On SDP- and CP-relaxations and on connections between SDP-, CP and SIP

Transcription

1 On SDP- and CP-relaations and on connections between SDP-, CP and SIP Georg Still and Faizan Ahmed University of Twente p 1/12

2 1. IP and SDP-, CP-relaations Integer program: IP) : min T Q s.t. a T j = b j, j J {0, 1} n tricks: Introduce: X = T 1 T X ) = 1 ) ) T 1 Use: i {0, 1} 2 i i = 0 X ii i = 0 redundant constraints: a T j ) 2 = b 2 j a T j T a j = b 2 j a T j Xa j = b 2 j p 2/12

3 IP is equivalent with: IP) : min,x Q, X s.t. aj T = b j, j J a j aj T, X = bj 2, j J X ii i = 0 i 0 ) 1 T = X 1 ) 1 ) T Note: 1 1 ) 1 ) 1 ) T S + n+1 = { l ) T Cn+1 = { SDP-relaation. rela in IP: CP-relaation. rela in IP: l v l v T l v l R n+1 } psd v l v T l v l R n+1 + } cp 1 T X 1 T X ) S + n+1 ) C n+1 p 3/12

4 Inclusion for feasible sets F IP F CP F SDP leads to the inequalities for the min-values v ): vsdp) vcp) vip) [Burer9] I.g. SDP gives approimations CP is EXACT for this) IP E. of NP-hard problems where CP is eact SDP is not): ma stable set problem [de Klerk/P02] graph 3-partitioning problem [Povh/Rendl07] quadratic assignment problem [Povh/Rendl09] mied) binary quadratic program [Burer09] few special quadratic programs with non-conve) quadratic constraints p 4/12

5 2 SDP-, CP-relaations of quadratic problems Quadratic problem: QP 0 : min c T 0 s.t. q j) 0, j J in CP-case also: 0 Notation for R n identify q j Q j q j ) = γ j + 2c T j + T C j Equivalent form QP : min,x ct 0 s.t. Q j, 1 T X 1 T γj c T j c j C j ) =: Q j ) X ) CP-case: 0 0, j J ) = 1 1 ) T p 5/12

6 original lifted) quadratic program QP : min,x ct 0 s.t. Q j, 1 T X 1 T ) X ) CP-case: 0 0, j J ) = 1 1 ) T leads to SDP and CP-relaations: SDP : same but ) 1 T S + n+1 X CP : same but ) 1 T C n+1 X psd completely pos. p 6/12

7 Notation: S = {Q j j J} {q j ) j J} in QP) F QP = F QP S) feasible set of QP given by S) Similarly: F SDP S) and F CP S) S) etc.: Projection of F QP S) into the -space. ) Q + := {Q = γ c T C 0 psd)} F QP c C Relaation properties: 0 in CP case) F QP S) conv F QP S) F CP S) F SDP S) p 7/12

8 SDP-relaation Motivation: Consider the feasible set F QP ) {Q}) given by one constraint with Q = γ c T and c C C 0 then F SDP {Q}) = R n Th1. Ko/Tu2000) conv [F QP S)] F QP cone S) Q + ) = F SDP S) Remark F QP cone S)) = F QP S) but F QP cone S) Q + ) F QP S Q + ) p 8/12

9 In Ko/Tu2000 an abstract non-constructive iterative SDP relaation procedure is discussed which in the limit leads to the eact original feasible set. This algorithm depends on a cutting quadratic constraint idea: Adding a redundant linear constraint does not lead to a smaller feasible set F SDP ). Adding a redundant quadratic constraint may lead to a smaller feasible set F SDP ). Ref. M. Kojima, L. Tuncel, Cones of matrices and successive conve relaations of nonconve sets, SIAM J.Optim. 10, No.3, , 2000) p 9/12

10 Etension to CP relaation C n = cone {vv T v R n + } completely positive matrices C n = {A S n T A 0 R n + } copositive matrices ) Q + := {Q = γ c T C C c C n } copositive quadratic functions Th3. conv [F QP S)] F QP cone S) Q + ) F CP S) p 10/12

