Compact relaxations for polynomial programming problems - SEA 2012

Compact relaxations for polynomial programming problems S. Cafieri (ENAC Toulouse, F) P. Hansen (GERAD & HEC, CA) L. L´etocart (LIPN Paris13, F) L. Liberti (LIX, Ecole Polytechnique, F) F. Messine (ENSEEIHT Toulouse, F) June 1, 2012

Compact relaxations

SEA 2012 – 1 / 40

⊲ Introduction The setting Exact reformulations ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review

Introduction

New developments Why bother? Thank you

Compact relaxations

SEA 2012 – 2 / 40

The setting Introduction The setting Exact reformulations

ReformulationLinearization Technique

⊲

Focus on (nonconvex) polynomial programming problems Aim to solve with sBB Need a tight convex (linear) relaxation at each node Reformulate before relaxing

RLT literature review Reduced RLT RRLT literature review New developments Why bother? Thank you

Compact relaxations

SEA 2012 – 3 / 40

Exact reformulations P Introduction The setting Exact reformulations

Q

⊲

ReformulationLinearization Technique RLT literature review

G

G

Reduced RLT RRLT literature review

F

New developments

L

φ|G φ|L

Why bother? Thank you

L F

φ

Compact relaxations

P harder than Q find optima in Q, map them back to P for each opt. in P ∃ corresponding opt. in Q

SEA 2012 – 4 / 40

Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

⊲

RLT literature review

Reformulation-Linearization Technique

Reduced RLT RRLT literature review New developments Why bother? Thank you

Compact relaxations

SEA 2012 – 5 / 40

Aim Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

⊲

RLT literature review Reduced RLT RRLT literature review

Reformulation-Linearization Technique (RLT): tightens the linear relaxation of mixed-integer (nonconvex) QCQPs  min c0 x + xQ0 x    x∈Rn1 ×Zn2  ∀1 ≤ i ≤ q ci x + xQi x ≤ 0  ∀1 ≤ i ≤ m ai x ≤ bi    L U x ∈ [x , x ].


Compact relaxations

SEA 2012 – 6 / 40

RLT constraints 1/2 Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

∀j, ℓ ≤ n = n1 + n2 , i ≤ m, all factors non-negative ⇒ constraints are valid

⊲


L )(x − x (xj − xL ℓ j ℓ) ≥ 0 U (xj − xL )(x j ℓ − xℓ ) ≥ 0 L (xU − x )(x − x j ℓ j ℓ) ≥ 0 U (xU j − xj )(xℓ − xℓ ) ≥ 0


(xj − xL j )(bi − ai x) ≥ 0

New developments

(xU j − xj )(bi − ai x) ≥ 0


Compact relaxations

SEA 2012 – 7 / 40

RLT constraints 2/2 Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

∀j, ℓ ≤ n, i ≤ m, get:

⊲

RLT literature review Reduced RLT RRLT literature review New developments

L L L ≥ 0 x − x x + x xj xℓ − xL j ℓ j j xℓ ℓ L L U −xj xℓ + xU x + x x − x j j ℓ j xℓ ℓ

≥ 0

U U L ≥ 0 x + x x − x −xj xℓ + xL j ℓ j xℓ ℓ j U U U xj xℓ − xU ℓ xj − xj xℓ + xj xℓ

≥ 0

L −xj ai x + xL a x + x b − x i j i j j bi ≥ 0 U xj a i x − xU a x − x b + x i j i j j bi ≥ 0


Compact relaxations

SEA 2012 – 8 / 40

Linearization Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

⊲

Replace xj xℓ by var. wjℓ (wj = (wj1 , . . . , wjn )): (McCormick’s con vex/concave envelopes L L L L wjℓ ≥ xℓ xj + xj xℓ − xj xℓ  for x x )  j ℓ  U L L U wjℓ ≤ xℓ xj + xj xℓ − xj xℓ U L U wjℓ ≤ xL ℓ xj + xj xℓ − xj xℓ    U U U U wjℓ ≥ xℓ xj + xj xℓ − xj xℓ 10

5

0


–5

0

–10

Reduced RLT

–4

1

–2

0 y

2

4

2

x

RRLT literature review New developments Why bother? Thank you

Compact relaxations

ai wj ai wj

≤ ≥

L xL j ai x + xj bi − xj bi U xU j a i x + x j bi − x j bi .

