THE SCENARIO APPROACH to STOCHASTIC OPTIMIZATION

Transcription

1 THE SCENARIO APPROACH to STOCHASTIC OPTIMIZATION Marco C. Campi University of Brescia Italy

2 What I like about experience is that it is such an honest thing. You may have deceived yourself, but experience is not trying to deceive you. C.S. Lewis

3 thanks to: Algo Care Simone Garatti Maria Prandini Giuseppe Calafiore Federico Ramponi Bernardo Pagnoncelli

4 optimization controller synthesis classification portfolio selection optimization program

5 optimization controller synthesis classification portfolio selection optimization program uncertain environment exercise caution

6 uncertain optimization:

7 uncertain optimization: not a valid mathematical formulation

8 uncertain optimization: not a valid mathematical formulation often, a description of uncertainty is not available, or it is only partially available

9 scenario-based knowledge:

10 scenario-based knowledge: knowledge about uncertainty can be acquired through experience, that is, we look at previous cases, or scenarios, of the same problem

11 example: portfolio optimization [with B. Pagnoncelli & D. Reich]

12 example: portfolio optimization 1$ [with B. Pagnoncelli & D. Reich]

13 example: portfolio optimization 1$ assets [with B. Pagnoncelli & D. Reich]

14 example: portfolio optimization 1$ assets = percentage of capital invested on asset [with B. Pagnoncelli & D. Reich]

15 example: portfolio optimization 1$ assets = percentage of capital invested on asset [with B. Pagnoncelli & D. Reich]

16 example: portfolio optimization 1$ assets = percentage of capital invested on asset [with B. Pagnoncelli & D. Reich]

17 example: portfolio optimization [with B. Pagnoncelli & D. Reich]

18 example: portfolio optimization record of past rate of returns: = return of asset over period (scenarios) [with B. Pagnoncelli & D. Reich]

19 example: portfolio optimization record of past rate of returns: = return of asset over period (scenarios) [with B. Pagnoncelli & D. Reich]

20 example: classification - defibrillation [with A. Caré]

21 example: classification - defibrillation decide whether a defibrillator has to be applied [with A. Caré]

22 example: classification - defibrillation (scenarios) [with A. Caré]

23 example: classification - defibrillation (scenarios) [with A. Caré]

24 scenario optimization (convex case)

25 min-max scenario optimization [with G. Calafiore]

26 min-max scenario optimization convex in [with G. Calafiore]

27 min-max scenario optimization convex in [with G. Calafiore]

28 min-max scenario optimization convex in [with G. Calafiore]

29 min-max scenario optimization convex in [with G. Calafiore]

30 min-max scenario optimization convex in [with G. Calafiore]

31 min-max scenario optimization convex in [with G. Calafiore]

32 what are the guarantees for unseen situations?

33 what are the guarantees for unseen situations? how guaranteed is for another?

34 what are the guarantees for unseen situations? how guaranteed is for another? from the visible to the invisible

35 about uncertainty

36 about uncertainty

37 about uncertainty

38 about uncertainty

39 about uncertainty

40 about uncertainty

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46 comments generalization need for structure good news: the structure we need is only convexity

47 more comments N depends on how complex the decision is via N does not depend on how complex the real world is

48 more comments N depends on how complex the decision is via N does not depend on how complex the real world is don t try to reconstruct the real world to answer easy questions!

49 more comments (IPM = Interval Prediction Model)

50 more comments (IPM = Interval Prediction Model)

51 more comments (IPM = Interval Prediction Model)

52 more comments (IPM = Interval Prediction Model) risk = prob. that next point is outside IPM

53 more comments (IPM = Interval Prediction Model)

54 more comments N is independent of Pr (distribution-free result)

55 more comments N is independent of Pr (distribution-free result) What I like about experience is that it is such an honest thing. You may have deceived yourself, but experience is not trying to deceive you. C.S. Lewis

56 a more general theoretical result

57

58 risk

59 risk

60 risk

61 risk

62 risk

63 risk

64 risk

65 risk

66 risk

67 application: classification - defibrillation

68 application: classification - defibrillation

69 application: classification - defibrillation

70 application: classification - defibrillation

71 generalizations and beyond

72 generalizations: risk-return tradeoff

73 generalizations: risk-return tradeoff

74 generalizations: risk-return tradeoff

75 generalizations: risk-return tradeoff

76 generalizations: risk-return tradeoff

77 generalizations: risk-return tradeoff

78 generalizations: risk-return tradeoff

79

80

81

82 performance - risk plot

83 performance - risk plot

84 generalizations

85 generalizations

86 generalizations

87 generalizations relevanto to: quantitative finance (minimum return) control with constraints (MPC)

88 Many have given a contribution: T. Álamo, G. Andersson, F. Borrelli, G. Calafiore, A. Carè, F. Dabbene, L. Fagiano, S. Garatti, P. Goulart, O. Granichin, J. Hespanha, H. Hjalmarsson, G. Iyengar, T. Kanamori, D. Kuhn, C. Lagoa, J. Luedtke, A. Luque, J. Lygeros, K. Margellos, M. Morari, J. Matuško, B. Pagnoncelli, M. Prandini, F. Ramponi, D, Reich, C. Rojas, B. Rustem, G. Schildbach, M. Sznaier, A. Takeda, R. Tempo, B. Van Parys, P. Vayanos, M. Vrakopoulou, J. Welsh, W. Wiesemann

89 the theory is still in its infancy

90 the theory is still in its infancy non-convex

91 the theory is still in its infancy margin

92 concluding the problem of extracting knowledge from observations is perhaps the most central issue of all science

93 concluding the problem of extracting knowledge from observations is perhaps the most central issue of all science the scenario approach is one way, and a lot of work remains to be done

94 concluding the problem of extracting knowledge from observations is perhaps the most central issue of all science the scenario approach is one way, and a lot of work remains to be done certainly: it is a wonderful world to explore!

95 concluding the problem of extracting knowledge from observations is perhaps the most central issue of all science the scenario approach is one way, and a lot of work remains to be done certainly: it is a wonderful world to explore! THANK YOU!

96 REFERENCES M.C. Campi and S. Garatti. The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs. SIAM J. on Optimization, 19, no.3: , M.C. Campi and S. Garatti. A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality. J. of Optimization Theory and Applications, 148: , B.K. Pagnoncelli, D. Reich and M.C. Campi Risk-Return Trade-off with the Scenario Approach: A Case study in Portfolio Selection. J. Of Optimization Theory and Applications, 155: , G. Calafiore and M.C. Campi. Uncertain Convex Programs: randomized Solutions and Confidence Levels. Mathematical Programming, 102: 25-46, G. Calafiore and M.C. Campi. The Scenario Approach to Robust Control Design. IEEE Trans. on Automatic Control, AC-51: , M.C. Campi, G. Calafiore and S. Garatti. Interval Predictor Models: Identification and Reliability. Automatica, 45: , M.C. Campi. Classification with guaranteed probability of error. Machine Learning, 80: 63-84, 2010.