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Sebastian Sager Optimization-based Analysis and Training of Human Decision Making Aussois, January 59, 2014 Michael Engelhart Interdisciplinary Center for Scientific Computing Ruprecht-Karls-Universitt Heidelberg 1 Sager |

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SLIDE 1

Sebastian Sager

Optimization-based Analysis and Training

  • f Human Decision Making

Aussois, January 5–9, 2014 Interdisciplinary Center for Scientific Computing Ruprecht-Karls-Universität Heidelberg

Michael Engelhart

1 Sager | Optimization-based Analysis and Training

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SLIDE 2

Theoretical advances New & better algorithms Open source software

Challenges for optimization Solution methodology Physician training Relevant scenarios Challenges for optimization Solution methodology

Decision support systems Simulators for diseases A L G O R I T H M S C L I N I C A L P R A C T I C E

MODEST: Mathematical Optimization for clinical DEcision Support and Training

Sebastian Sager, Magdeburg, Germany

Personalized medicine via optimization

Simulation: warnings and alerts what would happen if...? Optimization: fit models to patient data get patient-specific treatment get patient-specific diagnosis e.g., for cardiac arrhythmia Probability for Atrial Fibrillation: 85% Probability for Atrial Flutter: 93%

Clinical decision training

Simulation: Optimization: what would happen if... ? what would be best?

Mixed-integer nonlinear optimal control

Uncertainties, e.g., model-plant mismatch patient-specific parameters Integrality, e.g., which combination of drugs? Wenckebach or Mobitz block? Global optima needed MI(N)OCP MI(L)OCP OCP&MILP NLP&MILP MINLP MINLP OCP NLP

convexification discretization discretization relaxation discretization discretization relaxation initialization

OCP

Switching Time Optimization discretization

NLP

adaptive grid refinement

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SLIDE 3

Questions

Optimization in practice . . .

  • a key technology for 21st century, enabling progress&prosperity

3 Sager | Optimization-based Analysis and Training

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SLIDE 4

Questions

Optimization in practice . . .

  • a key technology for 21st century, enabling progress&prosperity
  • risks to increase the gap compared to human decision making

3 Sager | Optimization-based Analysis and Training

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SLIDE 5

Questions

Optimization in practice . . .

  • a key technology for 21st century, enabling progress&prosperity
  • risks to increase the gap compared to human decision making

This talk: can optimization also be used to train humans?

3 Sager | Optimization-based Analysis and Training

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SLIDE 6

Questions

Optimization in practice . . .

  • a key technology for 21st century, enabling progress&prosperity
  • risks to increase the gap compared to human decision making

This talk: can optimization also be used to train humans? Questions to you:

  • Who thinks to perform better (without algorithms) in finding a good

solution to a random combinatorial optimization problem compared to an average citizen?

3 Sager | Optimization-based Analysis and Training

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SLIDE 7

Questions

Optimization in practice . . .

  • a key technology for 21st century, enabling progress&prosperity
  • risks to increase the gap compared to human decision making

This talk: can optimization also be used to train humans? Questions to you:

  • Who thinks to perform better (without algorithms) in finding a good

solution to a random combinatorial optimization problem compared to an average citizen?

  • Who thinks this has to do with having seen optimal solutions and

sensitivities of similar optimization problems?

3 Sager | Optimization-based Analysis and Training

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Complex Problem Solving

  • Humans are asked to solve a given complex problem
  • Interest of psychologists: correlation to emotion regulation etc.
  • Gets more attention: included in future PISA evaluations

4 Sager | Optimization-based Analysis and Training

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Complex Problem Solving

  • Humans are asked to solve a given complex problem
  • Interest of psychologists: correlation to emotion regulation etc.
  • Gets more attention: included in future PISA evaluations
  • Most problems nowadays computer-based test-scenarios
  • Tailorshop: one of the most famous ones (fruitfly of CPS)
  • Developped in the 1980s (Dörner et al.)

