Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Joint Probability Maximization for Gaussian
Inequality Systems with Application to a
Probabilistic Model for Portfolio Problems
1
2
1
René Aid , Michel Minoux , Riadh Zorgati
1
EDF R&D, 2 University P. & M. Curie, FRANCE
ICSP, Buzios, June 28th, 2016
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Main Objectives of the talk
(a)
We investigate the class of (nonconvex) optimization problems which consist in
nding a solution of maximum likelihood (or "most robust solution") to a given
system of random linear inequalities, assuming nondegenerate gaussian
distribution of coecients of the system.
(b)
Conditions under which a given locally optimal solution can be guaranteed to be
globally optimal are proposed. They make essential use of a novel necessary and
sucient condition for local concavity of the probability functions to be
maximized.
(c)
The class of problems investigated is shown to provide a potentially interesting
(and, as far as we know, original) alternative to classical models for optimal
portfolio selection (e.g. the well-known Markowitz model, 1952)
(d)
Series of computational experiments will be presented for evaluating
the relevance of our global optimality certicate (simulated data)
the interest of our probabilistic porfolio problem (real data)
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Outline
1.
Problem Statement
2.
Local Concavity Condition for a Single-Sided Gaussian Inequality
3.
Case of m Independent Gaussian Inequalities
4.
Application : A New Probabilistic Model for Optimal Portfolio Selection
5.
Conclusions, Perspectives
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Joint Probability Maximization Problem
f (x)
maximizing
bi −µT
i x
Φ √ T
. Since
x Σi x
i=
is equivalent to maximizing log (f (x)) therefore
Let the following problem :(PMP)
f (x) > 0,
m
Q
max
x∈X
f (x) =
1
problem (PMP) can be stated in the equivalent form:
(
0
(PMP )
Φ
max
x∈X
m
X
1
ln
Φ
i=
bi − µT
i x
p
T
x Σi x
is the c.d.f. of the standard gaussian distribution
∀i = 1, . . . , m,
fi (x) = Φ
random inequality
ui x ≤ bi
bi −µT
i x
√
x T Σi x
where
!!
N (0, 1);
is the probability of meeting the
ui ∼ N (µi , Σi )
for given
µi ∈ Rn
and
Σi
(a given symmetric positive denite variance-covariance matrix)
X is a given polyhedron in Rn (in
> 0, X is assumed not to contain
case all the
bi
coecients are strictly
0, otherwise 0 is readily seen to be an
optimal solution).
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
General Characterization of Concavity
A key issue in the developpement of ecient methods for nding globally
optimal solutions concerns the concavity of the objective function.
Since the log function is concave increasing in its argument, it is easily seen that
this issue is strongly related to the conditions under which the functions
Φ
T
bi −µi
√
x
x T Σi x
are concave or at least locally concave.
2
Let:
f
θ=
(b−µT x )
x T Σx
. Up to a strictly positive mutiplicative factor, the hessian of
can be shown to be expressed as:
(θ − 1) ΣxµT + µx T Σ
(3 − θ)
T
√
− µµT − Σ
H̄ = T
Σxx Σ − √
x Σx
θ
x T Σx
Condition of concavity : set(s) of
x
satisfying
R. Aid, M. Minoux,R. Zorgati
(1)
H̄(x) 0
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
General Characterization of Concavity
Theorem
T
For a single probabilistic
constraint
of the form u x ≤ b with u ∼ N (µ, Σ) and
T
b−µ x
Σ positive denite : Φ √
has negative semi-denite hessian in x 6= 0,
x T Σx
T
such that b − µ x > 0, if and only if:
x T Σx
2
µT Σ−1 µ ≤ b + µT x + b 2
b − µT x
x T Σx
2
!
−3
(2)
2
√
µTk x
µTk x
T
T −1
2
x Σx µ Σ µ ≤ θ − 2θ + 2 θ(θ − 1) p
+ (θ + 1) T
x Σk x
xkT Σk x
(Proof: in (MZ 2016), "Convexity of Gaussian Chance Constraints and of Related
Probability Maximization Problems", Comput. Stat., 31-1, (2016) ).
