Optimization under Uncertainty

Stochastic optmization 

with applications to network 

planning under uncertainty 

Professor Alexei A. Gaivoronski 

Department of Industrial Economics and Technology Management 

Norwegian University of Science and Technology 

Alexei.Gaivoronski@iot.ntnu.no 

Alexei A.Gaivoronski 

July 2012, University of Cagliari 

Optimization under Uncertainty 

1

Topics of the course 

• Theory of stochastic programming 

– Decision models 

– Recourse problems 

– Approximations 

– Simulation and optimization 

• Algorithms 

– Benders decomposition 

– Stochastic dynamic programming 

– Stochastic dual dynamic programming 

– Stochastic gradients 




2

Topics of the course 

• Applied Decision Models with emphasis on 

networks 

– Electricity 

– Finance 

– Telecommunications 

– Production planning 

– Supply chain management 

• Software 

– Building blocks for implementation of stochastic 

programming models 

– Matlab, GAMS, MPL, CPLEX 




3

Optimal decisions under uncertainty 

• Worst case analysis 

– Too conservative 

• Scenario analysis 

– Solve deterministic problems for representative scenarios of 

uncertainty 

– Try to extract the common features from their solutions 

• Multiobjective optimization 

– Pareto - optimality 

• Stochastic optimization 

– Provides balanced solution that is not optimal for a given scenario, but 

optimal in average 




4

Uncertainty 

• Demand 

• Prices 

• Weather 

• Uncertainty on financial markets 

• Actions of competition 

Uncertainty is described by possible scenarios 

with given probabilities (probability 

distributions) 




5

Scenario analysis versus stochastic 

optimization 

• Example 1: two investment options in energy 

sector (per 1000 MW) 

• Uncertainty: two demand scenarios – 1000 and 

2000 MW with 0.5 probability each 




6

Comparison, continued 

• Scenario analysis: find optimal policy for each 

scenario and take robust elements which are 

good for all scenarios 

– hydro is better in both cases. 

• Stochastic optimization: minimize average costs 

and find flexibilities 

– build hydro for first 1000 MW and thermal for 

remaining 1000 MW. Expected cost of operation is 

smaller because there is no operating costs in the 

first scenario 




7

Example 2: Development of 

Telecommunication network 

• Three regions example 




8

Stochastic programming 

methodology for decision support under uncertainty 

expectation model 

x - decision variables 

- uncertain parameters 

maximize average profit 

- objective function 

- constraints 

X - admissible set of simple structure 




9

Example 3: constant rebalanced 

portfolios 




10

Example 3: constant rebalanced portfolios 

Blue lost 25%, black got 120% increase in value 

How we should divide our investment between these stocks? 




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Questions: 

• How should we invest? In blue? black? both? 

• What if we keep, say, 30% of our wealth during the 

whole period in blue stock? How this compares with 

portfolio only from black stock? Shall we obtain better 

results? Worse? The same? 

• What is the best division of our wealth between two 

stocks if our aim is to obtain as much wealth as 

possible at the end of the period? 




12

Change of portfolio value 

Black -ABF, blue - AEM, red - 30% AEM and 70% ABF 




13

Answers: 

• We should invest in both stocks 

• The final wealth increases by 27% compared to 

portfolio which contains only black stock 

• Solve fix-mix optimization problem 

- portfolio composition 

- relative change in asset prices between periods 




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Example 4: Markowitz portfolio 

Minimize variance of return subject to constraint 

on expected return 

- fraction of portfolio value invested in asset i 

- return of asset i 




15

performance 

Risk-return tradeoff 

risk 

- vector of average returns 

- covariance matrix 

Estimates of A and λ 

are needed 

mean/variance efficient portfolio, Markowitz (1952) 




16

Risk measures 

• Relative regret 

• Value at Risk 

• Conditional VaR 

Uryasev & Rockafellar (1999) 




17

Stochastic programming 

Probabilistic constraints, reliability constraints 

x - decision variables 

- uncertain parameters 

- objective function 

- constraints 

X - admissible set of simple structure 

maximize probability that 

profit exceeds some given 

level v 




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Example: Value at Risk 

- profit 

- Value at Risk 




19

Value at Risk (VaR) 

distribution of return (value) 

VaR 

the maximal value which can be lost in 

“almost” worst case 

• intuitive measure 

• industrial standard for risk management 




20

VaR, % 

Value at Risk: examples 

Sample VaR of Ford/IBM portfolio 

Gaivoronski & Pflug (1999) 

blue - 500 trading days, red - 2000 trading days 

Fraction of IBM stock 

portfolio 1: (0.51283, 0, 0.48717), portfolio 2: (0, 0.67798, 0.32202) 

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July 2012, University of Cagliari

Mean-Variance/VaR/CVaR efficient frontiers 




22

Mean-Variance/VaR/CVaR efficient frontiers 




23

Stochastic Dynamic Programming (SDP) 

v t – the state of the system 

u t – decision variables 

w t – uncertain (random) variables 

Goal: identify feasible decisions u that achieve the best 

system 

performance for a given state v and averaged over a 

range of 

uncertain inputs w. 

v t+1 =T t (v t ,u t ,w t ) 

C t (v t ,u t ,w t ) 

state equation 

costs at time t 

total costs 




24

Stochastic Dynamic Programming (SDP) 

This is a very difficult problem due to large number of 

variables and scenarios. It is substituted by a sequence 

of smaller problems 




25

System of connected reservours 




26

The hydrotermal dispatch problem 

(Pereira, 2000) 




27

Production cost 




28

Basic SDP step 




29

Discretizing states 




30

Last stage calculation 




31

Future cost function for stage T-1 




32

Cost calculation for stage T-1 




33

Implementation of SDP 




34

Implementation of SDP, continued 




35

System architecture 

MATLAB 

Data 

Spreadsheet 

model 

performance 

scenarios 

Modeler 

Top level algorithms 

Scenario generation 

Postprocessing 

Results 

User intervention 

Excel 

LP-solvers 

NLP solvers 

C,C++,Fortran 

NAG 

CPLEX 

XPRESS(XBSL) 

SQG 




36

LP of the last period T 


demand constraint 

bounds 




37

All other periods T-1,…,1 

Objective function 




38

LP of the all other periods T-1,…,1 


FCF approximation 

demand constraint 

bounds 




39

Implementation approach 

• Quick prototyping in Matlab possibly with 

additional toolboxes 

• Compile critical parts of code 

• Substitute Matlab solvers by faster compiled 

off-shelf solvers (may accelerate execution 

20-50 times) 

• Use EXCEL link for data input/visualization 




40

Alternative implementations: 

• Direct coding in Fortran/C/C++ 

– will be somewhat faster, but… for really fast implementation 

commercial solvers are needed anyway, so the speed gain 

will not be high 

• EXCEL and Visual Basic 

– possible approach but Matlab architecture is likely to be 

faster 

• AMPL 

– another interpretative language for optimization 

• Modeling libraries in MPL, CPLEX, XPRESS 

– good for heavier subproblems 




41

Representation of dependent 

inflows 

Markov chain approach 




42

SDP implementation with dependent 

inflows: additional state variable 




43

Continued 




44

Alternative process 




45

Alternative process, continued 




46

Limitations of Stochastic Dynamic 

Programming 

• Requires solution of optimization problem for 

all combination of states of the system 

Example: N reservours, M inflow states, K demand 

states, P price states, T periods 

number of problems T(MKP) N 

2 reservour, 3 periods, 10 scenarios for each random 

variable: 3000000 

- Aggregation and approximation 

- Better interpolation schemes 




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Optimization under Uncertainty

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