4 Sager | Optimization-based Analysis and Training

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SLIDE 10

Complex Problem Solving

  • Humans are asked to solve a given complex problem
  • Interest of psychologists: correlation to emotion regulation etc.
  • Gets more attention: included in future PISA evaluations
  • Most problems nowadays computer-based test-scenarios
  • Tailorshop: one of the most famous ones (fruitfly of CPS)
  • Developped in the 1980s (Dörner et al.)
  • Participant has to run shirt company
  • Round-based scenario
  • Aim: maximize overall capital of company

4 Sager | Optimization-based Analysis and Training

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SLIDE 11

The IWR Tailorshop

[Engelhart, Funke, S., Journal of Computational Science, 2013]

d i s t r i b u t i

  • n

& m a r k e t i n g m a n u f a c t u r i n g h u m a n r e s

  • u

r c e s g

  • a

l s maximize

  • verall balance

employees production sites shirts in stock shirt quality advertising reputation price per shirt demand sales distribution sites machine quality resources quality motivation of employees wages success maintenance (shirts) production + + + + + + + + + + +

  • +

+ + + + + +

  • +

m a t e r i a l s m a n a g e m e n t + +

  • Diamonds indicate influence of participant’s decisions.

5 Sager | Optimization-based Analysis and Training

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IWR Tailorshop web interface

  • implementation with AJAX, PHP using a MySQL database
  • adaptive interface for mobile devices

6 Sager | Optimization-based Analysis and Training

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Complex Problems: Complexity

Production Distribution Human Resources Resources Marketing

7 Sager | Optimization-based Analysis and Training

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Complex Problems: Interdependence

Production Distribution Human Resources Resources Marketing

8 Sager | Optimization-based Analysis and Training

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SLIDE 15

Complex Problems: Intransparency

Production Distribution Human Resources Resources Marketing

?

? ?? ? ? ?

9 Sager | Optimization-based Analysis and Training

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SLIDE 16

Complex Problems: Dynamics

Wages k k+1 k+2

10 Sager | Optimization-based Analysis and Training

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SLIDE 17

Complex Problems: Dynamics

Wages k k+1 k+2 Motivation

  • f Employees

Product Quality

10 Sager | Optimization-based Analysis and Training

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Complex Problems: mixed-integer decisions

  • Continuous decisions, e.g., wages
  • Discrete decisions, e.g., open/close a distribution site

11 Sager | Optimization-based Analysis and Training

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Optimization and Complex Problem Solving

  • First: use optimization to define interesting microworld
  • Bounded solution, multiple local maxima, important / unimportant

decisions, . . .

12 Sager | Optimization-based Analysis and Training

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Optimization and Complex Problem Solving

  • First: use optimization to define interesting microworld
  • Bounded solution, multiple local maxima, important / unimportant

decisions, . . .

  • Second:
  • ptimal solution as performance indicator!

12 Sager | Optimization-based Analysis and Training

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SLIDE 21

Optimization and Complex Problem Solving

  • First: use optimization to define interesting microworld
  • Bounded solution, multiple local maxima, important / unimportant

decisions, . . .

  • Second:
  • ptimal solution as performance indicator!
  • Simple test-scenarios (e.g. Tower of Hanoi):
  • ptimal solution known
  • Complex test-scenarios:
  • ptimal solution unknown
  • Third: can optimal solutions be used for training?

12 Sager | Optimization-based Analysis and Training

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Formulate abstract optimization problem

  • Same mathematical model (equations) for all tasks
  • Dynamic model with discrete time k = 0 . . . N
  • Decisions uk = u(k) and states xk = x(k)
  • Scenario specified by initial values x0 and parameters p

13 Sager | Optimization-based Analysis and Training

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Formulate abstract optimization problem

  • Same mathematical model (equations) for all tasks
  • Dynamic model with discrete time k = 0 . . . N
  • Decisions uk = u(k) and states xk = x(k)
  • Scenario specified by initial values x0 and parameters p
  • First: use optimization to define interesting microworld:

