Slides on my presentation of a Particle Swarm approach to portfolio selection, that introduces a new particle initialization procedure based on dynamic systems' theory . Held in the series "Applied Micro Seminar -- Graduate School of Economics" at Kyoto University, December 2016.

1. Particle Swarm Optimization for Portfolio Selection.
Giacomo di Tollo
Dipartimento di Economia, University Ca’ Foscari, Venezia
(giacomo.ditollo@unive.it)
Applied Micro Seminar
Graduate School of Economics, Kyoto University
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 1 / 29

2. Summary
1 Metaheuristics and Exact Methods
2 Particle Swarm Optimisation
3 Our Initialisation
4 Portfolio Selection
5 Experimental Analysis
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 2 / 29

3. Metaheuristics
Strategies to guide the action of subordinated heuristics.
General principles.
Large scale optimization problems typically require larger
computational resources, and both practitioners and theoreticians
claim for robust methods, often endowed also with theoretical
properties [Nocedal, Wright ’00].
Combining theoretical properties of exact methods and fast
progress of heuristics [Fasano et al. ’14].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 3 / 29

4. Metaheuristics
Strategies to guide the action of subordinated heuristics.
General principles.
Large scale optimization problems typically require larger
computational resources, and both practitioners and theoreticians
claim for robust methods, often endowed also with theoretical
properties [Nocedal, Wright ’00].
Combining theoretical properties of exact methods and fast
progress of heuristics [Fasano et al. ’14].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 3 / 29

5. Metaheuristics
Strategies to guide the action of subordinated heuristics.
General principles.
Large scale optimization problems typically require larger
computational resources, and both practitioners and theoreticians
claim for robust methods, often endowed also with theoretical
properties [Nocedal, Wright ’00].
Combining theoretical properties of exact methods and fast
progress of heuristics [Fasano et al. ’14].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 3 / 29

6. Metaheuristics
Strategies to guide the action of subordinated heuristics.
General principles.
Large scale optimization problems typically require larger
computational resources, and both practitioners and theoreticians
claim for robust methods, often endowed also with theoretical
properties [Nocedal, Wright ’00].
Combining theoretical properties of exact methods and fast
progress of heuristics [Fasano et al. ’14].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 3 / 29

7. Classification of Metaheuristics
Trajectory methods (Simulated Annealing, Tabu Search,
Threshold Accepting).
Population methods (Genetic Algorithms, Ant Colony
Optimisation, Particle Swarm Optimisation).
Intensification VS Diversification [Blum and Roli ’03].
Exploitation VS Exploration [di Tollo et al. ’11].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 4 / 29

8. Classification of Metaheuristics
Trajectory methods (Simulated Annealing, Tabu Search,
Threshold Accepting).
Population methods (Genetic Algorithms, Ant Colony
Optimisation, Particle Swarm Optimisation).
Intensification VS Diversification [Blum and Roli ’03].
Exploitation VS Exploration [di Tollo et al. ’11].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 4 / 29

9. Classification of Metaheuristics
Trajectory methods (Simulated Annealing, Tabu Search,
Threshold Accepting).
Population methods (Genetic Algorithms, Ant Colony
Optimisation, Particle Swarm Optimisation).
Intensification VS Diversification [Blum and Roli ’03].
Exploitation VS Exploration [di Tollo et al. ’11].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 4 / 29

10. Classification of Metaheuristics
Trajectory methods (Simulated Annealing, Tabu Search,
Threshold Accepting).
Population methods (Genetic Algorithms, Ant Colony
Optimisation, Particle Swarm Optimisation).
Intensification VS Diversification [Blum and Roli ’03].
Exploitation VS Exploration [di Tollo et al. ’11].
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 4 / 29

11. Basics on PSO
PSO [Kennedy-Eberhart ’95], for unconstrained global
optimization problem
min
x∈IRn
f(x) (1)
Population-based method.
f(x) is assumed to be nonlinear and non-convex.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 5 / 29

12. Basics on PSO
PSO [Kennedy-Eberhart ’95], for unconstrained global
optimization problem
min
x∈IRn
f(x) (1)
Population-based method.
f(x) is assumed to be nonlinear and non-convex.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 5 / 29

13. Basics on PSO
PSO [Kennedy-Eberhart ’95], for unconstrained global
optimization problem
min
x∈IRn
f(x) (1)
Population-based method.
f(x) is assumed to be nonlinear and non-convex.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 5 / 29

14. Basics on PSO
We have a set of particles. For each particle we define its velocity
and its position.
The random velocity flown the particle through the problem space.
Each particle is attracted to its previous best position.
Each particle is attracted to the global best position.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 6 / 29