valid linear relations between x and w

SEA 2012 – 9 / 40

Relaxation 1/2 Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

∀i ≤ q, replace terms xi xj in xQi x with linearizing variables wij   w11 . . . w1n  ..  .. w =  ... . .  wn1 . . . wnn

Get xQi x = Qi w Use IA on [xL , xU ] to compute ranges [wL , wU ] for w Adjoin commutativity constr. ∀j ≤ ℓ ≤ n wjℓ = wℓj Adjoin square constr. ∀i ≤ n s.t. xi is binary wii = xi

⊲



Compact relaxations

SEA 2012 – 10 / 40

Relaxation 2/2 Introduction ReformulationLinearization Technique Aim RLT constraints 1/2 RLT constraints 2/2 Linearization Relaxation 1/2 Relaxation 2/2

Relaxed MILP

⊲

RLT literature review Reduced RLT

min c0 x + Q0 w ∀i ≤ q ci x + Qi w ≤ 0 ∀i ≤ m ai x ≤ bi (RLT+Commutativity+Square constraints) x ∈ (Rn1 × Zn2 ) ∩ [xL , xU ] 2 n w ∈ R ∩ [wL , wU ]

              


Compact relaxations

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Introduction ReformulationLinearization Technique RLT literature

⊲ review

The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.



Compact relaxations

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The first paper Introduction ReformulationLinearization Technique RLT literature review The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.

⊲

Seminal paper Adams & Sherali, Mgt. Sci. 1986: x ∈ {0, 1}n MILP reformulation via Fortet’s inequalities

Reduced RLT

wjℓ ≤ min(xj , xℓ )

RRLT literature review New developments

wjℓ ≥ max(0, xj + xℓ − 1)


then continuous relaxation

Compact relaxations

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Mixed products Introduction ReformulationLinearization Technique RLT literature review The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.

⊲

Adams & Sherali, Op. Res. 1990: x ∈ Rn1 × {0, 1}n2 with products involving at least one binary variable MILP reformulation

Reduced RLT RRLT literature review New developments Why bother?


Thank you

Compact relaxations

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Relaxation of bilinear terms Introduction ReformulationLinearization Technique RLT literature review The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.

⊲

Sherali & Alameddine, JOGO 1992: x ∈ Rn with bilinear products LP relaxation

Reduced RLT RRLT literature review New developments

under special condition, LP is reformulation


Compact relaxations

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Mixed products again Introduction ReformulationLinearization Technique RLT literature review The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.

⊲


Adams & Sherali, Math. Prog. 1993: x ∈ Rn1 × {0, 1}n2 specialized to bilinear products involving one continuous and one binary variable MILP reformulation

New developments Why bother?


Thank you

Compact relaxations

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Relaxation hierarchy Sherali & Adams, DAM 1994: Introduction ReformulationLinearization Technique

x ∈ Rn1 × {0, 1}n2

RLT literature review The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.

Convex hull of MILP feasible region obtained through hierarchy of RLT constraints

⊲

Reduced RLT RRLT literature review New developments

Level d RLT: let J1 , J2 ⊆ {1, . . . , n2 } with |J1 ∪ J2 | = d and Y Y xj (1 − xj ); Fd (J1 , J2 ) = j∈J1

j∈J2


multiply RLT-(d − 1) constraints by all Fd (J1 , J2 ) the linearize to obtain RLT-d

Compact relaxations

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Etc. Introduction ReformulationLinearization Technique RLT literature review The first paper Mixed products Relaxation of bilinear terms Mixed products again Relaxation hierarchy Etc.

⊲

. . . and many more! extensions (polynomial and signomial programming, SDP, convex MINLP, disjunctive cuts) and applications


Compact relaxations

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Introduction ReformulationLinearization Technique RLT literature review

⊲ Reduced RLT The quadratic case RRLT-1 constraints Main result Geometric interpretation

Reduced RLT


Compact relaxations

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The quadratic case Introduction ReformulationLinearization Technique RLT literature review Reduced RLT The quadratic case RRLT-1 constraints Main result Geometric interpretation

⊲

RRLT literature review

Consider mixed-integer QCQP subject to linear equality constraints Ax = b (A has full rank)  c0 x + xQ0 x min   n n  x∈R 1 ×Z 2  ∀1 ≤ i ≤ q ci x + xQi x ≤ 0  Ax = b    L U x ∈ X ∩ [x , x ].

