Forward and Backward Deviation Measures and Robust Optimization

Peng Sun (Duke)

with Xin Chen (UIUC) and Melvyn Sim (NUS)

Agenda

Overview of Robust Optimization Forward and Backward Deviation Measures Uncertainty Sets and Probability Bounds Conclusions

Uncertain linear constraints Uncertain linear constraint

Chance constraint 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

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

Subset of worst case set, W : Budget of uncertainty

Control radius of uncertainty set, S

max: Worse case budget For reasonable probability bound, << max

Overview of Robust Optimization Robust Counterpart

Semi-infinite constraint Tractability Polyhedral Uncertainty Set

Linear Programming (LP) Conic Quadratic Uncertainty Set

Second Order Cone Programming (SOCP)

Overview of Robust Optimization

Ellipsoidal Uncertainty Set 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

Norm Uncertainty Set Robust Counterpart depends on the choice of norm Probability bound for symmetrically bounded

uncertainties:

Overview of Robust Optimization

Modeling limitations of Classical Uncertainty Sets 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 Set of forward deviation measures

Set of backward deviation measures

Alternatively,

Forward deviation

Backward deviation

Forward and Backward Deviation Measures Key idea

Important property

Subadditivity

Least conservative forward and backward deviation measures Suppose distribution is known

p* = q* if distribution is symmetrical p*, q* ¸ (standard deviation) p* = q* = if distribution is Normal

Least conservative forward and backward deviation measures p* , q* can be obtained numerically

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 Recall: Norm Uncertainty Set (Symmetrical)

Uncertainty Sets and Probability Bounds Generalizes Symmetrical Uncertainty Set

Uncertainty Sets and Probability Bounds

Uncertainty Sets and Probability Bounds

Uncertainty Sets and Probability Bounds Robust Counterpart

Uncertainty Sets and Probability Bounds Probability Bound

More information achieve less conservative solution while preserving the bound. E.g pj

= qj = 1 for symmetric distribution in [-1,1]

pj = qj = 0.58 for uniform distribution in [-1,1]

Conclusions RO framework

Affine uncertainty constraints Independent r.v.’s with asymmetric distributions

Forward and backward Deviation measures 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 Stochastic programming with chance constraints