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Sketch of proof
H̄ is NSD i H̃ = Σ−1/2 H̄Σ−1/2 is NSD, and setting:
1
/2
V = Σ x , W = Σ−1/2 µ, V T W = µT x , kV k2 = x T Σx , we get:
We observe that
T
T
(θ − 1) VW + WV
VV T
H̃ = (3 − θ)
− √
− WW T − I .
kV k2
kV ||
θ
an expression which can easily be shown to be equivalent to
H̃ = YY T − ZZ T − I
where:
H̃ is NSD if and only
YY T − ZZ T is not greater
Thus
i:
Y =
q
1
+ θ1 ×
V
kV k and
Z=
1
(θ− ) V
√
θ kV k
(3)
+W
.
if the largest eigenvalue of the (rank-2) matrix
than 1. An easy calculation shows that this is true
2
kY k2 − kZ k2 + kY k2 × kZ k2 − Z T Y
≤ 1.
(4)
and that the above condition implies (2).
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Case of m Independent Gaussian Inequalities
The previous result cannot guarantee concavity of the objective function of
(PMP/PMP') on the whole solution set X . However, under some conditions to be
specied below, we are going to show that, for a suciently hight probability value p ,
convexity of the set S = {x : ln (f (x)) ≥ ln (p)} ∩ X can be obtained.
Let X be a given nonempty closed convex subset of Rn \ {0}. Let
p ∈]0, 1[ be such that Φ−1 (p) ≥ 1 (⇔ p ≥ Φ(1) = 0.8413). For all k = 1, . . . m,
T
suppose that one can compute a lower bound value γk satisfying √µTk x ≥ γk for all
Proposition 1 :
2
x ∈ S = {x/f (x) ≥ p} ∩ X . Let θ̄ = Φ−1 (p) .
x Σk x
Then the set S is convex if the two following conditions (5) and (6) hold for all
k = 1, . . . , m:
γk
1
µTk Σ−
k µk
≥
0
(5)
≤
p
θ̄2 − 2θ̄ + 2 θ̄(θ̄ − 1)γk + (θ̄ + 1)γk2
(6)
In addition to this, ln (f (x)) is concave on S .
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Sketch of Proof
bi −µT
i x
√
f (x) is the product of the m values Φ
which are all ≤ 1 : any x satisfying
x T Σi x
√
bk −µT
x
f (x) ≥ p meets the condition √ T k ≥ θ̄ = Φ−1 (p) for all k = 1, . . . m. Now,
x Σk x
consider an arbitrary k ∈ [1, m] and suppose that both conditions (5) and (6) hold.
bk −µT
k x
≥ θ̄ ≥ 1. Since the function θ 7→ θ2 − 2θ is
Then, for any x ∈ S , it holds θ = √
T
x Σk x
nondecreasing for θ ≥ 1 and (5) implies µTk x ≥ 0 for all
x ∈ S = {x/ ln (f (x)) ≥ ln (p)} ∩ X , it follows that for all x ∈ S :
2
p
√
µTk x
µT x
+ (θ + 1) T
≥ θ̄2 − 2θ̄ + 2 θ̄(θ̄ − 1)γk + (θ̄ + 1)γk2 .
θ2 − 2θ + 2 θ(θ − 1) p k
T
x Σk x
xk Σk x
bk −µT
k x
Since (6) holds, the condition of Theorem 1 is satised, showing that Φ √
is
x T Σk x
bk −µT
k x
concave in x for all x ∈ S and so is ln Φ √
. We have shown that ln (f (x))
T
x Σk x
is concave in x for all x ∈ S and therefore the set S = {x/ ln (f (x)) ≥ ln (p)} ∩ X is
convex in x .