→ determine initial values x0 and parameters p

13 Sager | Optimization-based Analysis and Training

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SLIDE 24

Formulate abstract optimization problem

  • Same mathematical model (equations) for all tasks
  • Dynamic model with discrete time k = 0 . . . N
  • Decisions uk = u(k) and states xk = x(k)
  • Scenario specified by initial values x0 and parameters p
  • First: use optimization to define interesting microworld:

→ determine initial values x0 and parameters p

  • Second and Third: analysis and training

→ find decisions uk to maximize objective function → Compare participant’s performance to optimal solution → Provide feedback on better choice for learning

13 Sager | Optimization-based Analysis and Training

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IWR Tailorshop states

States Variable Unit employees xEM person(s) production sites xPS site(s) distribution sites xDS site(s) shirts in stock xSH shirt(s) production xPR shirt(s) sales xSA shirt(s) demand xDE shirt(s) reputation xRE — shirts quality xSQ — machine quality xMQ — motivation of employees xMO — capital xCA M.U.

M.U. means monetary units. 14 Sager | Optimization-based Analysis and Training

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IWR Tailorshop controls

Controls Variable Unit shirt price uSP

M.U./shirt

advertising uAD M.U. wages uWA

M.U./person

maintenance uMA M.U. resources quality uRQ — recruit/dismiss employees udEM/uDEM person(s) create/close production site udPS/uDPS site(s) create/close distribution site udDS/uDDS site(s)

M.U. means monetary units. 15 Sager | Optimization-based Analysis and Training

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SLIDE 27

IWR Tailorshop example model equations

State equations: xk+1 = G(xk, xk+1, uk, p)

xEM

k+1 = xEM k

− udEM

k

+ uDEM

k

xDE

k+1 = pDE,0 · exp

  • −pDE,1 · uSP

k

  • · log
  • pDE,2 · uAD

k

+ 1

  • ·
  • xRE

k

+ pDE,3 xSA

k+1 = min

  • pSA,0 · xDS

k+1 · log

  • pSA,1 · xEM

k+1

xPS

k+1 + xDS k+1 + pSA,2 + 1

  • ; xSH

k

+ xPR

k+1; pSA,3 · xDE k+1

  • . . .

16 Sager | Optimization-based Analysis and Training

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Second and Third: Optimization problem

max

x,u

F(xN) s.t. xk+1 = G(xk, uk, p), k = ns . . . N − 1, ≤ H(xk, uk, p), k = ns . . . N − 1, uk ∈ Ω, k = ns . . . N − 1, xns = xp

ns.

  • Dynamic model with discrete time k = 0 . . . N
  • Nonconvex mixed-integer nonlinear program

17 Sager | Optimization-based Analysis and Training

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Second and Third: Optimization problem

max

x,u

F(xN) s.t. xk+1 = G(xk, uk, p), k = ns . . . N − 1, ≤ H(xk, uk, p), k = ns . . . N − 1, uk ∈ Ω, k = ns . . . N − 1, xns = xp

ns.

  • Dynamic model with discrete time k = 0 . . . N
  • Nonconvex mixed-integer nonlinear program
  • Starting at month ns with same data xp

ns as participant

  • Need to solve N − 1 optimization problems per participant

17 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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SLIDE 37

Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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SLIDE 38

Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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SLIDE 39

Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solution 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant 18 Sager | Optimization-based Analysis and Training

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Optimal Solutions

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Objective function Month Capital □ Participant Optimal solutions 18 Sager | Optimization-based Analysis and Training

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SLIDE 42

How Much Is Still Possible

2 4 6 8 10 0,0 0,2 0,4 0,6 0,8 1,0 1,2 (x10

5)

Analysis function 2 Month Objective vs. how much is still possible □ Objective (participant) How much is still possible 19 Sager | Optimization-based Analysis and Training

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Second use of optimization: analysis

Have to work a little (N optimization problems) to get it. But:

