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Polyhedral Risk
Measures
Vadym Omelchenko,
Institute of Information Theory and Automation,
Academy of Sciences of the Czech Republic.
The presentation’s
structure
1. Definition of polyhedral risk measures (Two-stage)
2. Definition of polyhedral risk measures (Multi-stage)
3. Applications in the energy sector (CHP)
Definition of Polyhedral Risk
Measures (Two-Stage)
Polyhedral Risk Measures
• γ = 𝐿𝑝 (𝐹, ℝ) be the usual Banach space of real
random variables on some probability space
(Ω, F, P) for some p ∈ [1,∞).
Polyhedral Probability
Functionals
• Definition. A probability functional 𝑅: 𝛾 → ℝ is called
𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑎𝑙 if there exist 𝑘0 , 𝑘1 ∈ ℕ, 𝑤𝑖 , 𝑐𝑖 ∈ ℝ𝑘𝑖 , 𝑖 ∈
0,1 , and non-empty polyhedral sets 𝑉𝑖 ⊆ ℝ𝑘𝑖 , 𝑖 ∈ 0,1 ,
such that


v1  L p F , R k1 ,


R (Y )  sup  c0 , v0  E  c1 , v1  vi  Vi , i  0, 1 , 


w
,
v

w
,
v

Y


0
0
1 1




• for every Y ∈ 𝛾 . Here ⋅,⋅ denote scalar products on ℝ𝑘0
and ℝ𝑘1 .
• 𝑉1 ⊆ ℝ𝑘1 has to be understood in the sense a.s.
Linear Reformulation
• Definition. A probability functional 𝑅: 𝛾 → ℝ is called
𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑎𝑙 if there exist 𝑘0 , 𝑘1 ∈ ℕ, 𝑐𝑖 , 𝑤𝑖 ∈ ℝ𝑘𝑖 ,
matrices 𝐴𝑖 , and vectors 𝑏𝑖 ∈ ℝ𝑘𝑖 , 𝑖 ∈ 0,1 , such that



R (Y )  sup  E cT0  v0  c1T  v1







v1  L p F , R k1 , 


A1v1  b1 , a.s.,

A0 v0  b0 ,

w0  v0  w1  v1  Y 
Example
• We consider the functional R Y = E u Y
• on γ where u: ℝ → ℝ is of the form
𝑘
• 𝑢 𝑥 = sup 𝑐, 𝑣 : 𝑣 ∈ ℝ 𝑘+𝑟
+ , 𝑖=1 𝑣𝑖 = 1, 𝑤, 𝑣 = 𝑥 with
some 𝑐, 𝑣 ∈ ℝ 𝑘+𝑟 , 𝑘, 𝑟 ∈ ℕ and hence it is concave
and polyhedral in kinks 𝑤𝑖 , 𝑐𝑖 ∈ ℝ 2, 𝑖 = 1, … , 𝑘.
• According to Rockafellar and Wets (1998), Theorem
14.60, we can reverse the order of sup and E.
Example
• We consider the functional R Y = E u Y
• on γ where u: ℝ → ℝ is of the form
𝑘
• 𝑢 𝑥 = sup 𝑐, 𝑣 : 𝑣 ∈ ℝ 𝑘+𝑟
,
+
𝑖=1 𝑣𝑖 = 1, 𝑤, 𝑣 = 𝑥 with
some 𝑐, 𝑣 ∈ ℝ 𝑘+𝑟 , 𝑘, 𝑟 ∈ ℕ and hence 𝐑(⋅) is concave
and polyhedral in kinks 𝑤𝑖 , 𝑐𝑖 ∈ ℝ 2, 𝑖 = 1, … , 𝑘.
• See Rockafellar and Wets (1998), Theorem 14.60.
Theorem Rockafellar and Wets
Popular examples
• CV@R is a polyhedral risk measure.
• Every linear combination of CV@Rs are polyhedral
risk measures
• V@R is not polyhedral.
Properties of Polyhedral
Functionals
• Let R be a functional of the form:


R(Y )  sup  c0 , v0  E  c1 , v1







vi  Vi , i   0, 1 , 

w0 , v0  w1 , v1  Y 
v1  L p F , R k1 ,
• Let 𝑉𝑖 ⊆ ℝ𝑘𝑖 , 𝑖 ∈ 0,1 be polyhedral cones and assume:
1.
2.
𝑤1 , 𝑉1 = ℝ (complete recourse),
𝐷 ≔ 𝑢 ∈ ℝ: 𝑐0 − 𝑢𝑤0 ∈ 𝑉 0∗, 𝑐1 − 𝑢𝑤1 ∈ 𝑉 1∗ ≠ ∅ (dual feasibility.
Then R is finite, concave, and continuous on γ.
Properties of Polyhedral
Functionals
•
Let R be a functional of the form:


R(Y )  sup  c0 , v0  E  c1 , v1







vi  Vi , i   0, 1 , 

w0 , v0  w1 , v1  Y 
v1  L p F , R k1 ,
• Let 𝑉𝑖 ⊆ ℝ𝑘𝑖 , 𝑖 ∈ 0,1 be polyhedral cones and assume:
1.
𝑤1 , 𝑉1 = ℝ (complete recourse),
2. 𝐷 ≔ 𝑢 ∈ ℝ: 𝑐0 − 𝑢𝑤0 ∈ 𝑉 0∗, 𝑐1 − 𝑢𝑤1 ∈ 𝑉 1∗ ≠ ∅ (dual feasibility.
3.
1
1
𝑞 ∈ 1, ∞ given by 𝑝 + 𝑞 = 1
Then R admits the dual representation
𝑹 𝒀 = 𝒊𝒏𝒇 𝑬 𝒀 𝒁 : 𝒁 ∈ ℤ ∗
Where ℤ ∗ is a subset of ℤ = 𝐿𝑞 (𝐹, ℝ) given by
ℤ ∗= 𝑍 ∈ ℤ: 𝑐0 − 𝑤0 𝐸 𝑍 ∈ 𝑉 0∗, 𝑐1 − 𝑤1 𝑍 ∈ 𝑉 1∗ .
Definition of Polyhedral Risk
Measures (Multi-Stage)
Polyhedral Multi-Period
Acceptability Functionals
• Let us denote 𝛾 =×𝑇𝑡=1 𝐿𝑝 𝐹𝑡 , 𝑝 ∈ [1, ∞).
• Definition. A probability functional 𝑅: 𝛾 → ℝ is called
𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑎𝑙 if there are 𝑘𝑡 ∈ ℕ, 𝑐𝑡 ∈ ℝ𝑘𝑡 , and nonempty polyhedral sets 𝑉𝑡 ⊆ ℝ𝑘𝑡 , 𝑡 = 0, . . 𝑇, 𝑤𝑡,𝜏 ∈
ℝ𝑘𝑡−𝜏 , 𝜏 = 0, . . , 𝑡, 𝑡 = 0, . . 𝑇 such that

T
R (Y )  sup  E  ct , vt
 t 0





vt  L p F , R k1 , vt  Vt , t  0,.., T 

t

E wt , , vt   Yt , t  1,.., T 

 0


• holds for every Y ∈ 𝛾. Here ⋅,⋅ denotes scalar
products on every ℝ𝑘𝑡 .
Conditions for Supremal
Values
1.
2.
•
𝑉𝑡 is a polyhedral cone for 𝑡 = 0, . . , 𝑇 and 𝑤𝑡,0 , 𝑉𝑡 = ℝ holds for every
𝑡 = 1, . . , 𝑇 (complete recourse).
There exists 𝑢 ∈ ℝ𝑇 such that 𝑐0 − 𝑇𝜏=1 𝑤𝜏,𝜏−𝑡 𝑢𝜏 ∈ 𝑉0∗ , 𝑐𝑡 −
𝑇
∗
∗
𝜏=𝑡 𝑤𝜏,𝜏−𝑡 𝑢𝜏 ∈ 𝑉𝑡 , 𝑡 = 1, . . , 𝑇, hold, where sets 𝑉𝑡 are the polar cones to 𝑉𝑡 .
(dual feasibility)
If 1. and 2. and the polyhedral function is defined by:

T
R(Y )  sup  E  ct , vt
 t 0

•




vt  L p F , R k1 , vt Vt , t  0,.., T 

t

E wt , , vt   Yt , t  1,.., T 

 0


R is finite, positively homogeneous, concave, and continuous on 𝛾
Note on Multi-Stage



k1

v

L
F
,
R
, vt  Vt , t  0,.., T 
t
p
T


R (Y )  sup  E  ct , vt  t

E wt , , vt   Yt , t  1,.., T 

 t 0
 0


• The dual solutions that correspond to the constraint
is the slope of the R.