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Global Optimality Certicate
The following corollary, easily deduced from Proposition 1, then provides a
global optimality certicate when the conditions (5) and (6) are fulllled
∀k = 1, . . . m.
Corollary 1 :
max {ln (f
x̄ to the problem
p̄ = f (x̄) ≥ Φ (1) = 0.8413 and
Suppose that a locally optimal solution
(x))}
has been determined. If
x∈X
the conditions of Proposition 1,
x̄
under
is a globally optimal solution to (PMP') and
(PMP).
Proof :
where
Solving (PMP') is equivalent to solving (PMP) : max {ln (f
x∈S
S = {x : ln (f (x)) ≥ ln (f (x̄)) = ln (p̄)} ∩ X .
x̄ , as a local maximum of (PMP') is also a
Moreover,
local maximum of
(PMP). Under the conditions of Proposition 1 (taking
convex and ln (f
(x))
is concave on
(x))}
p = p̄ ),
the set
S
is
S.
Therefore (PMP) is a convex optimization problem and
x̄
is necessarily a global
optimum for (PMP), hence a global optimum for (PMP') and (PMP).
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Checking Global Optimality
Since (PMP') is a nonconvex optimization problem, the solution
x̄
produced is
a priori only locally optimal.
In order to check global optimality, using Proposition 1 and its corollary, we
γk
have to compute the lower bounds
for
k = 1 . . . m.
For each
k,
this is done
by solving the (convex) optimization problem:
(I )







and taking
ηk∗ =
T
min µk
x
s.t.:
√ p
θ̄ x T Σi x
µT
i x +
x ∈ X.
√
γk = ηk∗ θ̄/ (bk − ηk∗ ).
≤ bi
(i = 1, . . . , m)
Note that (I) is a SOCP problem, eciently
solved with an interior-point solver (CVX in the MATLAB environment).
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio Selection Problem : Statement
Given
m
n
j = 1, . . . n, each producing random return, we assume that
= 1, . . . m) provide, for each asset, estimates ωij and σij for
and standard deviation for the return uij .
assets
experts (i
the mean
The various assets and experts are supposed to be independent so that for
i, i 0
variables uij
any two experts
and any two assets
random
and
The decision variable
xj
ui 0 j 0
j, j 0
(such that
i 6= i 0
or
j 6= j 0 )
the
are independent.
represents the percentage of investment in asset
j
in the portfolio.
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio Selection Problem : Probabilistic
Formulation
We want to nd an optimal solution
x∗
to the "Portfolio Selection Problem"
which corresponds to a portfolio achieving a given common target return
r
with
maximum probability:
(PSP)















max
x
m
Q
i=
1
P
nP
n
j=1 uij xj ≥ r
o
subject to:
n
P
j=
1
xj = 1
xj ≥ 0 (∀j = 1, . . . , n).
We propose this class of models as a possible new way of formulating optimal
portfolio selection problems in terms of a well-dened stochastic optimization
problem.
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio Selection Problem : link with (PMP)
We observe that the above problem is a special case of (PMP) in which
2
Σi = diag σi21 , . . . , σin
: the components
uij
of each row
ui
of the random
inequality system are stochastically independent.
the polyhedron
X
is the
n-dimensional
simplex (in this case 0
∈
/X
)
” ≥ ” while
P
n
j=1 xj = 1
Pn
u
x
≥
r
i can be
j=1 ij j
The probabilistic portfolio problem has inequalities of the form
(PMP) has inequalities of the form
” ≤ ”.
Using the fact that
every feasible solution, each constraint of the form
transformed into the equivalent constraint
Pn
j=
1 (ρi − uij ) xj ≤ ρi − ri
for any
ρ i ∈ R.
Moreover,
ρi
can be chosen
in
>0
and big enough so that each coecient in the above constant is a Gaussian
random variable with nonnegative mean
σij
µij = ρi − ωij
(the standard deviations
are unchanged); in our experiments we have taken :
R. Aid, M. Minoux,R. Zorgati
ρi = maxj=1,...,n {ωij } .
