Moments, Sums of Squares and Semidefinite Programming Jean B. - PowerPoint PPT Presentation

Moments, Sums of Squares and Semidefinite Programming Jean B. LASSERRE LAAS-CNRS, and Institute of Mathematics, Toulouse, France ICMS- 2006, Castro-Urdiales, September 2006 1

• Semidefinite Programming • the Generalized Problem of Moments (GPM) • Some applications • Duality between moments and nonnegative polynomials • SDP-relaxations for the basic GPM. • s.o.s. vs nonnegative polynomials. Alternative SDP-relaxations • How to handle sparsity 2

Semidefinite Programming Consider the optimization problems n � x ∈ R n { c ′ x | → min A i x i � b } , P i =1 P ∗ → max { � b , Y � | Y � 0; � A i , Y � = c i , i = 1 , . . . , n } Y ∈S m • c ∈ R n whereas b, A i , Y ∈ S m are m × m symmetric matrices. • Y � 0 means Y semidefinite positive; � A, B � = trace ( AB ). P and its dual P ∗ are convex problems that are solvable in poly- nomial time to arbitrary precision ǫ > 0. = generalization to the convex cone S + m ( X � 0) of Linear Pro- gramming on the convex polyhedral cone R m + ( x ≥ 0). 3

• weak duality: � b , Y � ≤ c ′ x for all feasible x ∈ R n , Y ∈ S m . • strong duality: under “Slater interior point condition” n � ∃ x ∈ R n , Y ≻ 0; A i x i ≻ b ; � A i , Y � = c i i = 1 , . . . , n. i =1 Then there is no duality gap and sup P ∗ = max P ∗ = min P = inf P ∗ Several academic SDP software packages exist, (e.g. MATLAB “LMI toolbox”, SeduMi, SDPT3, ...). However, so far, size limitation is more severe than for LP software packages. Pioneer contributions by A. Nemirovsky, Y. Nesterov, N.Z. Shor, B.D. Yudin,... 4

The generalized problem of moments (GPM) � � = µ ∈ M ( K ) { min f 0 dµ | f j dµ ≥ b j , j = 1 , . . . , p } with K ⊆ R n and M ( K ) a convex set of finite Borel measures on K . We even consider the more general GPM � � = � � µ i ∈ M ( K i ) { min | ≥ b j , j = 1 , 2 , . . . } f oi dµ i f ji dµ i i ∈ I i ∈ I where for all i ∈ I , K i ⊆ R n i and M ( K i ) is a convex set of finite Borel measures on K i . The index set I may be countable. 5

• GPM has great modelling power, in various fields. Global Optimization (continuous, discrete), Control (Robust and optimal control), Nonlinear Equations, Probability and Statistics, Performance Evaluation (in e.g. Mathematical finance, Markov chains), Inverse Problems (cristallography, tomography), Numer- ical multivariate Integration, etc ... • GPM is a useful theoretical tool to prove existence and char- acterization of optimal solutions. • BUT ... in full generality .... GPM is unsolvable numerically. HOWEVER ... if the K i , ( ⊂ R n i ) are basic semi-algebraic sets and the f ij are polynomials (or even piecewise polynomials), then ... by using results of real algebraic geometry and on the problem of moments, one may now define efficient numerical approxima- tion chemes, based on Semidefinite Programming (SDP). 6

• Semidefinite Programming • the Generalized Problem of Moments (GPM) • Some applications • Duality between moments and nonnegative polynomials • SDP-relaxations for the basic GPM • s.o.s. vs nonnegative polynomials. Alternative SDP-relaxations • How to handle sparsity 7

A few examples: PROBLEM 1: Probability: Let K ⊆ R n , S ⊂ K be Borel subsets, and Γ ⊂ N n , Finding an upper bound (if possible optimal) on Prob ( X ∈ S ), the probability that a K -valued random variable X ∈ S , given some of its moments γ = { γ α } , α ∈ Γ ⊂ N n .... .... is equivalent to solving: � X α dµ = γ α , sup { µ ( S ) | α ∈ Γ } µ ∈ M ( K ) • M ( K ) is the (convex) set of probability measures on K ⊆ R n . • f α ≡ X α , α ∈ Γ (polynomial); f 0 = I S (piecewise polynomial) 8

PROBLEM 2: Moments problems in financial economics: Under no arbitrage, the price of an European Call Option with strike k , is given by E [( X − k ) + ] where E is the expectation operator w.r.t. the distribution of the underlying asset X . Hence, finding an ( optimal ) upper bound on the price of a Eu- ropean Call Option with strike k , given the first p moments { γ j } , reduces to solving: � � ( X − k ) + dµ X j dµ = γ j , sup { | j = 1 , . . . , p } µ ∈ M ( K ) with K = R + , and M ( K ) the set of probability measures on K . f j ≡ X j (polynomials), and f 0 ≡ ( X − k ) + (piecewise polynomial) 9

PROBLEM 3: Global Optimization: Let K ⊆ R n , f : R n → R , and consider the optimization problem f ∗ := inf x { f ( x ) | x ∈ K } with f ∗ being the global minimum. Finding f ∗ is equivalent to solving � inf f dµ µ ∈ M ( K ) with M ( K ) being the set of probability measures on K . 10

