Moments and nonnegative polynomials

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Moments and nonnegative polynomials

• Presented by Kai Ye
• Supervisor Berç Rustem

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Introduction

• Moment problems in probability theory mainly are
  concerned with the bounds, and Markov, Chebyshev
  and Chernoff provide some classical and widely
  used results.
• Nature questions follows

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Introduction

• Are such bounds best possible? In the univariate
  case, Is Chebyshev inequality best possible?
• Can such bounds be generalized in multivariate
  case?
• Can we develop a general theory based on
  optimization methods to address moment problems
  in probability theory?

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Introduction

• To answer these questions, D.Bertsimas,
  I.Popescu, and J.B. Lasserre have done many
  valuable works on this.
• First, D.Bertsimas et al. show a connection
  between moment problems and semidefinite
  relaxations in discrete optimization (convex).

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Introduction

• Then J.B.Lasserre is inspired to expand to a
  semi- algebraic set (nonconvex), and multivariate.

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Structure

• Moment space and related polynomials
• Representing positive polynomials
• Application focus on J.B.Lauseere paper Bounds
  on measures satisfying moment conditions
• Further study and references

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Moment Space

• Definition (1-dimension)
• The Moment space is given by
• The nth-moment space is defined by
  truncating the sequence to
• or by projecting
  to its
• first n coordinates.

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Moment Space

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Moment Space
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Moment Space

• rn1.
• is compact and convex.
• a is n-dimensional body.

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Moment Space

• The moment space when n2.

1,1
0,0
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Moment Space

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Moment Space
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Moment Space

• Corollary 1

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Moment Space

• Explanation

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Moment Space

• About m-dimension.

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Moment Space
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Moment Space

• Corollary 2
• The results of corollary 1 also holds for the
  m-dimension situation (multivariate).
• Many properties can be explored using corollary
  1. The most important one is as follows.

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Hankel determinants

• From corollary 1,we can always find a nonnegative
  polynomial to represent .

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Hankel determinants

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Hankel determinants
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Hankel determinants
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Hankel determinants
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Hankel determinants
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Hankel determinants
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Hankel determinants
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Representation of positive polynomials

• It is then quite nature to think the problem as
  the title.
• In one dimension, we have been working on it, and
  it seems that a positive polynomial can be
  represented by a sum of squares (s.o.s.), but in
  n-dimension, s.o.s. is not unique.

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Representation of positive polynomials

• Interesting thing is that we can check
• whether some given polynomial
• is s.o.s. reduces to solving a SDP.

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Representation of positive polynomials
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Representation of positive polynomials

• Then, checking whether a polynomial
• is s.o.s. reduces to checking the LMI

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Applications

• Bounds on measures satisfying moment conditions
  by J.B.Lasserre3

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Applications

• The duality is following from Smith4 p.825 and
  it is worthy noticing that the duality result
  does not depend on the structure of .

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Applications

• Before representing the polynomial ,
• we need to recall a theorem.

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Applications
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Applications
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Applications
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Applications
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Discussion and further study

• We can see that in the paper the only thing not
  perfect is about an assu-mption about s.o.s. in
  transformation
  
• but what can we do about it?
• Further study.

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Bibliography

• 1 H.Dette, W.J.Studden, the theory of Canonical
  Moments with application in Statistics,
  Probability, and Analysis.1997
• 2 Karlin and Studden, Tchebycheef systems with
  application in analysis and statistics,1966
• 3 J.B.Lasserre, Bounds on measures satisfying
  moment conditions, The Annals of Applied
  probability,2002,Vol.12,No.3, 1114-1137.
• 4 J.E.Smith, Generalized Chebychev
  Inequalities Theory and Application in Decision
  Analysis, Operations Research, Vol.43, No.5,
  807-825
• 5 D.Bertsimas, I.Popescu, On the relation
  between option and stock prices A convex
  optimization approach, Operations Research,
  Vol.5-,No.2, 358-374
• 6 D.Bertsimas and J.Sethuraman, Moments
  problems and semidefinite optimization. In
  Handbook of Semidefinite Programming,2000,311-319
• 7 L.E.Ghaoui,et al. Worst-case Value-At-Risk
  and Robust portfolio optimization
• A Conic programming approach, 2003,
  Operations Research, Vol. 51, No.4, 543-556

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Thank you!!!