  • provides objective performance measure
  • allows time– and decision–specific analysis of what went wrong
  • Details in [S., Barth, Diedam, Engelhart, Funke, Optimization as an analysis tool for human complex problem

solving, SIAM Journal of Optimization, 2011] 20 Sager | Optimization-based Analysis and Training

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SLIDE 44

Third use: Optimization-based Feedback

F E E D B A C K O P T I M I Z A T I O N

4 Bar chart 1 Highlight variables

55

2 Show arrows

55 55

3 Toggle values

38 55

B Start optimization in xk, fix decisions uk with constraints

Artificial constraints for uk yield sensitivities

A Start optimization in xk+1

Identical to the start values, the participant will have for next decisions uk+1

xk+1 xk uk uk+1 uk-1

21 Sager | Optimization-based Analysis and Training

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SLIDE 45

Web-based feedback study

  • study conducted Nov/Dec 2013
  • IWR Tailorshop web interface
  • participants recruited in lectures and social networks
  • 100 complete datasets

22 Sager | Optimization-based Analysis and Training

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SLIDE 46

Web-based feedback study

  • study conducted Nov/Dec 2013
  • IWR Tailorshop web interface
  • participants recruited in lectures and social networks
  • 100 complete datasets
  • 4 rounds of 10 “months” each, different initial values
  • 2 rounds (= 20 months) with feedback (goal: learning)
  • 2 rounds (= 20 months) without (goal: performance)

22 Sager | Optimization-based Analysis and Training

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SLIDE 47

Web-based feedback study

  • study conducted Nov/Dec 2013
  • IWR Tailorshop web interface
  • participants recruited in lectures and social networks
  • 100 complete datasets
  • 4 rounds of 10 “months” each, different initial values
  • 2 rounds (= 20 months) with feedback (goal: learning)
  • 2 rounds (= 20 months) without (goal: performance)
  • Feedback in 6 randomized groups:

control, highscore, highlight, arrows, value, chart

F E E D B A C K O P T I M I Z A T I O N 4 Bar chart 1 Highlight variables

55

2 Show arrows 55 55 3 Toggle values

38 55

B Start optimization in xk, fix decisions uk with constraints Artificial constraints for uk yield sensitivities A Start optimization in xk+1 Identical to the start values, the participant will have for next decisions uk+1 xk+1 xk uk uk+1 uk-1

22 Sager | Optimization-based Analysis and Training

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SLIDE 48

Study results: feedback groups

  • −5.0 × 10+5

−2.5 × 10+5 0.0 × 10+0 2.5 × 10+5 all co hs in tr va ch

Score

Round 1

  • −2 × 10+5

−1 × 10+5 0 × 10+0 1 × 10+5 all co hs in tr va ch

Round 2

  • −3 × 10+5

−2 × 10+5 −1 × 10+5 0 × 10+0 1 × 10+5 2 × 10+5 3 × 10+5 all co hs in tr va ch

Group Score

Round 3

  • −2 × 10+5

0 × 10+0 2 × 10+5 4 × 10+5 all co hs in tr va ch

Group

Round 4

(co=control, hi=highscore, in=highlight variable, tr=show arrows, va=show values, ch=sensitivity chart) 23 Sager | Optimization-based Analysis and Training

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SLIDE 49

Study results: use of potential

10 20 30

  • 6
  • 4
  • 2

(x10

4)

Analysis function 7 Month Potential for all rounds Control Highscore Indicate Trend Value Chart 24 Sager | Optimization-based Analysis and Training

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SLIDE 50

Hypothesis Proved (A) participants with opt.-based feedback perform better

  • verall
  • (B)

participants with opt.-based feedback perform better in feedback rounds

  • (C)

participants with opt.-based feedback perform better in performance rounds

  • (D)

control group performs worst — (E) control group performs worse than opt.-based groups in performance rounds

  • (F)

trend group performs best overall — (G) trend group performs best in performance rounds — (H) value group performs best in feedback rounds