E w  , v    Y


t
0
t,
t
t
• This problem is solved by means of cost-to-go
functions and bellman’s equation.
Note on Multi-Stage



k1

v

L
F
,
R
, vt  Vt , t  0,.., T 
t
p
T


R (Y )  sup  E  ct , vt  t

E wt , , vt   Yt , t  1,.., T 

 t 0
 0


• The dual solutions that correspond to the constraint
is the slope of the R.

E w  , v    Y


t
0
t,
t
t
• This problem is solved by means of cost-to-go
functions and bellman’s equation.
Vt (Y )  maxCt (Y , x)  E Vt 1 (Yt 1 (Yt , x)) | Y 
x
Note on V@R
• If we use V@R, many problems will cease to be
linear and convex. However, replacing V@R with
CV@R enables us to preserve the convexity of the
underlying problem because this measure is
polyhedral.
Applications in the
Energy Sector (CHP)
Liberalization/Deregulation
of the Energy Markets
• The deregulation of energy markets has lead to an
increased awareness of the need for profit
maximization with simultaneous consideration of
financial risk, adapted to individual risk aversion
policies of market participants.
• More requirements on Risk management.
Liberalization/Deregulation
of the Energy Markets
• Mathematical modeling of such optimization
problems with uncertain input data results in mixedinteger large-scale stochastic programming models
with a risk measure in the objective.
• Often Multi-Stage problems are solved in the
framework of either dynamic or stochastic
programming.
• Simultaneous optimization of profits and risks.
Applications of
polyhedral Risk Measures
The problem of finding a strategy that yields the
optimal (or near optimal) profit under taking into
account technical constraint and risks.
min →
1 − 𝛾 ∗ 𝑅𝑖𝑠𝑘 − 𝛾 ∗ 𝑃𝑟𝑜𝑓𝑖𝑡
Specification of the
Problem
• The multi-stage stochastic optimization models are
tailored to the requirements of a typical German
municipal power utility, which has to serve an
electricity demand and a heat demand of
customers in a city and its vicinity.
• The power utility owns a combined heat and power
(CHP) facility that can serve the heat demand
completely and the electricity demand partly.
Stochasticity of the Model
Sources:
1. Electricity spot prices
2. Electricity forward prices
3. Electricity demand (load)
4. Heat demand.
Stochasticity of the Model
Multiple layers of seasonality
1. Electricity spot prices (daily, weekly, monthly)
2. Electricity demand (daily, weekly, monthly)
3. Heat demand (daily, weekly, monthly)
The seasonality is captured by the deterministic part.
Interdependency between the
Data (prices&demands)
• Prices depend on demands and vice versa
• Tri-variate ARMA models (demand for heat&electricity
and spot prices).
• Spot prices AR-GARCH.
• The futures prices are calculated aposteriori from the
spot prices in the scenario tree. (month average)
Parameters
Decision Variables
Objective
Objective – Cash Values
• Cash values are what we earn from producing heat
and electricity. We of course take into account
technical constraints.
Objective
Simulation Results
• The best strategy is to not use any contracts.
• Minimizing without a risk measure causes high spread for
the distribution of the overall revenue.
• The incorporation of the (one-period) CV@R applied to
z(T) reduces this spread considerably for the price of high
spread and very low values for z(t) at time t<T.
Simulation Results
Simulation Results
Simulation Results
Simulation Results
Simulation Results
Simulation Results
Conclusion
• Polyhedral risk measures enable us to incorporate
more realistic features of the problem and to
preserve its convexity and linearity.
• Hence, they enable the tractability of many
problems.
• V@R is a less sophisticated risk measure, but many
problems cannot be solved by using V@R unlike
CV@R.
Bibliography
• A. Philpott, A. Dallagi, E. Gallet. On Cutting Plane Algorithms
and Dynamic Programming for Hydroelectricity Generation.
Handbook of Risk Management in Energy Production and
Trading International Series in Operations Research &
Management Science , Volume 199, 2013, pp 105-127.
• A. Shapiro, W. Tekaya, J.P. da Costa, and M.P. Soares. Risk
neutral and risk averse Stochastic Dual Dynamic Programming
method. 2013.
• G. C Pflug, W. Roemisch. Modeling, Measuring and Managing
Risk. 2010.
• A. Eichhorn, W. Römisch, Mean-risk optimization of electricity
portfolios using multiperiod polyhedral risk measures. 2005