X is a polyhedron


Compact relaxations

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RRLT-1 constraints Introduction ReformulationLinearization Technique

Generate RLT constraints from Ax = b Ax = b

RLT literature review Reduced RLT The quadratic case RRLT-1 constraints Main result Geometric interpretation

∀ℓ ≤ n ∀ℓ ≤ n

xℓ Ax = xℓ b

⇒ ⇒

Awℓ = xℓ b

⊲


Compact relaxations

Consider homogeneous system ∀ℓ ≤ n (Awℓ = 0) and a set N of nonbasic variable index pairs (j, ℓ); let: C = {(x, w) | Ax = b ∧ ∀j, ℓ ≤ n (wjℓ = xj xℓ )} RN

= {(x, w) | Ax = b ∧ ∀ℓ ≤ n (Awℓ = bxℓ ) ∧ ∀(j, ℓ) ∈ N (wjℓ = xj xℓ )}

SEA 2012 – 21 / 40

Main result Introduction ReformulationLinearization Technique RLT literature review Reduced RLT The quadratic case RRLT-1 constraints Main result Geometric interpretation

⊲


Thm. Let [n] = {1, . . . , n}; ∃N ⊆ [n] × [n] C = RN Proof (RRLT system) ∀ℓ ≤ n Awℓ − xℓ b = 0 ⇒ (replace b by Ax) ∀ℓ ≤ n Awℓ − xℓ Ax = 0 ⇒ (z = wℓ − xℓ x are vars. of hom. sys.) ∀ℓ ≤ n A(wℓ − xℓ x) = 0.(1)    get square nonsing. subsyst.   (1) is homogeneous  ′ A z = 0 of (1)  N ⊆ [n] × [n]: nonbasic of (1)  ⇒ corresponding to basic cols.   ∀(j, ℓ) ∈ N wjℓ = xj xℓ  B = [n] × [n] r N

by basic linear algebra, A′ z = 0 implies ∀(j, ℓ) ∈ B (wjℓ = xj xℓ ).

Cor. RRLT constraints ⇒ exact ref. with fewer quadratic terms Proof Only need quadratic terms indexed by N , RRLT implies those in B

Compact relaxations

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Geometric interpretation 10

10

8

Introduction ReformulationLinearization Technique

6 5

4 2 0

0 -2 -4


-6

-5

-8 -10

Reduced RLT The quadratic case RRLT-1 constraints Main result Geometric interpretation

⊲


0

-10 0

0.2 0.4 0.6 0.8

1 x 1.21.4 1.6 1.8

4

2

-4

-2

0 y

2

C = {(x, y, w) | x = 1 ∧ w = xy}

0.2 0.4 0.6 0.8

1 x 1.21.4 1.6 1.8

2

-4

-2

0 y

2

4

McCormick’s rel. of w = xy restricted to x = 1

10

5

Notice C = R (straight red segment)

0

Why bother?

Equation w = y can be obtained via RRLT: multiply equation x = 1 by y and linearize via w

-5

Thank you -10 0

0.2 0.4 0.6 0.8

1 x 1.21.4 1.6 1.8

4 2 -2 2

-4

0 y

R = {(x, y, w) | x = 1 ∧ w = y)}

Compact relaxations

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Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature

⊲ review

Announcement General theory Automatic reformulation Application to quantum chemistry



Compact relaxations

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Announcement Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review Announcement General theory Automatic reformulation Application to quantum chemistry

⊲

New developments

L. , ITOR 2004: RRLT constraints are linearly independent Preliminary results on pooling problems


Compact relaxations

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General theory Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review Announcement General theory Automatic reformulation Application to quantum chemistry

⊲

New developments

L. , JOGO 2005: General theory of RRLT constraints Reformulation proofs


Compact relaxations

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Automatic reformulation Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review Announcement General theory Automatic reformulation Application to quantum chemistry

⊲

New developments

L. & Pantelides, JOGO 2006: Graph-based automatic reformulation algorithm Full computational results on pooling and blending problems


Compact relaxations

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Application to quantum chemistry Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review Announcement General theory Automatic reformulation Application to quantum chemistry

⊲

L. , Lavor, Maculan, Chaer Nascimento, DAM 2009: Application of an RRLT-2 subset to solving Hartree-Fock systems


Compact relaxations

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Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲

New developments


Compact relaxations

SEA 2012 – 29 / 40

Optimal RRLT Introduction ReformulationLinearization Technique

Feasible region of QCQP: use RN instead of C

RLT literature review Reduced RLT RRLT literature review New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲


Compact relaxations

SEA 2012 – 30 / 40

Optimal RRLT Introduction ReformulationLinearization Technique

Feasible region of QCQP: use RN instead of C RN relies on quadratic constraints ∀(j, ℓ) ∈ N (wjℓ = xj xℓ )


⊲


Compact relaxations

SEA 2012 – 30 / 40

Optimal RRLT Introduction ReformulationLinearization Technique RLT literature review

Feasible region of QCQP: use RN instead of C RN relies on quadratic constraints ∀(j, ℓ) ∈ N (wjℓ = xj xℓ ) Degree of freedom: choice of basic/nonbasic partition B, N of [n] × [n]

Reduced RLT RRLT literature review New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲


Compact relaxations

SEA 2012 – 30 / 40

Optimal RRLT Introduction ReformulationLinearization Technique RLT literature review Reduced RLT

Feasible region of QCQP: use RN instead of C RN relies on quadratic constraints ∀(j, ℓ) ∈ N (wjℓ = xj xℓ ) Degree of freedom: choice of basic/nonbasic partition B, N of [n] × [n] (j, ℓ) ↔ volume Vjℓ of conv. env. of xj xℓ

RRLT literature review New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲


Compact relaxations

SEA 2012 – 30 / 40


Reduced RLT


New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲

Feasible region of QCQP: use RN instead of C RN relies on quadratic constraints ∀(j, ℓ) ∈ N (wjℓ = xj xℓ ) Degree of freedom: choice of basic/nonbasic partition B, N of [n] × [n] (j, ℓ) ↔ volume Vjℓ of conv. P env. of xj xℓ Vjℓ Convexity gap: V(N ) = (j,ℓ)∈N


Compact relaxations

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Reduced RLT



⊲


Let N ∗ = arg minN V(N )


Compact relaxations

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Reduced RLT



⊲



Smaller gap ⇒ tight bound more likely


Compact relaxations

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Reduced RLT



⊲




B, N partition [n] × [n] ⇒ N ∗ = [n] × [n] r arg maxB V(B)


Compact relaxations

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Reduced RLT



⊲




B, N partition [n] × [n] ⇒ N ∗ = [n] × [n] r arg maxB V(B) Reduces to finding a max-weight basis of a linear system


Compact relaxations

SEA 2012 – 30 / 40


Reduced RLT



⊲




B, N partition [n] × [n] ⇒ N ∗ = [n] × [n] r arg maxB V(B) Reduces to finding a max-weight basis of a linear system Greedy algorithm solves problem optimally


Compact relaxations

SEA 2012 – 30 / 40

RRLT for polynomial programming 1/3 Introduction ReformulationLinearization Technique

Consider general polynomial programming MINLP g0 (x)

∀i ≤ q

gi (x) ≤ 0 Ax = b x ∈ X ∩ [xL , xU ]

x∈R


min n

1 ×Zn2

where gi ∈ Q[x] for all i ∈ {0, . . . , q}

        

⊲


Compact relaxations

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Reformulation: for all J ⊆ [n − 1] multiply Ax = b by Q xj j∈J


⊲


Compact relaxations

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Reformulation: for all J ⊆ [n − 1] multiply Ax = b by Q xj j∈J Q Linearization: replace each term xj by the added j∈J

variable wJ (for all J ⊆ [n])


⊲


Compact relaxations

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j∈J


Reformulation: for all J ⊆ [n − 1] multiply Ax = b by Q xj j∈J Q Linearization: replace each term xj by the added

variable wJ (for all J ⊆ [n]) Q Adjoin defining constraints wJ = xj j∈J


⊲


Compact relaxations

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j∈J




⊲

Define natural extensions of C, RN : C

= {(x, w) | Ax = b ∧ ∀J ⊆ [n − 1] (wJ =

Y

xj }

j∈J

RN

= {(x, w) | Ax = b ∧ ∀J ⊆ [n − 1] (AwJ = bwJ ) ∧ Y xj )} ∀J ∈ N (wJ = j∈J

where wJ = (w(J,1) , . . . , w(J,n) )


Compact relaxations

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j∈J




⊲


Compact relaxations

Define natural extensions of C, RN : C

= {(x, w) | Ax = b ∧ ∀J ⊆ [n − 1] (wJ =

Y

xj }

j∈J

RN

= {(x, w) | Ax = b ∧ ∀J ⊆ [n − 1] (AwJ = bwJ ) ∧ Y xj )} ∀J ∈ N (wJ = j∈J

where wJ = (w(J,1) , . . . , w(J,n) ) Main result C = RN still holds SEA 2012 – 32 / 40