15. Basics on PSO
We have a set of particles. For each particle we define its velocity
and its position.
The random velocity flown the particle through the problem space.
Each particle is attracted to its previous best position.
Each particle is attracted to the global best position.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 6 / 29

16. Basics on PSO
We have a set of particles. For each particle we define its velocity
and its position.
The random velocity flown the particle through the problem space.
Each particle is attracted to its previous best position.
Each particle is attracted to the global best position.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 6 / 29

17. Basics on PSO
We have a set of particles. For each particle we define its velocity
and its position.
The random velocity flown the particle through the problem space.
Each particle is attracted to its previous best position.
Each particle is attracted to the global best position.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 6 / 29

18. Basics on PSO
xk+1
j = xk
j + vk+1
j
vk+1
j = vk
j + α ⊗ (pk
j − xk
j ) + β ⊗ (pk
g − xk
j )
(2)
α = cj
k rk
j
β = cg
k rk
g
cj
k , cg
k ∈ (0, 2.5)
(3)
xk+1
j = xk
j + vk+1
j
vk+1
j = wj
k vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
(4)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 7 / 29

19. Basics on PSO
xk+1
j = xk
j + vk+1
j
vk+1
j = vk
j + α ⊗ (pk
j − xk
j ) + β ⊗ (pk
g − xk
j )
(2)
α = cj
k rk
j
β = cg
k rk
g
cj
k , cg
k ∈ (0, 2.5)
(3)
xk+1
j = xk
j + vk+1
j
vk+1
j = wj
k vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
(4)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 7 / 29

20. Basics on PSO
xk+1
j = xk
j + vk+1
j
vk+1
j = vk
j + α ⊗ (pk
j − xk
j ) + β ⊗ (pk
g − xk
j )
(2)
α = cj
k rk
j
β = cg
k rk
g
cj
k , cg
k ∈ (0, 2.5)
(3)
xk+1
j = xk
j + vk+1
j
vk+1
j = wj
k vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
(4)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 7 / 29

21. Basics on PSO
xk+1
j = xk
j + vk+1
j
vk+1
j = vk
j + α ⊗ (pk
j − xk
j ) + β ⊗ (pk
g − xk
j )
(2)
α = cj
k rk
j
β = cg
k rk
g
cj
k , cg
k ∈ (0, 2.5)
(3)
xk+1
j = xk
j + vk+1
j
vk+1
j = wj
k vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
(4)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 7 / 29

22. Basics on PSO
xk+1
j = xk
j + vk+1
j
vk+1
j = vk
j + α ⊗ (pk
j − xk
j ) + β ⊗ (pk
g − xk
j )
(2)
α = cj
k rk
j
β = cg
k rk
g
cj
k , cg
k ∈ (0, 2.5)
(3)
xk+1
j = xk
j + vk+1
j
vk+1
j = wj
k vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
(4)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 7 / 29

23. Basics on PSO
xk+1
j = xk
j + vk+1
j
vk+1
j = vk
j + α ⊗ (pk
j − xk
j ) + β ⊗ (pk
g − xk
j )
(2)
α = cj
k rk
j
β = cg
k rk
g
cj
k , cg
k ∈ (0, 2.5)
(3)
xk+1
j = xk
j + vk+1
j
vk+1
j = wj
k vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
(4)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 7 / 29

24. Basics on PSO
vk+1
j = χ wj
k
vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
xk+1
j = xk
j + vk+1
j
(5)
vk
j is the velocity (search direction) of the j-th particle at step k.
xk
j is the position of the j-th particle at step k.
f(pk
j ) = min
0≤ℓ≤k
{f(xℓ
j }, j = 1, . . . , P.
f(pk
g) = min
0≤ℓ≤k;j=1,...,P
{f(xℓ
j }.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 8 / 29

25. Basics on PSO
vk+1
j = χ wj
k
vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
xk+1
j = xk
j + vk+1
j
(5)
vk
j is the velocity (search direction) of the j-th particle at step k.
xk
j is the position of the j-th particle at step k.
f(pk
j ) = min
0≤ℓ≤k
{f(xℓ
j }, j = 1, . . . , P.
f(pk
g) = min
0≤ℓ≤k;j=1,...,P
{f(xℓ
j }.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 8 / 29

26. Basics on PSO
vk+1
j = χ wj
k
vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
xk+1
j = xk
j + vk+1
j
(5)
vk
j is the velocity (search direction) of the j-th particle at step k.
xk
j is the position of the j-th particle at step k.
f(pk
j ) = min
0≤ℓ≤k
{f(xℓ
j }, j = 1, . . . , P.
f(pk
g) = min
0≤ℓ≤k;j=1,...,P
{f(xℓ
j }.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 8 / 29

