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

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 What if such a decompositjon is not given? Hiriart-Urruty, 1985 Tuy, 1995 2

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?

Existence of dc decompositjon (1/4) • 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 (2/4) sos convex SOS-convexity 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. 5

Existence of dc decompositjon (3/4) • Lemma: Let be a full dimensional cone in a vector space Thenany can be writuen as with . Proof sketch: such that E K 6

Existence of dc decompositjon (4/4) • • 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: can be shown to be in the interior of • Also shows that a decompositjon can be obtained effjciently : solving is an SDP. sos-convex • In fact, we show that a decompositjon can be found via LP and SOCP (not covered here).

Uniqueness of dc decompositjon • • Dc decompositjon : given a polynomial , fjnd and convex polynomials such that • Questjons:   Yes • Does such a decompositjon always exist? Through sos-convexity • Can I obtain such a decompositjon effjciently? • Is this decompositjon unique? Alternatjve decompositjons Initjal decompositjon x convex “Best decompositjon?”

Convex-Concave Procedure (CCP) • Heuristjc for minimizing DC programming problems. • Idea: Convexify by linearizing Input Solve convex subproblem x Take to be the solutjon of x initjal point convex convex affjne 9

Convex-Concave Procedure (CCP) • Toy example: , where • Convexify to obtain Initjal point: Minimize and obtain Reiterate 10

Picking the “best” decompositjon for CCP Algorithm Linearize around a point to obtain convexifjed version of Idea Pick such that it is as close as possible to affjne around Mathematjcal translatjon Minimize curvature of at Worst-case curvature* Average curvature* s.t. s.t. convex convex * * 11

Undominated decompositjons (1/4) 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 DOMINATED BY Convexify around to get If dominates then the next iterate in CCP obtained using always beats the one obtained using 12

Undominated decompositjons (2/4) Recover locatjon of the points in Incomplete distance matrix Solve: There is a realizatjon in ifg opt value 13

Undominated decompositjons (3/4) is an undominated dcd of the objectjve functjon. 14

Undominated decompositjons (4/4) • 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. 15

Comparing difgerent decompositjons (1/2) • Solving the problem: , where has and • • Decompose run CCP for 4 minutes and compare objectjve value. Feasibility Feasibility Undominated Undominated s.t. s.t. sos-convex sos-convex sos-convex 16

Comparing difgerent decompositjons (2/2) • Average over 30 iteratjons • Solver: Mosek Feasibility • Computer: 8Gb RAM, 2.40GHz processor Undominated Conclusion: Rate of convergence of CCP strongly afgected by initjal decompositjon. 17

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). 18

Thank you for listening Questjons? Want to learn more? htup://scholar.princeton.edu/ghall 7/31/16 19