constraintand incentives for financial innovation, Annals of Finance , 345–367(2008).[FS12] Claudio Fontana, Martin Schweizer, Simplified mean-variance portfoliooptimisation, Math Finan Econ , 125–152, DOI 10.1007/s11579-012-0067-4 (2012).[M52] Harry Markowitz, Portfolio Selection, The Journal of Finance (No. 1),77–91 (1952).[M10] Franck Moraux, Finance de march´e, Pearson Education France (2010).[N09] Anna Nagurney, Portfolio Optimization, University of Massachusetts(2009).[PP14] Patrice Poncet, Roland Portait, Finance de march, Dalloz , Paris, 4edition (2014).[PR19] Sarah Perrin, Thierry Roncalli, Machine Learning Optimization Algo-rithms & Portfolio Allocation, Papers 1909.10233, arXiv.org (2011).[R17] Aude Rondepierre, M´ethodes num´eriques pour l’optimisation non lin´eaired´eterministe, INSA de Toulouse (2017).[S63] William Forsyth Sharpe, A Simplified Model for Portfolio Analysis, Man-agement Science9