27. Basics on PSO
vk+1
j = χ wj
k
vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
xk+1
j = xk
j + vk+1
j
(5)
vk
j is the velocity (search direction) of the j-th particle at step k.
xk
j is the position of the j-th particle at step k.
f(pk
j ) = min
0≤ℓ≤k
{f(xℓ
j }, j = 1, . . . , P.
f(pk
g) = min
0≤ℓ≤k;j=1,...,P
{f(xℓ
j }.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 8 / 29

28. Basics on PSO
vk+1
j = χ wj
k
vk
j + ck
j rk
j ⊗ (pk
j − xk
j ) + ck
g rk
g ⊗ (pk
g − xk
j )
xk+1
j = xk
j + vk+1
j
(5)
vk
j is the velocity (search direction) of the j-th particle at step k.
xk
j is the position of the j-th particle at step k.
f(pk
j ) = min
0≤ℓ≤k
{f(xℓ
j }, j = 1, . . . , P.
f(pk
g) = min
0≤ℓ≤k;j=1,...,P
{f(xℓ
j }.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 8 / 29

29. Our Approach
Ortogonal particles’ initialization in PSO.
Deterministic PSO.
Experimental Analysis to prove the effectiveness of our proposal.
Portfolio Selection Problem.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 9 / 29

30. Our Approach
Ortogonal particles’ initialization in PSO.
Deterministic PSO.
Experimental Analysis to prove the effectiveness of our proposal.
Portfolio Selection Problem.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 9 / 29

31. Our Approach
Ortogonal particles’ initialization in PSO.
Deterministic PSO.
Experimental Analysis to prove the effectiveness of our proposal.
Portfolio Selection Problem.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 9 / 29

32. Our Approach
Ortogonal particles’ initialization in PSO.
Deterministic PSO.
Experimental Analysis to prove the effectiveness of our proposal.
Portfolio Selection Problem.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 9 / 29

33. Dynamic Systems
xk = A ·xk−1 + bk−1
x1 = A ·x0 + b0
x2 = A ·x1 + b1
x2 = A ·(A·x0 + b0) + b1
x2 = A2 ·x0 + A ·b0 + b1
x3 = A ·x2 + b2
x3 = A ·(A2 ·x0 + A ·b0 + b1) + b2
x3 = A3 ·x0 + A2 ·b0 + A ·b1 + b2
xk = Ak ·x0 + k−1
t=0 Ak−t−1·bt
FREE REPONSE FORCED RESPONSE
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 10 / 29

34. Dynamic Systems
xk = A ·xk−1 + bk−1
x1 = A ·x0 + b0
x2 = A ·x1 + b1
x2 = A ·(A·x0 + b0) + b1
x2 = A2 ·x0 + A ·b0 + b1
x3 = A ·x2 + b2
x3 = A ·(A2 ·x0 + A ·b0 + b1) + b2
x3 = A3 ·x0 + A2 ·b0 + A ·b1 + b2
xk = Ak ·x0 + k−1
t=0 Ak−t−1·bt
FREE REPONSE FORCED RESPONSE
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 10 / 29

35. Dynamic Systems
xk = A ·xk−1 + bk−1
x1 = A ·x0 + b0
x2 = A ·x1 + b1
x2 = A ·(A·x0 + b0) + b1
x2 = A2 ·x0 + A ·b0 + b1
x3 = A ·x2 + b2
x3 = A ·(A2 ·x0 + A ·b0 + b1) + b2
x3 = A3 ·x0 + A2 ·b0 + A ·b1 + b2
xk = Ak ·x0 + k−1
t=0 Ak−t−1·bt
FREE REPONSE FORCED RESPONSE
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 10 / 29

36. Dynamic Systems
xk = A ·xk−1 + bk−1
x1 = A ·x0 + b0
x2 = A ·x1 + b1
x2 = A ·(A·x0 + b0) + b1
x2 = A2 ·x0 + A ·b0 + b1
x3 = A ·x2 + b2
x3 = A ·(A2 ·x0 + A ·b0 + b1) + b2
x3 = A3 ·x0 + A2 ·b0 + A ·b1 + b2
xk = Ak ·x0 + k−1
t=0 Ak−t−1·bt
FREE REPONSE FORCED RESPONSE
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 10 / 29

