Detecting optimality and extracting minimizers in polynomial optimization based on the Lasserre relaxation and the truncated GNS construction María López Quijorna University of Konstanz Graz, 7 September 2018 1/12
Notation: Let n , k ∈ N 0 then: X := ( X 1 , . . . , X n ), R [ X ] := R [ X 1 , . . . , X n ] R [ X ] k real polynomials with degree less or equal to k R [ X ] = k real forms of degree k R [ X ] ∗ k := { L : R [ X ] k → R | L is R -linear } m � � R [ X ] 2 g 2 k := { i | m ∈ N 0 , g i ∈ R [ X ] k } i =0 1/12
Notation: Let n , k ∈ N 0 then: X := ( X 1 , . . . , X n ), R [ X ] := R [ X 1 , . . . , X n ] R [ X ] k real polynomials with degree less or equal to k R [ X ] = k real forms of degree k R [ X ] ∗ k := { L : R [ X ] k → R | L is R -linear } m � � R [ X ] 2 g 2 k := { i | m ∈ N 0 , g i ∈ R [ X ] k } i =0 The polynomial optimization problem Let f , p 1 , . . . , p m ∈ R [ X ] and m ∈ N 0 , � minimize f ( x ) ( P ) : x ∈ S := { y ∈ R n | p 1 ( y ) ≥ 0 , . . . , p m ( y ) ≥ 0 } s.t. : P ∗ := inf { f ( x ) | x ∈ S } ∈ {−∞} ∪ R ∪ {∞} S ∗ := { x ∗ ∈ S | for all x ∈ S , f ( x ∗ ) ≤ f ( x ) } 1/12
Strategy Polynomial Optimization Problem (POP) 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 2.Check if a matrix is generalized Hankel 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 k:=k+1 and 2.Check if a matrix is generalized Hankel go to step 1 No 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 k:=k+1 and 2.Check if a matrix is generalized Hankel go to step 1 No Yes: Step 3 3.Translate the solution from the space R [ X ] ∗ k to a set of points N ⊆ R n , via the truncated-GNS construction. 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 k:=k+1 and 2.Check if a matrix is generalized Hankel go to step 1 No Yes: Step 3 Truncated 3.Translate the solution from the space R [ X ] ∗ k to a set Moment of points N ⊆ R n , via the truncated-GNS construction. Problem 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 k:=k+1 and 2.Check if a matrix is generalized Hankel go to step 1 No Yes: Step 3 Truncated 3.Translate the solution from the space R [ X ] ∗ k to a set Moment of points N ⊆ R n , via the truncated-GNS construction. Problem Step 4 4. N ⊆ S and deg f < k ? 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 k:=k+1 and 2.Check if a matrix is generalized Hankel go to step 1 No Yes: Step 3 Truncated 3.Translate the solution from the space R [ X ] ∗ k to a set Moment of points N ⊆ R n , via the truncated-GNS construction. Problem Step 4 k:=k+1 and 4. N ⊆ S and deg f < k ? go to step 1 No 2/12
Strategy Polynomial Optimization Problem (POP) Step 1 1.Relaxation of the problem in the space R [ X ] ∗ k , for relatively small k . Solve a SDP problem and find a solution in R [ X ] ∗ k . Step 2 k:=k+1 and 2.Check if a matrix is generalized Hankel go to step 1 No Yes: Step 3 Truncated 3.Translate the solution from the space R [ X ] ∗ k to a set Moment of points N ⊆ R n , via the truncated-GNS construction. Problem Step 4 k:=k+1 and 4. N ⊆ S and deg f < k ? go to step 1 No Yes We have reached optimality and N contains minimizers of the original POP. 2/12
First attempt Linearize the polynomial optimization problem: X α := X α 1 1 · · · X α n �− → y α , new real variable n 3/12
First attempt Linearize the polynomial optimization problem: X α := X α 1 1 · · · X α n �− → y α , new real variable n Second attempt Add redundant inequalities and after linearize the polynomial opti- mization problem. 3/12
The 2d-truncated quadratic module Let p 1 , . . . , p m ∈ R [ X ] 2 d with d ∈ N 0 ∪ {∞} . We define the 2 d - truncated quadratic module generated by p 1 , . . . , p m as: � R [ X ] 2 � � � � � R [ X ] 2 p 1 M 2 d ( p 1 , . . . , p m ) := R [ X ] 2 d ∩ + R [ X ] 2 d ∩ � � � R [ X ] 2 p m + · · · + R [ X ] 2 d ∩ ⊆ R [ X ] 2 d 4/12
