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

Robust Counterpart

Uncertainty Sets and Probability Bounds Probability Bound More information achieve less conservative solution while preserving the bound. E.g  p j = q j = 1 for symmetric distribution in [-1,1]  p j = q j = 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