37. Dynamic Systems
xk = A ·xk−1 + bk−1
x1 = A ·x0 + b0
x2 = A ·x1 + b1
x2 = A ·(A·x0 + b0) + b1
x2 = A2 ·x0 + A ·b0 + b1
x3 = A ·x2 + b2
x3 = A ·(A2 ·x0 + A ·b0 + b1) + b2
x3 = A3 ·x0 + A2 ·b0 + A ·b1 + b2
xk = Ak ·x0 + k−1
t=0 Ak−t−1·bt
FREE REPONSE FORCED RESPONSE
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 10 / 29

38. Dynamic Systems
xk = A ·xk−1 + bk−1
x1 = A ·x0 + b0
x2 = A ·x1 + b1
x2 = A ·(A·x0 + b0) + b1
x2 = A2 ·x0 + A ·b0 + b1
x3 = A ·x2 + b2
x3 = A ·(A2 ·x0 + A ·b0 + b1) + b2
x3 = A3 ·x0 + A2 ·b0 + A ·b1 + b2
xk = Ak ·x0 + k−1
t=0 Ak−t−1·bt
FREE REPONSE FORCED RESPONSE
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 10 / 29

39. Ortogonal Initialisation
Assumption
We assume in (5) that ck
j = c, rk
j = r for any j = 1, ..., P, ck
g = ¯c,
rk
g = ¯r and wk
j = w, for any k ≥ 0.
PSO iteration (5) is equivalent to the following discrete stationary
(time-invariant) system (i.e., X(k + 1) = AX(k) + b(k)):
Xj (k + 1) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In

 Xj (k) +


χ crpk
j + ¯c¯rpk
g
χ crpk
j + ¯c¯rpk
g


where
Xj (k) =


vk
j
xk
j

 ∈ IR2n
, k ≥ 0.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 11 / 29

40. Ortogonal Initialisation
Assumption
We assume in (5) that ck
j = c, rk
j = r for any j = 1, ..., P, ck
g = ¯c,
rk
g = ¯r and wk
j = w, for any k ≥ 0.
PSO iteration (5) is equivalent to the following discrete stationary
(time-invariant) system (i.e., X(k + 1) = AX(k) + b(k)):
Xj (k + 1) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In

 Xj (k) +


χ crpk
j + ¯c¯rpk
g
χ crpk
j + ¯c¯rpk
g


where
Xj (k) =


vk
j
xk
j

 ∈ IR2n
, k ≥ 0.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 11 / 29

41. Ortogonal Initialisation
Assumption
We assume in (5) that ck
j = c, rk
j = r for any j = 1, ..., P, ck
g = ¯c,
rk
g = ¯r and wk
j = w, for any k ≥ 0.
PSO iteration (5) is equivalent to the following discrete stationary
(time-invariant) system (i.e., X(k + 1) = AX(k) + b(k)):
Xj (k + 1) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In

 Xj (k) +


χ crpk
j + ¯c¯rpk
g
χ crpk
j + ¯c¯rpk
g


where
Xj (k) =


vk
j
xk
j

 ∈ IR2n
, k ≥ 0.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 11 / 29

42. PSO reformulation
Using a standard notation for linear systems, we can split Xj(k) into the
free response XjL(k) and the forced response XjF (k), so that
Xj (k) = XjL(k) + XjF (k)
where
XjL(k) = Φ(k)Xj (0), XjF (k) =
k−1
τ=0
H(k − τ)Uj(τ)
and
Φ(k) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k
∈ IR2n×2n
H(k − τ) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k−τ−1
∈ IR2n×2n
Uj (τ) =


χ crpτ
j + ¯c¯rpτ
g
χ crpτ
j + ¯c¯rpτ
g

 ∈ IR2n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 12 / 29

43. PSO reformulation
Using a standard notation for linear systems, we can split Xj(k) into the
free response XjL(k) and the forced response XjF (k), so that
Xj (k) = XjL(k) + XjF (k)
where
XjL(k) = Φ(k)Xj (0), XjF (k) =
k−1
τ=0
H(k − τ)Uj(τ)
and
Φ(k) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k
∈ IR2n×2n
H(k − τ) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k−τ−1
∈ IR2n×2n
Uj (τ) =


χ crpτ
j + ¯c¯rpτ
g
χ crpτ
j + ¯c¯rpτ
g

 ∈ IR2n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 12 / 29

44. PSO reformulation
Using a standard notation for linear systems, we can split Xj(k) into the
free response XjL(k) and the forced response XjF (k), so that
Xj (k) = XjL(k) + XjF (k)
where
XjL(k) = Φ(k)Xj (0), XjF (k) =
k−1
τ=0
H(k − τ)Uj(τ)
and
Φ(k) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k
∈ IR2n×2n
H(k − τ) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k−τ−1
∈ IR2n×2n
Uj (τ) =