Choice of optimal N extends from quadratic case, but:


⊲


Compact relaxations

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Choice of optimal N extends from quadratic case, but: Added complication:


⊲


Compact relaxations

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Choice of optimal N extends from quadratic case, but: Added complication: –

Vij and Vijk are expressed in different units of measure


⊲


Compact relaxations

SEA 2012 – 33 / 40



Choice of optimal N extends from quadratic case, but: Added complication: – –

Vij and Vijk are expressed in different units of measure Summing up VJ for J’s of different sizes may not make much sense


⊲


Compact relaxations

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Vij and Vijk are expressed in different units of measure Summing up VJ for J’s of different sizes may not make much sense P Multi-objective problem: ∀p ∈ [n] max VJ – –



|J|=p

⊲


Compact relaxations

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|J|=p

Thm. P Efficient solution is an optimum of max VJ J⊆[n]

⊲


Compact relaxations

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⊲





|J|=p

Thm. P Efficient solution is an optimum of max VJ J⊆[n]

Greedy is still OK


Compact relaxations

SEA 2012 – 33 / 40

Sparsity 1/4 Introduction ReformulationLinearization Technique

Polynomial programs are never dense in practice


RRLT needs B ∪ N = P([n])

Need to introduce exponentially many new monomials


?

⊲


Compact relaxations

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Sparsity 2/4 Introduction

β =set of multi-indices for monomials already in problem

ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲


Compact relaxations

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β =set of multi-indices for monomials already in problem Every new monomial J 6∈ β yields a new variable wJ


⊲


Compact relaxations

SEA 2012 – 35 / 40


β =set of multi-indices for monomials already in problem Every new monomial J 6∈ β yields a new variable wJ Sometimes ∃J 6∈ β s.t. wJ yields > 1 new RRLT constr.


⊲


Compact relaxations

SEA 2012 – 35 / 40

Sparsity 2/4 Introduction ReformulationLinearization Technique RLT literature review


β =set of multi-indices for monomials already in problem Every new monomial J 6∈ β yields a new variable wJ Sometimes ∃J 6∈ β s.t. wJ yields > 1 new RRLT constr. E.g.: x1 + x2 = 1 ∧ 2x1 − x2 = 3 ∧ β = {(1, 3)} one new monomial (x2 x3 ) ⇒ two new RRLT constraints

⊲


Compact relaxations

SEA 2012 – 35 / 40


Reduced RLT

x1 + x2 = 1 ∧ 2x1 − x2 = 3 ∧ β = {(1, 3)}


⊲

β =set of multi-indices for monomials already in problem Every new monomial J 6∈ β yields a new variable wJ Sometimes ∃J 6∈ β s.t. wJ yields > 1 new RRLT constr. E.g.:

one new monomial (x2 x3 ) ⇒ two new RRLT constraints x1 + x2 = 1 (×x3 =) w13 + w23 = x3 2x1 − x2 = 3 (×x3 =) 2w13 − w23 = 3x3

Principle: one new equation, one fewer degrees of freedom


Compact relaxations

SEA 2012 – 35 / 40


Reduced RLT

x1 + x2 = 1 ∧ 2x1 − x2 = 3 ∧ β = {(1, 3)}


⊲

β =set of multi-indices for monomials already in problem Every new monomial J 6∈ β yields a new variable wJ Sometimes ∃J 6∈ β s.t. wJ yields > 1 new RRLT constr. E.g.:

one new monomial (x2 x3 ) ⇒ two new RRLT constraints x1 + x2 = 1 (×x3 =) w13 + w23 = x3 2x1 − x2 = 3 (×x3 =) 2w13 − w23 = 3x3


Compact relaxations

Principle: one new equation, one fewer degrees of freedom Create fewer J’s than new RRLT constraints

SEA 2012 – 35 / 40

Sparsity 3/4

Problem:

Introduction ReformulationLinearization Technique RLT literature review

Look for subset ρ of rows of Ax = b to be multiplied by a subset σ of P([n − 1]) such that the number of new vars wJ is < number of new RRLT constraints


⊲


Compact relaxations

SEA 2012 – 36 / 40

Sparsity 3/4

Problem:

Introduction


ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review


Formalization: consider bipartite graph (U, V, E) wL × (a1 x = b1 ) wH × (a1 x = b1 )