The 2d-truncated quadratic module Let p 1 , . . . , p m ∈ R [ X ] 2 d with d ∈ N 0 ∪ {∞} . We define the 2 d - truncated quadratic module generated by p 1 , . . . , p m as: � R [ X ] 2 � � � � � R [ X ] 2 p 1 M 2 d ( p 1 , . . . , p m ) := R [ X ] 2 d ∩ + R [ X ] 2 d ∩ � � � R [ X ] 2 p m + · · · + R [ X ] 2 d ∩ ⊆ R [ X ] 2 d The 2d-degree Lasserre relaxation Let p 1 , . . . , p m ∈ R [ X ] 2 d with d ∈ N 0 ∪ {∞} . The Lasserre relax- ation (or Moment relaxation) of ( P ) of degree 2 d is the following problem: minimize L ( f ) L ∈ R [ X ] ∗ subject to: 2 d ( P 2 d ) : L (1) = 1 and L ( M 2 d ( p 1 ,..., p m )) ⊆ R ≥ 0 4/12
The 2d-truncated quadratic module Let p 1 , . . . , p m ∈ R [ X ] 2 d with d ∈ N 0 ∪ {∞} . We define the 2 d - truncated quadratic module generated by p 1 , . . . , p m as: � R [ X ] 2 � � � � � R [ X ] 2 p 1 M 2 d ( p 1 , . . . , p m ) := R [ X ] 2 d ∩ + R [ X ] 2 d ∩ � � � R [ X ] 2 p m + · · · + R [ X ] 2 d ∩ ⊆ R [ X ] 2 d The 2d-degree Lasserre relaxation Let p 1 , . . . , p m ∈ R [ X ] 2 d with d ∈ N 0 ∪ {∞} . The Lasserre relax- ation (or Moment relaxation) of ( P ) of degree 2 d is the following problem: minimize L ( f ) L ∈ R [ X ] ∗ subject to: 2 d ( P 2 d ) : L (1) = 1 and L ( M 2 d ( p 1 ,..., p m )) ⊆ R ≥ 0 the optimal value of ( P 2 d ) is denoted by P ∗ 2 d ∈ {−∞} ∪ R ∪ {∞} . 4/12
Notation: r d := dim R [ X ] d Generalized Hankel matrix (or Moment matrix) Every matrix M ∈ R r d × r d indexed by a basis of R [ X ] d is called a generalized Hankel matrix (of order d ). 5/12
Notation: r d := dim R [ X ] d Generalized Hankel matrix (or Moment matrix) Every matrix M ∈ R r d × r d indexed by a basis of R [ X ] d is called a generalized Hankel matrix (of order d ). Example: n = 2 1 1 X 1 X 2 X 1 X 2 1 X 1 X 2 y (0 , 0) y (1 , 0) y (0 , 1) 1 1 X 2 X 1 X 1 X 2 − → y (1 , 0) y (2 , 0) y (1 , 1) X 1 X 1 1 X 2 X 2 X 1 X 2 y (0 , 1) y (1 , 1) y (0 , 2) X 2 X 2 2 A matrix of this form is a generalized hankel matrix (of order 2). 5/12
Notation: r d − 1 := dim R [ X ] d − 1 and s d := dim R [ X ] = d A Smul’jan result (1959) Let L ∈ R [ X ] ∗ 2 d be a feasible solution of ( P 2 d ). Set the Moment matrix associated to L: M L := ( L ( X α + β )) | α | , | β |≤ d Then there exists W ∈ R r d − 1 × s d and X ∈ R s d × s d such that: R [ X ] d − 1 R [ x ] = d � � A L A L W R [ X ] d − 1 M L = W T A L W T A L W + XX T R [ X ] = d 6/12
Notation: r d − 1 := dim R [ X ] d − 1 and s d := dim R [ X ] = d A Smul’jan result (1959) Let L ∈ R [ X ] ∗ 2 d be a feasible solution of ( P 2 d ). Set the Moment matrix associated to L: M L := ( L ( X α + β )) | α | , | β |≤ d Then there exists W ∈ R r d − 1 × s d and X ∈ R s d × s d such that: R [ X ] d − 1 R [ x ] = d � � A L A L W R [ X ] d − 1 M L = W T A L W T A L W + XX T R [ X ] = d Observation and definition Moreover: � � A L A L W � M L := � 0 W T A L W T A L W the modified Moment matrix associated to L is well defined, i.e it does not depend of the choice of W . 6/12
First result 2 d be an optimal solution of ( P 2 d ) and suppose � Let L ∈ R [ X ] ∗ M L is a generalized Hankel matrix. Then there are a 1 , . . . , a r ∈ R n pairwise different points and λ 1 > 0 , . . . ,λ r > 0 weights such that: r � L ( p ) = λ i p ( a i ) for all p ∈ R [ X ] 2 d − 1 i =1 where r = rank A L . 7/12
First result 2 d be an optimal solution of ( P 2 d ) and suppose � Let L ∈ R [ X ] ∗ M L is a generalized Hankel matrix. Then there are a 1 , . . . , a r ∈ R n pairwise different points and λ 1 > 0 , . . . ,λ r > 0 weights such that: r � L ( p ) = λ i p ( a i ) for all p ∈ R [ X ] 2 d − 1 i =1 where r = rank A L . Moreover if { a 1 , . . . , a r } ⊆ S and f ∈ R [ X ] 2 d − 1 then a 1 , . . . , a r are global minimizers of ( P ) and P ∗ = P ∗ 2 d = f ( a i ) for all i ∈ { 1 , . . . , r } . 7/12
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