χ crpτ
j + ¯c¯rpτ
g
χ crpτ
j + ¯c¯rpτ
g

 ∈ IR2n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 12 / 29

45. PSO reformulation
Using a standard notation for linear systems, we can split Xj(k) into the
free response XjL(k) and the forced response XjF (k), so that
Xj (k) = XjL(k) + XjF (k)
where
XjL(k) = Φ(k)Xj (0), XjF (k) =
k−1
τ=0
H(k − τ)Uj(τ)
and
Φ(k) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k
∈ IR2n×2n
H(k − τ) =


χwIn −χ(cr + ¯c¯r)In
χwIn [1 − χ(cr + ¯c¯r)] In


k−τ−1
∈ IR2n×2n
Uj (τ) =


χ crpτ
j + ¯c¯rpτ
g
χ crpτ
j + ¯c¯rpτ
g

 ∈ IR2n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 13 / 29

46. Near Orthogonality of particles’ trajectories
The free
response XjL(k) only depends on the initial point Xj(0), and
not on the vectors pτ
j , pτ
g, with τ ≥ 0.
The velocity vk
j of the j-th particle at iteration k may be regarded
as a search direction from the current position xk
j .
lim
k→∞
XjL(k) = 0 when Φ(k) eigenvalues are ≤ 1 (modulus).
We can enforce diversification / intensification by setting Φ(k) in
an appropriated way:
0 < χw < 1;
0 < χ(cr + ¯c¯r) < 2(χw + 1).
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 14 / 29

47. Near Orthogonality of particles’ trajectories
The free
response XjL(k) only depends on the initial point Xj(0), and
not on the vectors pτ
j , pτ
g, with τ ≥ 0.
The velocity vk
j of the j-th particle at iteration k may be regarded
as a search direction from the current position xk
j .
lim
k→∞
XjL(k) = 0 when Φ(k) eigenvalues are ≤ 1 (modulus).
We can enforce diversification / intensification by setting Φ(k) in
an appropriated way:
0 < χw < 1;
0 < χ(cr + ¯c¯r) < 2(χw + 1).
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 14 / 29

48. Near Orthogonality of particles’ trajectories
The free
response XjL(k) only depends on the initial point Xj(0), and
not on the vectors pτ
j , pτ
g, with τ ≥ 0.
The velocity vk
j of the j-th particle at iteration k may be regarded
as a search direction from the current position xk
j .
lim
k→∞
XjL(k) = 0 when Φ(k) eigenvalues are ≤ 1 (modulus).
We can enforce diversification / intensification by setting Φ(k) in
an appropriated way:
0 < χw < 1;
0 < χ(cr + ¯c¯r) < 2(χw + 1).
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 14 / 29

49. Near Orthogonality of particles’ trajectories
The free
response XjL(k) only depends on the initial point Xj(0), and
not on the vectors pτ
j , pτ
g, with τ ≥ 0.
The velocity vk
j of the j-th particle at iteration k may be regarded
as a search direction from the current position xk
j .
lim
k→∞
XjL(k) = 0 when Φ(k) eigenvalues are ≤ 1 (modulus).
We can enforce diversification / intensification by setting Φ(k) in
an appropriated way:
0 < χw < 1;
0 < χ(cr + ¯c¯r) < 2(χw + 1).
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 14 / 29

50. Near Orthogonality of particles’ trajectories
The free
response XjL(k) only depends on the initial point Xj(0), and
not on the vectors pτ
j , pτ
g, with τ ≥ 0.
The velocity vk
j of the j-th particle at iteration k may be regarded
as a search direction from the current position xk
j .
lim
k→∞
XjL(k) = 0 when Φ(k) eigenvalues are ≤ 1 (modulus).
We can enforce diversification / intensification by setting Φ(k) in
an appropriated way:
0 < χw < 1;
0 < χ(cr + ¯c¯r) < 2(χw + 1).
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 14 / 29

51. Near Orthogonality of particles’ trajectories
The free
response XjL(k) only depends on the initial point Xj(0), and
not on the vectors pτ
j , pτ
g, with τ ≥ 0.
The velocity vk
j of the j-th particle at iteration k may be regarded
as a search direction from the current position xk
j .
lim
k→∞
XjL(k) = 0 when Φ(k) eigenvalues are ≤ 1 (modulus).
We can enforce diversification / intensification by setting Φ(k) in
an appropriated way:
0 < χw < 1;
0 < χ(cr + ¯c¯r) < 2(χw + 1).
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 14 / 29