11 Proof: = = = Use that F QP cone S) Q + ) equals { ) } 1 T, Q 0 Q [cone S) Q T + { ) } ] 1 T [cone S) Q T + { } ] { Note that 1 T ) [S + Q T + ) ] ) } 1 T [S T T 0 C n )] Q + ) := {P P, Q 0 Q = ) = { 0 0 T } use: F CP S) = { 0 C n 1 T X ) γ c T c C } C n+1 S, some X ), C C n } p 11/12

12 In Th1 we have = because in SDP: ) 1 X T ) S n + T S + n+1 X In Th2 we have because in CP: X T ) C n ) 1 T C n+1 X Eample for : ) T 1 T = X ) with X = 2 0 and = 1, 1). Then 0 2 ) 1 1 X T = 0 so / C n ) T C n+1 p 12/12

13 3. Connections between SDP,CP and SIP primal/dual conic program: P) ma ct s.t. B R n n i A i i=1 K D) min Y Y, B s.t. Y, A i = c i, i I, Y K. with a cone K and its dual K. We obtain: SDP: for K = K = S + m CP: for K = C m and K = C m With: S + m := {A S m : z T Az 0 for all z R m }, psd C m := {A S m : z T Az 0 for all z R m, z 0 } copositive matrices p 13/12

14 Linear semi-infinite programs SIP): SIP P ) SIP D ) ma ct s.t. bz) az) T 0 z Z, R n min y z bz)y z z Z s.t. y z az) = c, y z 0. z Z with Z R m is an infinite) compact inde set In SIP D min is taken over all finite sums p 14/12

15 Use/define: S + m = {A : zt Az 0, z = 1}, C m = {A : zt Az 0, z = 1 z 0 } z T B i ia i ) z ) az) = z T A 1 z,..., z T A n z ), bz) = z T Bz quadratic in z SIP-form of SDP, CP: SIP with ) and Z = {z R m : z = 1} for SDP Z = {z R m Gain of SIP-form: : z = 1, z 0 } for CP KKT-type conditions for ma z j Z y zj az j ) = c, Lagrangian mult. y zj 0, active inde-set: Z = {z Z : bz) = az) T } p 15/12

16 For eample: KKT-conditions give for SDP,CP the solution Y of the dual: Y = y zj z j zj T with n terms in z j Z optimality conditions and duality results: SIP SDP,CP p 16/12

17 Application of SIP: discretization/echange methods: CP in SIP-form: P) ma R n ct s.t. z T B n ) i A i z 0 z Z i=1 } {{ } :=F ) Z = {z R m + j z j = 1} unit simple Def. Discretization proposed in Bundfuss/Duer09) partition of Z into sub-simplices i Z = i i, int i int j =, i j Z d = {v ν v ν is a verte of an i } mesh size d := ma { v ν v µ v µ, v µ vertices in same i } p 17/12

18 Discretized program an LP) P d ) ma R n ct s.t. z T F )z 0 z Z d Notation: solution, vp) value, F feasible set of P d solution, vp d ) value, F d feasible set of P d Known from SIP: If Z d also covers all boundary parts of all dimensions then for the discretization error we have: Od 2 ) for feasible set and value Od) or Od 2 ) for the difference d of maimizers depending on the order of maimizer if CQ holds) p 18/12

19 Using for F S m, z, u R m z T Fu = 1 [ 2 z T Fz + u T Fu + z u) T F z u) ] for z = ν λ νv ν Z, ν λ ν = 1, λ ν 0: z T Fz = ν µ λ ν λ µ v ν Fv µ γ } {{ } if γ we obtain with F ) = B i ia i ) Lemma. Suppose CQ holds for 0 F: Then z T F 0 )z s 0 > 0 z Z z T F d )z 1 2 d 2 F d z Z ˆ d = d + ρd 2 0 d ) F if ρ F 0) 2s 0 0 vp d ) vp) [c T d 0 )ρ] d 2 p 19/12

20 Conclusions: Applying SIP to SDP/CP is promising. further research active inde sets, order of maimizers in SDP/CP genericity results and more efficient Newton-echange methods p 20/12