Forward and Backward
Deviation Measures and Robust
Optimization
Peng Sun (Duke)
with Xin Chen (UIUC) and Melvyn Sim (NUS)
Agenda
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Overview of Robust Optimization
Forward and Backward Deviation Measures
Uncertainty Sets and Probability Bounds
Conclusions
Uncertain linear constraints
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Uncertain linear constraint
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Chance constraint
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Hard to solve when  is small
Linear constraints with uncertainty

Affine Uncertainty
: zero mean, independent but not necessarily
identically distributed
Overview of Robust Optimization

Worst case
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Relies only on the distribution support
Easy to solve (Soyster 1973)
Extremely conservative
Overview of Robust Optimization

Goal of Robust Optimization:
 Easy to obtain feasible solutions that satisfy
chance constraint
 Not as conservative as worst case
Overview of Robust Optimization

Design of Uncertainty Set
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Subset of worst case set, W
: Budget of uncertainty
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Control radius of uncertainty set, S
max: Worse case budget
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For reasonable probability bound,  << max
Overview of Robust Optimization
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Robust Counterpart
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Semi-infinite constraint
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Tractability
Polyhedral Uncertainty Set
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Linear Programming (LP)
Conic Quadratic Uncertainty Set
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Second Order Cone Programming (SOCP)
Overview of Robust Optimization
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Ellipsoidal Uncertainty Set
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Ben-Tal and Nemirovski (1997), El-Ghaoui et al.
(1996)
Robust Counterpart is a SOCP
Probability bound for symmetrically bounded
uncertainties:
Overview of Robust Optimization

Polyhedral Uncertainty Set
 Bertsimas and Sim (2000)
 Robust Counterpart is a LP
 Probability bound for symmetrically bounded uncertainties:

More conservative compared to Ellipsoidal Uncertainty Set
Overview of Robust Optimization
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Norm Uncertainty Set
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Robust Counterpart depends on the choice of norm
Probability bound for symmetrically bounded
uncertainties:
Overview of Robust Optimization
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Modeling limitations of Classical
Uncertainty Sets
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Uncertainty set is symmetrical, but
distributions are generally asymmetrical
Can we use more information on
distributions to obtain less conservative
solution in achieveing the same probability
bound?
--- Forward and Backward Deviation Measures
Forward and Backward Deviation
Measures
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Set of forward deviation measures

Set of backward deviation measures
Alternatively,

Forward deviation

Backward deviation
Forward and Backward Deviation
Measures
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Key idea
Important property

Subadditivity
Least conservative forward and backward
deviation measures
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Suppose distribution is known

p* = q* if distribution is symmetrical
p*, q* ¸  (standard deviation)
p* = q* =  if distribution is Normal
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Least conservative forward and backward
deviation measures
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p* , q* can be obtained numerically
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E.g: p* = q* = 0.58 for uniform distribution in [-1,1]
E.g:
Forward and backward deviation measures
– two point distribution example
Distribution not known

Suppose distribution is bounded in [-1,1]
(but not necessarily symmetrical)
Only know mean and support
Forward and Backward Deviation
Measures: Function g()
General Deviation Measures
General Deviation Measure
Agenda




Overview of Robust Optimization
Forward and Backward Deviation Measures
Uncertainty Sets and Probability Bounds
Conclusions
Uncertainty Sets and Probability Bounds

Model of Data Uncertainty

Recall: Affine Uncertainty
: zero mean, independent but not necessarily
identically distributed
Uncertainty Sets and Probability Bounds

Model of Data Uncertainty
Uncertainty Sets and Probability Bounds

New Uncertainty Set (Asymmetrical)
Uncertainty Sets and Probability Bounds
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Recall: Norm Uncertainty Set (Symmetrical)
Uncertainty Sets and Probability Bounds
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Generalizes Symmetrical Uncertainty Set
Uncertainty Sets and Probability Bounds
Uncertainty Sets and Probability Bounds
Uncertainty Sets and Probability Bounds
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Robust Counterpart
Uncertainty Sets and Probability Bounds
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
Probability Bound
More information achieve less conservative
solution while preserving the bound. E.g
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pj = qj = 1 for symmetric distribution in [-1,1]
pj = qj = 0.58 for uniform distribution in [-1,1]
Conclusions
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RO framework
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Forward and backward Deviation measures
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Affine uncertainty constraints
Independent r.v.’s with asymmetric distributions
Defined from moment generating functions – implying
probability bound
Sub-additivity – linear combinations
Easy to calculate and approximate from support
information
Advantage -- less conservative solution
Next
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Stochastic programming with chance constraints