52. Near Orthogonality of particles’ trajectories
We set the initial position and velocity of the particles, so that the
subvectors {νk
j } (first n entries of the free responses) {XjL(k)},
are mutually orthogonal:
We assign 1 to one variable, and 0 to all other variables if P ≤ 2n.
If P > 2n, then set the initial position/velocity of the first 2n
particles as stated, while the other particles may have whatever
initial position/velocity.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 15 / 29

53. Near Orthogonality of particles’ trajectories
We set the initial position and velocity of the particles, so that the
subvectors {νk
j } (first n entries of the free responses) {XjL(k)},
are mutually orthogonal:
We assign 1 to one variable, and 0 to all other variables if P ≤ 2n.
If P > 2n, then set the initial position/velocity of the first 2n
particles as stated, while the other particles may have whatever
initial position/velocity.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 15 / 29

54. Basics of Portfolio Selection
Given a set of assets, the aim is to decide in which assets to invest
and by how much in order to optimise some specific criterion.
Minimise the risk given an expected return [Markowitz ’52].
Different risk measures can be used.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 16 / 29

55. Basics of Portfolio Selection
Given a set of assets, the aim is to decide in which assets to invest
and by how much in order to optimise some specific criterion.
Minimise the risk given an expected return [Markowitz ’52].
Different risk measures can be used.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 16 / 29

56. Basics of Portfolio Selection
Given a set of assets, the aim is to decide in which assets to invest
and by how much in order to optimise some specific criterion.
Minimise the risk given an expected return [Markowitz ’52].
Different risk measures can be used.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 16 / 29

57. Markowitz
Information about future prices is contained in historical series.
We are given a set (Universe) of assets {a1 . . . an}. Each asset ai
has associated a mean return ri and a return variance σ2
i .
For each pair of assets (ai, aj ) we know the return covariance σij .
A portfolio is a vector of real values P = x1 . . . xn.
rp = n
i=1 rixi .
σp = n
i=1
n
j=1 σijxi xj .
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 17 / 29

58. Markowitz
Information about future prices is contained in historical series.
We are given a set (Universe) of assets {a1 . . . an}. Each asset ai
has associated a mean return ri and a return variance σ2
i .
For each pair of assets (ai, aj ) we know the return covariance σij .
A portfolio is a vector of real values P = x1 . . . xn.
rp = n
i=1 rixi .
σp = n
i=1
n
j=1 σijxi xj .
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 17 / 29

59. Markowitz
Information about future prices is contained in historical series.
We are given a set (Universe) of assets {a1 . . . an}. Each asset ai
has associated a mean return ri and a return variance σ2
i .
For each pair of assets (ai, aj ) we know the return covariance σij .
A portfolio is a vector of real values P = x1 . . . xn.
rp = n
i=1 rixi .
σp = n
i=1
n
j=1 σijxi xj .
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 17 / 29

60. Markowitz
Information about future prices is contained in historical series.
We are given a set (Universe) of assets {a1 . . . an}. Each asset ai
has associated a mean return ri and a return variance σ2
i .
For each pair of assets (ai, aj ) we know the return covariance σij .
A portfolio is a vector of real values P = x1 . . . xn.
rp = n
i=1 rixi .
σp = n
i=1
n
j=1 σijxi xj .
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 17 / 29

61. Markowitz
Information about future prices is contained in historical series.
We are given a set (Universe) of assets {a1 . . . an}. Each asset ai
has associated a mean return ri and a return variance σ2
i .
For each pair of assets (ai, aj ) we know the return covariance σij .
A portfolio is a vector of real values P = x1 . . . xn.
rp = n
i=1 rixi .
σp = n
i=1
n
j=1 σijxi xj .
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 17 / 29

62. Markowitz
Information about future prices is contained in historical series.
We are given a set (Universe) of assets {a1 . . . an}. Each asset ai
has associated a mean return ri and a return variance σ2
i .
For each pair of assets (ai, aj ) we know the return covariance σij .
A portfolio is a vector of real values P = x1 . . . xn.
rp = n
i=1 rixi .
σp = n
i=1
n
j=1 σijxi xj .
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 17 / 29

63. Markowitz
we impose a minimum required return re
min n
i=1
n
j=1 σij xi xj
n
i=1 ri xi ≥ re
n
i=1 xi = 1
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 18 / 29

64. Markowitz
we impose a minimum required return re
min n
i=1
n
j=1 σij xi xj
n
i=1 ri xi ≥ re
n
i=1 xi = 1
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 18 / 29