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Illustration of the nonconvexity
Observe that nonconvexity makes the problem dicult to solve: consider the
following (purely numeric) instance with

ω=
5
3
9

11
11
12
,
6
12
12
n = m = 3, r = 10 and


3.5
4
1
σ =  0.3 0.15 1 
0.6
1.9
0.8
Using fmincon (from MATLAB optimization toolbox) started at various initial
solutions yields distinct locally optimal solutions:
x0 =
x0 =
0.2
0.1
0.6
0.2
0.2
0.7
T
T
→
→
local opt.
x̄ =
local opt.
x̄ 0 =
R. Aid, M. Minoux,R. Zorgati
0
0
1
0
0
1
T
T
(joint proba 0.0342).
(joint proba 0.1541).
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Generation of Instances
As a useful consequence of our results we will obtain
a guarantee of global optimality for a signicant part of the (PSP) instances
solved. To the best of our knowledge, no similar result could be found in the
literature so far. The instances considered in our computational experiments
have been drawn at random as follows.
[ωmin = 5, ωmax = 15]
ωij
values in
σij
values (standard deviations) in
Higher values of
σij
σij
(average returns between 5% and 15%);
[σmin = 2, σmax = 5
correspond to higher values of
correspond to lower values of
ωij
ωij ,
to 10];
and lower values of
(assets having higher average return
are usually more "risky")
The
r
value is chosen close to
rmin = mini minj {ωij }.
it can be interpreted
as a "target return" for the portfolio which is to be achieved with
maximum probability.
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Computational Results
initial solution used to run fmincon taken as
v
x0 = v /
P
n
j=
1 vj
n
vector is randomly chosen from the uniform distribution in [0, 1] .
rmin :
Min average return and
r = ρ ∗ rmin :
Target return.
P0 : value of the joint probability function for x0 (column 6)
P̄ : value of the joint probability function for solution produced
x̄
, where the
by fmincon
(column 7)
Checking Global optimality :
n
o
−1
2 − 2θ̄ + 2√θ̄(θ̄ − 1)γk + (θ̄ + 1)γ 2 .
G = mink=1,...,m µT
Σ
µ
−
θ̄
k
k k
k
Since the problem: maxx∈X {ln(f
a priori
not convex,
guaranteed to be a
the solution
global
(x))} (where X is the n-dimensional simplex)
x produced by the fmincon solver is not
optimum solution, it is only a
R. Aid, M. Minoux,R. Zorgati
local
is
maximum.
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Computational Results : m = 5, n = 20 (1/3)
Exp. σmin
1
2
3
4
2
2
2
2
2
2
2
2
2
2
2
2
σmax
rmin
ρ
P0
P̄
5
5
5
7
7
7
9
9
9
11
11
11
4.18
7.12
7.54
4.45
6.28
6.70
4.35
5.44
5.86
4.3
4.60
5.02
1
1.7
1.8
1
1.5
1.6
1
1.3
1.4
1
1.1
1.2
0.999999
0.980717
0.965746
0.999972
0.990487
0.973886
0.999230
0.991809
0.989036
0.994511
0.985983
0.981858
0.999999
0.999729
0.998816
0.999997
0.998937
0.996626
0.999921
0.998373
0.995415
0.999436
0.998538
0.995839
R. Aid, M. Minoux,R. Zorgati
Global
Optimality ? (G)
YES ( 2037.78)
YES (94.19)
NO (-8.09)
YES ( 584.14)
YES (22.41)
NO (-40.42)
YES (321.37)
YES (10.79)
NO (-44.30)
YES ( 136.51)
YES (29.55)
NO (-33.92)
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Computational Results : m = 20, n = 50 (2/3)
Exp. σmin
1
2
3
4
2
2
2
2
2
2
2
2
2
2
2
2
σmax
rmin
ρ
P0
P̄
5
5
5
7
7
7
9
9
9
11
11
11
4.65
7.91
8.84
4.67
7.21
7.44
4.69
6.51
6.98
4.63
6.02
6.51
1
1.7
1.9
1
1.55
1.6
1
1.4
1.5
1
1.3
1.4
1
0.911510
0.317833