Important particular case : Solving Polynomial Equations K := { x ∈ R n : g j ( x ) = 0 , j = 1 , . . . , m } with g j ∈ R [ X 1 , . . . , X n ] for all j = 1 , . . . , m . Finding a solution x ∗ ∈ K that minimizes f ( X ) on K is equivalent to solving � inf f dµ µ ∈ M ( K ) with M ( K ) being the set of probability measures on K . 11

PROBLEM 4: Convex envelope: Let K ⊆ R n , f : K → R (= + ∞ outside K ) and with x fixed, consider the optimization problem � � � f ( x ) := µ ∈ M ( K ) { inf f dµ | X j dµ = x j , j = 1 , . . . , n } with M ( K ) being the set of probability measures on K . � f is convex and is the convex envelope of f , defined on the convex hull co ( K ) of K . 12

PROBLEM 5: Measures with given marginals: Let K j ⊂ R n j , j = 1 , . . . , p , and K := K 1 × K 2 · · · × K p ⊂ R n , and with natural projections π j : K : → K j , j = 1 , . . . , p . Let ν j be a given Borel measure on K j , j = 1 , . . . , p , For a measure µ on K , denote π j µ its marginal on K j , i.e. π j µ ( B ) := µ ( π − 1 ( B )) = µ ( { x ∈ K : π j x ∈ B } ) , B ∈ B ( K j ) j � µ ∈ M ( K ) { inf f dµ | π j µ = ν j , j = 1 , . . . , p } with M ( K ) being the set of finite Borel measures on K . Generalization of the famous Monge-Kantorovich transportation problem, with many other interesting applications, particularly in Probability. 13

•• If K j is compact then the constraint on marginal π j µ = ν j is equivalent to the countably many linear equalities � � X α dµ X α dν j , ∀ α ∈ N n j = between moments of µ and ν j ... because the space of polynomials is dense (for the sup-norm) in the space C ( K j ) of continuous functions on K j . 14

PROBLEM 6: Deterministic Optimal Control: � T j ∗ := min 0 h ( s, x ( s ) , u ( s )) ds + H ( x ( T )) u x ( s ) ˙ = f ( s, x ( s ) , u ( s )) , s ∈ [0 , T ] (1) ( x ( s ) , u ( s )) ∈ X × U, s ∈ [0 , T ) ∈ x ( T ) X T , and with initial condition x (0) = x 0 ∈ X , and - X, X T ⊂ R n and U ⊂ R m are basic semi-algebraic sets. - h, f ∈ R [ t, x, u ] , H ∈ R [ x ] 15

Let u = { u ( t ) , 0 ≤ t < T } be an admissible control. Introduce the probability measure ν u on R n , and the measure µ u on [0 , T ] × R n × R m , defined by ν u ( B ) := I B [ x ( T )] , B ∈ B n � µ u ( A × B × C ) := [0 ,T ] ∩ A I B × C [( x ( s ) , u ( s ))] ds, for all hyper-rectangles ( A, B, C ). The measure µ u is called the occupation measure of the state- action (deterministic) process ( s, x ( s ) , u ( s )) up to time T , whereas ν u is the occupation measure of the state x ( T ) at time T . 16

• Observe that for an admissible trajectory ( s, x ( s ) , u ( s )) x ( t ) = f ( t, x ( t ) , u ( t )) , ˙ t ∈ [0 , T ) implies that for suitable g : [0 , T ] × X → R , the time integration � T ∂g ( s, x ( s )) + ∂g ( s, x ( s )) g ( x ( T )) = g (0 , x (0))+ f ( s, x ( s ) , u ( s )) ds ∂t ∂x 0 is equivalent to the spatial integration � ∂g � � � ∂t + ∂g g T dν u = g (0 , x 0 ) + dµ u ∂x f [0 ,T ] × X × U X T with g T ( x ) := g ( T, x ) for all x . 17

� T • Similarly, the criterion 0 h ( s, x ( s ) , u ( s )) ds + H ( x ( T )) reads � � H dν u + [0 ,T ] × X × U h dµ u = L y ( H ) + L z ( h ) . X T The so-called weak formulation is the infinite-dimensional LP  � �  ρ ∗ = min µ,ν  H dν + h dµ    � ∂g  �  ∂t + ∂g    s.t. g T dν − ∂x f dµ = g (0 , x 0 ) , ∀ g ∈ R [ t, x ]       µ : measure supported on [0 , T ] × X × U     ν : prob. measure supported on X T • Theorem: [R. Vinter]. If X, X T , U are compact, f ( s, x, U ) is convex for all ( s, x ) ∈ [0 , T ] × X , and h, H are convex, then ρ ∗ = j ∗ . 18

• Semidefinite Programming • the Generalized Problem of Moments (GPM) • Some applications • Duality between moments and nonnegative polynomials • SDP-relaxations for the basic GPM • s.o.s. vs nonnegative polynomials. Alternative SDP-relaxations • How to handle sparsity 19

Duality With M ( K ) the space of Borel prob. measures on K , the GPM � � µ ∈ M ( K ) { min f 0 dµ | f j dµ = b j , j = 1 , . . . , p } is the infinite-dimensional LP µ ∈ M { � f 0 , µ � | min � f j , µ � = b j , j = 1 , . . . , p ; � 1 , µ � = 1; µ ≥ 0 } where M is the vector space of finite signed Borel measures on K . The dual LP reads: p � λ ∈ R p ,γ ∈ R { max | f 0 − λ j ( f j − b j ) ≥ γ on K } γ j =1 20