65. Markowitz
we impose a minimum required return re
min n
i=1
n
j=1 σij xi xj
n
i=1 ri xi ≥ re
n
i=1 xi = 1
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 18 / 29

66. Markowitz
we impose a minimum required return re
min n
i=1
n
j=1 σij xi xj
n
i=1 ri xi ≥ re
n
i=1 xi = 1
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 18 / 29

67. Coherence [Artzner et al. ’99]
Monotonicity: x ≤ y implies ρ(x) ≥ ρ(y);
Sub-additivity: ρ(x + y) ≤ ρ(x) + ρ(y) (no new investments
increase risk);
Positive homogeneity: ρ(λx) = λρ(x) (liquidity);
Translation invariance: ρ(x + αr0) = ρ(x) − α.
Markowitz is not coherent
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 19 / 29

68. Coherence [Artzner et al. ’99]
Monotonicity: x ≤ y implies ρ(x) ≥ ρ(y);
Sub-additivity: ρ(x + y) ≤ ρ(x) + ρ(y) (no new investments
increase risk);
Positive homogeneity: ρ(λx) = λρ(x) (liquidity);
Translation invariance: ρ(x + αr0) = ρ(x) − α.
Markowitz is not coherent
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 19 / 29

69. Coherence [Artzner et al. ’99]
Monotonicity: x ≤ y implies ρ(x) ≥ ρ(y);
Sub-additivity: ρ(x + y) ≤ ρ(x) + ρ(y) (no new investments
increase risk);
Positive homogeneity: ρ(λx) = λρ(x) (liquidity);
Translation invariance: ρ(x + αr0) = ρ(x) − α.
Markowitz is not coherent
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 19 / 29

70. Coherence [Artzner et al. ’99]
Monotonicity: x ≤ y implies ρ(x) ≥ ρ(y);
Sub-additivity: ρ(x + y) ≤ ρ(x) + ρ(y) (no new investments
increase risk);
Positive homogeneity: ρ(λx) = λρ(x) (liquidity);
Translation invariance: ρ(x + αr0) = ρ(x) − α.
Markowitz is not coherent
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 19 / 29

71. Coherence [Artzner et al. ’99]
Monotonicity: x ≤ y implies ρ(x) ≥ ρ(y);
Sub-additivity: ρ(x + y) ≤ ρ(x) + ρ(y) (no new investments
increase risk);
Positive homogeneity: ρ(λx) = λρ(x) (liquidity);
Translation invariance: ρ(x + αr0) = ρ(x) − α.
Markowitz is not coherent
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 19 / 29

72. Coherence [Artzner et al. ’99]
Monotonicity: x ≤ y implies ρ(x) ≥ ρ(y);
Sub-additivity: ρ(x + y) ≤ ρ(x) + ρ(y) (no new investments
increase risk);
Positive homogeneity: ρ(λx) = λρ(x) (liquidity);
Translation invariance: ρ(x + αr0) = ρ(x) − α.
Markowitz is not coherent
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 19 / 29

73. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

74. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

75. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

76. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

77. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

78. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

79. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

80. Our Optimisation Problem [Chen and Wang ’08]
ρa,p(x) = a (x − E[x])+
1 + (1 − a) (x − E[x])−
p − E[x]
rP ≥ re
n
i=1 xi = 1
Kd ≤ n
i=1 zi ≤ Ku
zi d ≤ xi ≤ zi u, i = 1, . . . , n
zi (zi − 1) = 0, i = 1, . . . , n
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 20 / 29

81. Exact Penalty Approach [di Tollo and Roli, 2008]
minx,z ρa,p(x) +
1
ε
7
ℓ=1
pℓ
p1 = max(0, re − rP)
p2 =
n
x=1
xi − 1
p3 + p4 =
n
x=1
max (0, zi · d − xi ) +
n
x=1
max (0, (xi − zi · u))
p5 + p6 = max 0, (Kd −
n
x=1
zi ) + max 0, (
n
x=1
zi − Ku)
p7 =
n
x=1
|zi · (1 − zi )|
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 21 / 29

82. Exact Penalty Approach [di Tollo and Roli, 2008]
minx,z ρa,p(x) +
1
ε
7
ℓ=1
pℓ
p1 = max(0, re − rP)
p2 =
n
x=1
xi − 1
p3 + p4 =
n
x=1
max (0, zi · d − xi ) +
n
x=1
max (0, (xi − zi · u))
p5 + p6 = max 0, (Kd −
n
x=1
zi ) + max 0, (
n
x=1
zi − Ku)
p7 =
n
x=1
|zi · (1 − zi )|
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 21 / 29