0.999999
0.994600
0.976564
0.999999
0.990472
0.981397
0.999923
0.995545
0.927700
1
0.999991
0.999567
0.999999
0.999943
0.999832
0.999999
0.999789
0.998698
0.999996
0.999536
0.996165
R. Aid, M. Minoux,R. Zorgati
Global
Optimality ? (G)
YES (+∞))
YES (175.43)
NO (-1.35)
YES ( 2979.67
YES (86.46)
NO (-5.12)
YES (654.78)
YES (20.33)
NO (-101.90)
YES (461.23)
YES (4.80)
N0 (-127.34)
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Computational Results : m = 40, n = 100 (3/3)
Exp. σmin
1
2
3
4
2
2
2
2
2
2
2
2
2
2
2
2
σmax
rmin
ρ
P0
P̄
5
5
5
7
7
7
9
9
9
11
11
11
4.81
7.21
8.65
4.81
6.73
8.65
4.81
7.21
7.69
4.81
6.73
7.69
1
1.5
1.7
1
1.4
1.8
1
1.5
1.6
1
1.4
1.6
1
0.999999
0.957583
1
0.999999
0.509131
0.999999
0.996499
0.922291
0.999999
0.998323
0.945066
1
0.999999
0.999969
1
0.999999
0.999873
0.999999
0.999990
0.999907
0.999999
0.999970
0.997714
R. Aid, M. Minoux,R. Zorgati
Global
Optimality ? (G)
YES (+∞))
YES (691.40)
NO (-248.24)
YES (+∞))
YES (358.97)
NO (-205.61)
YES (4339.61)
YES (116.94)
NO (-105.89)
YES (1484.44)
YES (86.03)
NO (-257.21)
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Comments on Results of Global Optimality Test
Global optimality of the solution is guaranteed in a signicant proportion of
instances as long as the target return does not exceed too much
rmin .
Guaranteeing global optimality tends to become progressively harder as the
σmax
σmin increases (i.e. as the condition number of the covariance
matrices increases).
ratio
As
m
and
n
increase, the probability levels required to guarantee the global
optimality turn out to become closer to 1.
when global optimality is conrmed, the values of the optimal joint
probability obtained are always close to 1 ( at least 0.99).
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Computational Results : Average CPU Times.
m
n
5
20
40
100
150
20
50
150
200
250
CPU Time (s)
for Solving (PMP')
0.08
0.22
0.5
2.43
6.19
Total CPU Time (s)
(including global optimality test)
1.25
18.22
459.5
2345.43
11019.19
Computations have been carried out on Intel Xeon E3 (3.4 GHz, 16 Gbits RAM).
The computational approach proposed leads to small computation times for
instances with m ≤ 40, n ≤ 100, but is also capable of handling larger problems
(up to m = 150, n = 250 within acceptable computation times.
The time required by fmincon alone is only a very small fraction of the total CPU
time. This is consistent with the fact that guaranteeing global optimality for
nonconvex problems is known to be a nontrivial task.
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P1" (m = 1, n = 20) : Data and Results
Target return : r = 4.5%
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P1" (m = 1, n = 20) : Data and Results
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P3" (m = 3, n = 15) : Data
Target return :
r = 4.5%
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P3" (m = 3, n = 15) : Results
Exp.
1
2
3
Return
Return
Volatility
Volatility
P
P
Markowitz
MaxP
Markowitz
MaxP
Markowitz
MaxP
4.50
6.92
2.35
5.53
0.50
0.67
4.50
6.75
2.79
6.48
0.50
0.63
4.50
6.32
2.86
6.73
0.50
0.60
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P6" (m = 7, n = 9) : Data
Target return :
r = 0.045
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P6" (m = 7, n = 9) : Results
Exp.