  • 25

Sager | Optimization-based Analysis and Training

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SLIDE 51

Hypothesis Proved (I) value group performs better in feedback rounds, worse in performance rounds () (J) participants with high BFI-10 values perform worse/better — (K) participants who play computer games regularly perform better

  • (L)

participants interested in economics perform better

  • (M)

participants who solve problems systematically perform better

  • (N)

control group needs more time than opt.-based feedback groups — (O) well-performers know more about the model

  • 26

Sager | Optimization-based Analysis and Training

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SLIDE 52

Hypothesis Proved (P) participants who know much about the model, perform well

  • (Q)

value group knows less, trend group knows most about the model —/ (1) participants learn to control the model

  • (2)

learning function is approximately logarithmic — (3)

  • ptimization-based feedback groups learn faster

() (4) value group does almost not learn in feedback rounds

  • (5)

trend group learns fastest

  • 27

Sager | Optimization-based Analysis and Training

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SLIDE 53

Hypothesis Proved (6) participants who learn much, perform well ? (7) participants who perform well, learned much ? (8) participants with high model knowledge learned more

  • (9)

initial performance is not important for final performance

  • (10)

chart group suffered from feedback ()

28 Sager | Optimization-based Analysis and Training

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SLIDE 54

IWR Tailorshop: global solutions?

  • Nonconvex mixed-integer nonlinear program
  • Need global solutions! Can we use Couenne or Baron?

29 Sager | Optimization-based Analysis and Training

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SLIDE 55

IWR Tailorshop: global solutions?

  • Nonconvex mixed-integer nonlinear program
  • Need global solutions! Can we use Couenne or Baron?
  • But: for N = 1: 0.9 sec,

for N = 2: 12 sec, for N = 3: ≫ 10 min . . .

  • Interesting effects (investment paying off) for N ≥ 5.

29 Sager | Optimization-based Analysis and Training

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SLIDE 56

IWR Tailorshop: global solutions?

  • Nonconvex mixed-integer nonlinear program
  • Need global solutions! Can we use Couenne or Baron?
  • But: for N = 1: 0.9 sec,

for N = 2: 12 sec, for N = 3: ≫ 10 min . . .

  • Interesting effects (investment paying off) for N ≥ 5.
  • Developed tailored decomposition approach for tight bounds (fast)
  • [Engelhart, Funke, S., A Decomposition Approach for a New Test-Scenario in Complex Problem Solving, Journal of

Computational Science, 2013] 29 Sager | Optimization-based Analysis and Training

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SLIDE 57

Discussion

Optimization and Human Decision Making: Related work

  • Armin Fügenschuh (“System Dynamics”)
  • Buchheim, Engell, Michaels, . . .

in Dortmund Research Training Group “Discrete optimization of technical systems under uncertainty”

  • Christian Kirches
  • . . .
  • Who else?

30 Sager | Optimization-based Analysis and Training

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SLIDE 58

Summary

  • Systematic & synergetic modeling and optimization approach
  • Three uses: get good microworld, analysis, and training

31 Sager | Optimization-based Analysis and Training

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SLIDE 59

Summary

  • Systematic & synergetic modeling and optimization approach
  • Three uses: get good microworld, analysis, and training
  • Challenging MINLPs solved by tailored decomposition
  • Web-based study with 100 complete datasets:
  • ptimization-based feedback can make a significant difference
  • Details can be found in:
  • [Engelhart, Funke, S., A Decomposition Approach for a New Test-Scenario in Complex Problem Solving, Journal of

Computational Science, 2013]

  • [S., Barth, Diedam, Engelhart, Funke, Optimization as an analysis tool for human complex problem solving, SIAM

Journal of Optimization, 2011]

  • [Engelhart, PhD thesis, to be submitted soon]