83. Exact Penalty Approach [di Tollo and Roli, 2008]
minx,z ρa,p(x) +
1
ε
7
ℓ=1
pℓ
p1 = max(0, re − rP)
p2 =
n
x=1
xi − 1
p3 + p4 =
n
x=1
max (0, zi · d − xi ) +
n
x=1
max (0, (xi − zi · u))
p5 + p6 = max 0, (Kd −
n
x=1
zi ) + max 0, (
n
x=1
zi − Ku)
p7 =
n
x=1
|zi · (1 − zi )|
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 21 / 29

84. Exact Penalty Approach [di Tollo and Roli, 2008]
minx,z ρa,p(x) +
1
ε
7
ℓ=1
pℓ
p1 = max(0, re − rP)
p2 =
n
x=1
xi − 1
p3 + p4 =
n
x=1
max (0, zi · d − xi ) +
n
x=1
max (0, (xi − zi · u))
p5 + p6 = max 0, (Kd −
n
x=1
zi ) + max 0, (
n
x=1
zi − Ku)
p7 =
n
x=1
|zi · (1 − zi )|
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 21 / 29

85. Exact Penalty Approach [di Tollo and Roli, 2008]
minx,z ρa,p(x) +
1
ε
7
ℓ=1
pℓ
p1 = max(0, re − rP)
p2 =
n
x=1
xi − 1
p3 + p4 =
n
x=1
max (0, zi · d − xi ) +
n
x=1
max (0, (xi − zi · u))
p5 + p6 = max 0, (Kd −
n
x=1
zi ) + max 0, (
n
x=1
zi − Ku)
p7 =
n
x=1
|zi · (1 − zi )|
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 21 / 29

86. Exact Penalty Approach [di Tollo and Roli, 2008]
minx,z ρa,p(x) +
1
ε
7
ℓ=1
pℓ
p1 = max(0, re − rP)
p2 =
n
x=1
xi − 1
p3 + p4 =
n
x=1
max (0, zi · d − xi ) +
n
x=1
max (0, (xi − zi · u))
p5 + p6 = max 0, (Kd −
n
x=1
zi ) + max 0, (
n
x=1
zi − Ku)
p7 =
n
x=1
|zi · (1 − zi )|
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 21 / 29

87. Benchmarks
FTSE MIB (32 assets, 1396 days)
DJIA (32 assets, 9312 days)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 22 / 29

88. Benchmarks
FTSE MIB (32 assets, 1396 days)
DJIA (32 assets, 9312 days)
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 22 / 29

89. Best Objective over time
Figure: Experiments with ORTHOinit
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 23 / 29

90. Best Objective over time
Figure: Experiments without ORTHOinit
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 24 / 29

91. Results
Exact Solver
re ρ Constraints Violated Computational time (sec)
min(ri ) 0.0046 — 25693
max(ri ) 0.021 — 7064
PSO with ORTHOinit
re ρ Constraints Violated Computational time (sec)
min(ri ) 0.0048 Lower Bound 5375
max(ri ) 0.0238 Lower Bound, Capital, re 7021
PSO without ORTHOinit
re ρ Constraints Violated Computational time (sec)
min(ri ) 0.0042 Lower Bound 6530
max(ri ) 0.0238 Lower Bound, Capital, re 6135
Table: Experimental Results, Instance DJIA, basic PSO
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 25 / 29

92. Results
ρ Success Ratio
Strategy Min Max Std Mean
PSO-newinit-REVAC 0.00683077 0.00872689 0.00109750 46.6
PSO-standard 0.00691271 0.15525469 0.07204026 23.3
NEOS 0.00658258 0.00658258
Table: Experimental Results, Instance FTSI MIB, enhanced PSO.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 26 / 29

93. Conclusions
ORTHOinit fosters a better diversification;
Computational times;
The exact penalty approach should be revised, trying different
reformulations of our portfolio selection problem, where possibly
the simple constraints (i.e. linear constraints) are not moved to the
penalty function.
G. di Tollo (et al.) PSO for Portfolio Selection 6th December,2016 27 / 29

94. References
[Artzner et al ’99]: P. Artzner, F. Delbaen, J.M. Eber, D. Heath (1999),
Coherent measures of risk, Mathematical Finance, vol. 9, pp. 203–228.
[Blum and Roli ’03]: C. Blum, A. Roli (2003), Metaheuristics in
combinatorial optimization: Overview and conceptual comparison, ACM
Computing Surveys, vol. 35 (3), pp. 268–308.
[Chen and Wang ’08]: Z. Chen, Y. Wang (2008), Two-sided coherent risk
measures and their application in realistic portfolio optimization, Journal
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