Polynomial DC decompositjons Georgina Hall Princeton, ORFE Joint - - PowerPoint PPT Presentation

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Polynomial DC decompositjons Georgina Hall Princeton, ORFE Joint - - PowerPoint PPT Presentation

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Polynomial DC decompositjons Georgina Hall Princeton, ORFE Joint work with Amir Ali Ahmadi Princeton, ORFE 7/31/16 DIMACS Distance geometry workshop 1 Difgerence of convex (dc) programming Problems of the form where , convex for

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SLIDE 1

Polynomial DC decompositjons

Georgina Hall Princeton, ORFE Joint work with Amir Ali Ahmadi Princeton, ORFE

1 7/31/16 DIMACS – Distance geometry workshop

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SLIDE 2

Difgerence of convex (dc) programming

Problems of the form where , convex for

  • 2

Hiriart-Urruty, 1985 Tuy, 1995

What if such a decompositjon is not given?

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SLIDE 3

Difgerence of convex (dc) decompositjon

  • Difgerence of convex (dc) decompositjon: given a polynomial , fjnd

and such that where convex polynomials.

  • Questjons:
  • Does such a decompositjon always exist?
  • Can I obtain such a decompositjon effjciently?
  • Is this decompositjon unique?
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  • A polynomial is a sum of squares (sos) if , polynomials, s.t.
  • A polynomial of degree 2d is sos if and only if such that

where is the vector of monomials up to degree

  • Testjng whether a polynomial is sos is a semidefjnite program.
  • 4

Existence of dc decompositjon (1/4)

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Theorem: Any polynomial can be writuen as the difgerence of two sos-convex polynomials. Corollary: Any polynomial can be writuen as the difgerence of two convex polynomials.

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Existence of dc decompositjon (2/4)

convex sos SOS-convexity

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SLIDE 6

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Lemma: Let be a full dimensional cone in a vector space Thenany can be writuen as with. Proof sketch:

  • K

E

such that

Existence of dc decompositjon (3/4)

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SLIDE 7
  • Here, {polynomials of degree 2d, in n variables}

sos-convex polynomials of degree 2d and in n variables

  • Remains to show that is full dimensional:
  • Also shows that a decompositjon can be obtained effjciently:
  • In fact, we show that a decompositjon can be found via LP and SOCP (not

covered here).

  • Existence of dc decompositjon (4/4)

can be shown to be in the interior of

sos-convex

is an SDP. solving

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Uniqueness of dc decompositjon

  • Dc decompositjon: given a polynomial , fjnd and convex polynomials

such that

  • Questjons:
  • Does such a decompositjon always exist?
  • Can I obtain such a decompositjon effjciently?
  • Is this decompositjon unique?
  •  

Yes Through sos-convexity

Initjal decompositjon

x

Alternatjve decompositjons convex

“Best decompositjon?”

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Convex-Concave Procedure (CCP)

  • Heuristjc for minimizing DC programming problems.
  • Idea:

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Input

x initjal point

Convexify by linearizing

x convex affjne convex

Solve convex subproblem

Take to be the solutjon of

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Convex-Concave Procedure (CCP)

  • Toy example: , where
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Initjal point: Convexify to obtain Minimize and

  • btain

Reiterate

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Picking the “best” decompositjon for CCP

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Algorithm Linearize around a point to obtain convexifjed version of Mathematjcal translatjon Minimize curvature of at Worst-case curvature* s.t. convex Average curvature* s.t. convex Idea Pick such that it is as close as possible to affjne around

* *

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Undominated decompositjons (1/4)

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Defjnitjon: is an undominated decompositjon of if no other decompositjon of can be obtained by subtractjng a (nonaffjne) convex functjon from

Convexify around to get Convexify around to get DOMINATED BY If dominates then the next iterate in CCP obtained using always beats the one obtained using

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Incomplete distance matrix Recover locatjon of the points in Solve: There is a realizatjon in ifg opt value

Undominated decompositjons (2/4)

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is an undominated dcd of the objectjve functjon.

Undominated decompositjons (3/4)

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Theorem: Given a polynomial, consider min, (where ) s.t. convex, convex Any optjmal solutjon is an undominated dcd of (and an optjmal solutjon always exists). Theorem: If has degree 4, it is strongly NP-hard to solve . Idea: Replace convex by sos-convex.

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Undominated decompositjons (4/4)

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Comparing difgerent decompositjons (1/2)

  • Solving the problem: , where has and
  • Decompose run CCP for 4 minutes and compare objectjve value.
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Feasibility Undominated Feasibility Undominated sos-convex s.t. sos-convex s.t. sos-convex

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Comparing difgerent decompositjons (2/2)

Conclusion: Rate of convergence of CCP strongly afgected by initjal decompositjon.

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Undominated Feasibility

  • Average over 30 iteratjons
  • Solver: Mosek
  • Computer: 8Gb RAM, 2.40GHz

processor

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Main messages

  • We studied the questjon of decomposing a polynomial into the

difgerence of two convex polynomials.

  • This decompositjon always exists and is not unique.
  • Choice of decompositjon can impact convergence rate of the CCP

algorithm.

  • Dc decompositjons can be effjciently obtained using the notjon of

sos-convexity (SDP)

  • Also possible to use LP or SOCP-based relaxatjons to obtain dc

decompositjons (not covered here).

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Thank you for listening

Questjons?

19 7/31/16

Want to learn more? htup://scholar.princeton.edu/ghall