31 Sager | Optimization-based Analysis and Training

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SLIDE 60

IWR Tailorshop objective function

xCA

k+1 = pCA,0 ·

  • xCA

k

+

  • xSA

k+1 · uSP k

  • −
  • xEM

k+1 · uWA k

  • − uAD

k

−

  • xSH

k+1 · pCA,6

−

  • xPR

k+1 · uRQ k

· pCA,3 − uMA

k

−

  • xPS

k

· pCA,4 −

  • xDS

k

· pCA,5 +

  • udPS

k

· pCA,1 +

  • udDS

k

· pCA,2 −

  • uDPS · pCA,7

−

  • uDDS · pCA,8

Objective function: F(xN) = xCA

N

32 Sager | Optimization-based Analysis and Training

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SLIDE 61

IWR Tailorshop objective function

˜ xCA

k+1 = pCA,0 ·

  • xCA

k

+

  • xSA

k+1 · uSP k

  • −
  • xEM

k+1 · uWA k

  • − uAD

k

−

  • xSH

k+1 · pCA,6

+ f1

  • xPR

k+1, uSQ k

  • + f2
  • xPS

k , xDS k , xPR k+1, xEM k+1

  • +
  • udPS

k

· pCA,1 +

  • udDS

k

· pCA,2 −

  • uDPS · pCA,7

−

  • uDDS · pCA,8

Objective function: ˜ F(xN) = ˜ xCA

N

33 Sager | Optimization-based Analysis and Training

slide-62
SLIDE 62

The IWR Tailorshop: reducing the model

d i s t r i b u t i

  • n

& m a r k e t i n g m a n u f a c t u r i n g h u m a n r e s

  • u

r c e s g

  • a

l s maximize

  • verall balance

employees production sites shirts in stock shirt quality advertising reputation price per shirt demand sales distribution sites machine quality resources quality motivation of employees wages success maintenance (shirts) production + + + + + + + + + + +

  • +

+ + + + + +

  • +

m a t e r i a l s m a n a g e m e n t + +

  • Diamonds indicate (influence of) free variables.

34 Sager | Optimization-based Analysis and Training

slide-63
SLIDE 63

The IWR Tailorshop: reducing the model

d i s t r i b u t i

  • n

& m a r k e t i n g m a n u f a c t u r i n g h u m a n r e s

  • u

r c e s g

  • a

l s maximize

  • verall balance

employees shirts in stock shirt quality advertising reputation price per shirt demand sales sites machine quality resources quality wages maintenance (shirts) production + + + + + + + + +

  • +

+ + + + + m a t e r i a l s m a n a g e m e n t

  • Diamonds indicate (influence of) free variables.

35 Sager | Optimization-based Analysis and Training

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SLIDE 64

Decomposition approach

  • Idea: split problem up to get (at least) good upper bound
  • Comparable to Lagrangian Relaxation approaches
  • One master problem, several decoupled problems
  • Coupled via the newly introduced cost functions f1 and f2 for the

decoupled problems

36 Sager | Optimization-based Analysis and Training

slide-65
SLIDE 65

Decomposition approach

  • riginal problem

max F(xN) s.t. xk+1 = G(xk,xk+1,uk,p) 0 ≤ H(xk,xk+1,uk,p)

37 Sager | Optimization-based Analysis and Training

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SLIDE 66

Decomposition approach

master problem

max F(xN) s.t. xk+1 = G(xk,xk+1,uk,p) 0 ≤ H(xk,xk+1,uk,p) ~ ~ ~ min c1(·) s.t. ...

decoupled problems

min c2(·) s.t. ...

cost function f1(·) cost function f2(·) input variables input variables

38 Sager | Optimization-based Analysis and Training

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SLIDE 67

Decomposition approach

max F(xN) s.t. xk+1 = G(xk,xk+1,uk,p) 0 ≤ H(xk,xk+1,uk,p) ~ ~ ~ min c1(·) s.t. ... min c2(·) s.t. ...

  • riginal problem

max F(xN) s.t. xk+1 = G(xk,xk+1,uk,p) 0 ≤ H(xk,xk+1,uk,p)

≤

decomposition

39 Sager | Optimization-based Analysis and Training