1
2
3
4
5
6
7
Return
Markowitz
4.50
4.50
4.50
4.81
5.48
4.50
4.50
Return
MaxP
6.24
6.37
6.95
6.38
7.47
6.41
5.90
Volatility
Markowitz
3.54
3.10
2.79
2.97
2.80
3.55
3.21
R. Aid, M. Minoux,R. Zorgati
Volatility
MaxP
6.88
5.84
6.44
5.70
5.43
6.89
6.19
P
Markowitz
0.50
0.50
0.50
0.54
0.63
0.50
0.50
P
MaxP
0.60
0.62
0.64
0.63
0.70
0.61
0.59
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "P6" (m = 7, n = 9) : Results
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Experiments 1: Relevance of the Global Optimality Certicate
Experiments 2: Interest of our Probabilistic Porfolio Approach
Portfolio "Rockafellar & Uryasev" (m = 1, n = 3) :
Data and Results
Target return :
r = 0.0110

M=
0.0101
Allocation
Markowitz
MaxP
0.0044
S&P
0.45
0
0.0137
Gov Bond
0.11
0

0.0005
0.0002

0.0002
0.0076
0.0002
Σ =  0.0002
0.0042
Small Cap
0.43
1
R. Aid, M. Minoux,R. Zorgati
0.0042
0.0032
*
*
*
Return
0.0110
0.0137
σ
0.0615
0.0874
P
0.50
0.5123
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Conclusions, Perspectives
A new necessary and sucient condition for convexity of solutions sets dened by
the probability to satisfy a random inequality with Gaussian distribution has been
established. Conditions have been derived for guaranting global optimality and
have been shown to open the way to solving exactly some instances of joint
probability maximization problems of signicant size (up to 250 variables and 150
inequalities).
A probabilistic model of portfolio, based on probability maximization has been
proposed. Based on rst experiments, the method seems to provide higher
returns with higher risks.
Search for improved, more powerful, global optimality tests, possibly addressing
the case of joint two-sided chance constraints is an ongoing research.
Possible extension of (at least part of) the results obtained to nongaussian
probability distributions would be worth considering.
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Acknowledgement
Dr Marie BOURROUSSE, researcher at department Optimization, Simulation,
Risks and Statistics, group of mathematical nance, of Research &
Development Division of EDF, is gratefully acknowledged for her precious help
in providing real data and for various insightful remarks.
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A
Outline
Problem Statement
Local Concavity Condition
Case of m Independent Gaussian Inequalities
Application: A New Probabilistic Model for Optimal Portfolio Selection
Conclusions, Perspectives
Bibliography
Charnes V., Cooper W., 1959, "Chance-Constrained Programming", Management
Science, 6:73-79.
Henrion R., 2007, "Structural Properties of Linear Probabilistic Constraints",
Optimization, 56, 4:425-440.
Henrion R., Strugarek C., 2008, "Convexity of Chance Constraints with Independent
Random Variables", Computational Optimization and Applications, 41:263-276.
Henrion R., Strugarek C., 2011, "Convexity of Chance Constraints with Dependent
Random Variables: the Use of Copulae", Optimization Methods in Finance and Energy,
International Series in OR-MS, Vol. 163, Springer: 427-439.
Minoux M., Zorgati R., 2016, "Convexity of Gaussian Chance Constraints and of Related
Probability Maximization Problems", Computational Statistics, ISSN 0943-4062, Volume
31, number 1, Comput Stat (2016) 31:387-408, DOI 10.1007/s00180-015-0580-z,
Springer.
THANK YOU ! OBRIGADO !
R. Aid, M. Minoux,R. Zorgati
Joint Probability Maximization for Gaussian Inequality Systems with A