Basic Properties and Theory

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Basic Properties and Theory

• Probabilistic or Chance Constraints

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Percentile v.s. Probabilistic constraints

• Proposition 1.
• Let f (x,y) be a loss functions and ?a(x) be
  a-percentile (a-VaR) then
• ?a(x) ? ? ? Pr f
  (x,y) ? ? ? a
• Proof follows from the definition of
  a-percentile ?a(x)
• ?a(x) min ? Pr f (x,y) ? ? ?
  a
• Generally, ?a(x) is nonconvex (e.g., discrete
  distributions), therefore ?a(x) ? ? as well as
  Pr f (x,y) ? ? ? a may be nonconvex
  constraints
• Probabilistic constraints were considered by
  Prekopa, Raik, Szantai, Kibzun, Uryasev, Lepp,
  Mayer, Ermoliev, Kall, Pflug, Gaivoronski,

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Non-Percentile Risk Measures

• Low partial moment constraint (considered in
  finance literature from 70-th)
• E((f (x,y) - ? ))a ? b , a ?
  0, g max 0, g
• special cases
• a 0 gt Pr f (x,y) - ? ? 0
  
• a 1 gt E (f (x,y) - ?)
• a 2 , ? E f (x,y) gt
  semi-variance E((f (x,y) - ?))2
• Regret (King, Dembo) is a variant of low partial
  moment with
• ? 0 and f (x,y)
  performance - benchmark
• Various variants of low partial moment were
  successfully applied in stochastic optimization
  by Ziemba, Mulvey, Zenios, Konno, King, Dembo,
  Mausser, Rosen,
• Haneveld and Prekopa considered a special case of
  low partial moment with a 1, ? 0 integrated
  chance constraints

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Percentile v.s. Low Partial Moment

• Low partial moment with a gt 0 does not control
  percentiles. It is applied when loss can be
  hedged at additional cost
• total expected value expected cost
  without high losses
• 
  expected cost of high losses
• expected cost of high losses p
  E(f (x,y) - ?)
• Percentiles constraints control risks explicitly
  in percentile terms.
• Testury and Uryasev1 established equivalence
  between CVaR approach (percentile measure) and
  low partial moment, a 1 (non-percentile
  measure) in the following sense
• a) Suppose that a decision is optimal in an
  optimization problem with a CVaR constraint, then
  the same decision is optimal with a low partial
  moment constraint with some ? gt 0
• b) Suppose that a decision is optimal in an
  optimization problem with a low partial moment
  constraint, then the same decision is optimal
  with a CVaR constraint at some confidence level
  a.
• 1Testuri, C.E. and S. Uryasev. On Relation
  between Expected Regret and Conditional
  Value-At-Risk. Research Report 2000-9. ISE Dept.,
  University of Florida, August 2000. Submitted to
  Decisions in Economics and Finance journal.
  (www.ise.ufl.edu/uryasev/Testuri.pdf)

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Probabilistic or Chance Constraints
Probability or chance constraints with linear
functions
Definition
Deterministic equivalent program (DEP) if each
i corresponds to a distinct linear constraint,
Ai is a fixed row vector, and Fi is
distribution of hi, then
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An Example with Disconnected Sets
Suppose
x ? ? ,
Indeed, for the first atom we have
Similar, we can obtain inequalities for the
second atom.
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Quasi and Logarithmically Concave Measures
A probability measure is quasi-concave if for
any convex measurable sets U and V and any 0 ?
? ? 1
Definition
A probability measure is Log-concave if for any
convex measurable sets U and V and any 0 ? ? ?
1
Definition
Statement Log-concave ? quasi-concave
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Convexity And Closeness of the Set
Suppose A is fixed and h has an associated
quasi-concave probability measure P. Then K1
(?) is a closed convex set for 0 ? ? ? 1.
Theorem 14
Proof
Convexity let
, suppose
Closure suppose
. If for some
subsequence of ,
then , and, consequently,
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Quasi-Concave Measures
If f is the density of a continuous probability
distribution in ?m and is convex
on ?m, then the probability measure defined
for all Borel sets B in ?m is quasi-concave.
Theorem 15
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Convexity of Feasibility Set
Suppose A is fixed and the components hi , i
1, , m1, of h are stochastically independent
random variables with logarithmically concave
probability measures, Pi , and distribution
functions, Fi , then K1 (?) is convex, and
Theorem 16
Proof from independence
Convexity
the logarithm of Fi (Ai? x) is a concave
function, and, hence, K1 (?) is convex.
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Convexity of Feasibility Set (cont'd)
If A1? , , An1? , h have a joint normal
distribution with a common covariance structure,
a matrix C, such that where rij, si are
constants for all i and j, then K1 (?) is
convex for ? ? .
Theorem 17
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Deterministic Equivalents
Suppose that m1 1, h1 0, and A1? has
mean and covariance matrix C1 , then
where ? is the standard normal distribution
function.
Theorem 18
Proof
Substitution in the definition of K1 (?) yields
the result.
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Deterministic Equivalents (cont'd)
If Pi is the probability measure of hi and
Fi is the distribution function for hi then
using substitution , the
problem with probabilistic constraints
(1)
is reduced to the deterministic equivalent
problem
(2)
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Deterministic Equivalents (cont'd)
Suppose x and ?, ? are optimal and
optimal dual solutions of (2), respectively,
where cTx bT ? h T?. Define
Equivalent stochastic program with simple
recourse to (1) is
(3)
The following theorem gives conditions for
one-side equivalence of (1) and (3)
For the defined as a function of some
optimal ? for the dual to (1), if x is
optimal in (1), there exists y ? 0 almost
surely such that ( x, y) is optimal in (3).
Theorem 19