J¯ ¯ H

wL × (a2 x = b2 )

¯ L

wJ × (ai x = bi )

⊲


Compact relaxations

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Sparsity 3/4

Problem:

Introduction




J¯ ¯ H

wL × (a2 x = b2 )

¯ L

wJ × (ai x = bi )

U =row i by var. wJ (indexed by (i, J))

⊲


Compact relaxations

SEA 2012 – 36 / 40

Sparsity 3/4

Problem:

Introduction



⊲


J¯ ¯ H

wL × (a2 x = b2 )

¯ L

wJ × (ai x = bi )

U =row i by var. wJ (indexed by (i, J)) ¯ V =var. wJ¯ with J¯ 6∈ β (indexed by J)


Compact relaxations

SEA 2012 – 36 / 40

Sparsity 3/4

Problem:

Introduction



⊲


J¯ ¯ H

wL × (a2 x = b2 )

¯ L

wJ × (ai x = bi )

U =row i by var. wJ (indexed by (i, J)) ¯ V =var. wJ¯ with J¯ 6∈ β (indexed by J) Edges: E=incidence of added vars in RRLT constrs


Compact relaxations

SEA 2012 – 36 / 40

Sparsity 3/4

Problem:

Introduction



⊲



J¯ ¯ H

wL × (a2 x = b2 )

¯ L

wJ × (ai x = bi )

U =row i by var. wJ (indexed by (i, J)) ¯ V =var. wJ¯ with J¯ 6∈ β (indexed by J) Edges: E=incidence of added vars in RRLT constrs

Aim: find induced subgraph (U ′ , V ′ , E ′ ) such that |U ′ | is maximum, |U ′ | > |V ′ |, and V ′ =neighb(U ′ )

Compact relaxations

SEA 2012 – 36 / 40

Sparsity 4/4 Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review

Mathematical Programming formulation: P ui,J ′ max (i,J ′ )∈U P P vJ + 1 ui,J ′ ≥


(i,J ′ )∈U

∀{(i, J ′ ), J} ∈ E

J6∈β

vJ ≥ ui,J ′ u ∈ {0, 1}|U | v ∈ {0, 1}|P([n])rβ| .

                

⊲


Compact relaxations

SEA 2012 – 37 / 40

Sparsity 4/4 Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review New developments Optimal RRLT RRLT for polynomial programming 1/3 RRLT for polynomial programming 2/3 RRLT for polynomial programming 3/3 Sparsity 1/4 Sparsity 2/4 Sparsity 3/4 Sparsity 4/4

⊲

Mathematical Programming formulation: P ui,J ′ max (i,J ′ )∈U P P vJ + 1 ui,J ′ ≥ (i,J ′ )∈U

∀{(i, J ′ ), J} ∈ E

Thm. This problem is in P Proof Use matching-based algorithm

J6∈β

vJ ≥ ui,J ′ u ∈ {0, 1}|U | v ∈ {0, 1}|P([n])rβ| .

                


Compact relaxations

SEA 2012 – 37 / 40

Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review New developments

Why bother?

⊲ Why bother? Use within sBB Thank you

Compact relaxations

SEA 2012 – 38 / 40

Use within sBB Introduction ReformulationLinearization Technique RLT literature review Reduced RLT


CPU time in sBB: number of nodes, time to solve each node Need few, small convex relaxation LPs Usually concentrate on few (tight bound) but large (valid cuts) Different approach: slacken bound, aim to solve each LP faster

New developments Why bother? Use within sBB

⊲

Thank you

Compact relaxations

SEA 2012 – 39 / 40

Use within sBB Introduction ReformulationLinearization Technique RLT literature review Reduced RLT


CPU time in sBB: number of nodes, time to solve each node Need few, small convex relaxation LPs Usually concentrate on few (tight bound) but large (valid cuts) Different approach: slacken bound, aim to solve each LP faster Outcome:

Why bother? Use within sBB

⊲

– –

Thank you

Compact relaxations

bound quality: 0.07% worse; CPU improvement: 40%

Future work: embed in sBB

SEA 2012 – 39 / 40

Introduction ReformulationLinearization Technique RLT literature review Reduced RLT RRLT literature review New developments

Thank you

Why bother?

⊲ Thank you

Compact relaxations

SEA 2012 – 40 / 40

Compact relaxations for polynomial programming problems - SEA 2012

Compact relaxations for polynomial programming problems - SEA 2012

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