An algorithm for semi-infinite polynomial optimization
aa r X i v : . [ m a t h . O C ] J a n AN ALGORITHM FOR SEMI-INFINITE POLYNOMIALOPTIMIZATION
J.B. LASSERRE
Abstract.
We consider the semi-infinite optimization problem: f ∗ := min x ∈ X { f ( x ) : g ( x , y ) ≤ , ∀ y ∈ Y x } , where f, g are polynomials and X ⊂ R n as well as Y x ⊂ R p , x ∈ X , arecompact basic semi-algebraic sets. To approximate f ∗ we proceed in two steps.First, we use the “joint+marginal” approach of the author [9] to approximatefrom above the function x Φ( x ) = sup { g ( x , y ) : y ∈ Y x } by a polynomialΦ d ≥ Φ, of degree at most 2 d , with the strong property that Φ d converges toΦ for the L -norm, as d → ∞ (and in particular, almost uniformly for somesubsequence ( d ℓ ), ℓ ∈ N ). Therefore, to approximate f ∗ one may wish to solvethe polynomial optimization problem f d = min x ∈ X { f ( x ) : Φ d ( x ) ≤ } viaa (by now standard) hierarchy of semidefinite relaxations, and for increasingvalues of d . In practice d is fixed, small, and one relaxes the constraint Φ d ≤ d ( x ) ≤ ǫ with ǫ >
0, allowing to change ǫ dynamically. As d increases, thelimit of the optimal value f ǫd is bounded above by f ∗ + ǫ . Introduction
Consider the semi-infinite optimization problem:(1.1) P : f ∗ := min x ∈ X { f ( x ) : g ( x , y ) ≤ , ∀ y ∈ Y x } , where X ⊂ R n , Y x ⊂ R p for every x ∈ X , and some functions f : R n → R , g : R n × R p : → R .Problem P is called a semi-infinite optimization problem because of the infin-itely many constraints g ( x , y ) ≤ y ∈ Y x (for each fixed x ∈ X ). It hasmany applications and particularly in robust control.In full generality P is a very hard problem and most methods aiming at com-puting (or at least approximating) f ∗ use discretization to overcome the difficultsemi-infinite constraint g ( x , y ) ≤ y ∈ Y x . Namely, in typical approacheswhere Y x ≡ Y for all x ∈ X (i.e. no dependence on x ), the set Y ⊂ R p is dis-cretized on a finite grid and if the resulting nonlinear programming problems Key words and phrases.
Polynomial optimization; min-max optimization; robust optimiza-tion; semidefinite relaxations.This work was performed during a visit in November 2010 at CRM (Centre de RecercaMatematica), a Mathematics center from UAB (Universidad Autonoma de Barcelona), and theauthor wishes to gratefully acknowledge financial support from CRM. re solved to global optimality, then convergence to a global optimum of thesemi-infinite problem occurs as the grid size vanishes (see e.g. the discussionand the many references in [10]). Alternatively, in [10] the authors provide lowerbounds on f ∗ by discretizing Y and upper bounds via convex relaxations of theinner problem max y ∈ Y { g ( x , y ) } ≤
0. In [11] the authors also use a discretiza-tion scheme of Y but now combined with a hierarchy of sum of squares convexrelaxations for solving to global optimality. Contribution.
We restrict ourselves to problem P where : • f, g are polynomials, and • X ⊂ R n and Y x ⊂ R p , x ∈ X , are compact basic semi-algebraic sets.For instance many problems of robust control can be put in this framework; seee.g. their description in [4]. Then in this context we provide a numerical schemewhose novelty with respect to previous works is to avoid discretization of the set Y x . Instead we use the “joint+marginal” methodology for parametric polynomialoptimization developed by the author in [9], to provide a sequence of polynomials(Φ d ) ⊂ R [ x ] (with degree 2 d , d ∈ N ) that approximate from above the functionΦ( x ) := max y { g ( x , y ) : y ∈ Y x } , and with the strong property that if d → ∞ then Φ d → Φ in the L -norm. (In particular, Φ d ℓ → Φ almost uniformly on X for some subsequence ( d ℓ ), ℓ ∈ N .) Then, ideally, one could solve the nestedsequence of polynomial optimization problems:(1.2) P d : f ∗ d = min { f ( x ) : Φ d ( x ) ≤ } , d = 1 , , . . . For fixed d , one may approximate (and often solve exactly) (1.2) by solvinga hierarchy of semidefinite relaxations, as defined in [6]. However, as the size O ( d n ) of these semidefinite relaxations increases very fast with d , in practice onerather let d be fixed, small, and relax the constraint Φ d ( x ) ≤ d ( x ) ≤ ǫ for some scalar ǫ > d increases, the resulting optimal value f ǫd is bounded above by f ∗ + ǫ . Theapproach is illustrated on a sample of small problems taken from the literature.2. Notation, definitions and preliminary results
Let R [ x ] (resp. R [ x , y ]) denote the ring of real polynomials in the variables x = ( x , . . . , x n ) (resp. x and y = ( y , . . . , y p )), whereas Σ[ x ] (resp. Σ[ x , y ])denote its subset of sums of squares.Let R [ y ] k ⊂ R [ y ] denote the vector space of real polynomials of degree at most k . For every α ∈ N n the notation x α stands for the monomial x α · · · x α n n andfor every d ∈ N , let N nd := { α ∈ N n : P j α j ≤ d } with cardinal s ( d ) = (cid:0) n + dn (cid:1) .Similarly N pd := { β ∈ N p : P j β j ≤ d } with cardinal (cid:0) p + dp (cid:1) . A polynomial f ∈ R [ x ] is written x f ( x ) = X α ∈ N n f α x α , nd f can be identified with its vector of coefficients f = ( f α ) in the canonicalbasis. For a real symmetric matrix A the notation A (cid:23) A is positivesemidefinite.A real sequence z = ( z α ), α ∈ N n , has a representing measure if there existssome finite Borel measure µ on R n such that z α = Z R n x α dµ ( x ) , ∀ α ∈ N n . Given a real sequence z = ( z α ) define the linear functional L z : R [ x ] → R by: f (= X α f α x α ) L z ( f ) = X α f α z α , f ∈ R [ x ] . Moment matrix.
The moment matrix associated with a sequence z = ( z α ), α ∈ N n , is the real symmetric matrix M d ( z ) with rows and columns indexedby N nd , and whose entry ( α, β ) is just z α + β , for every α, β ∈ N nd . If z has arepresenting measure µ then M d ( z ) (cid:23) h f , M d ( z ) f i = Z f dµ ≥ , ∀ f ∈ R s ( d ) . Localizing matrix.
With z as above and g ∈ R [ x ] (with g ( x ) = P γ g γ x γ ), the localizing matrix associated with z and g is the real symmetric matrix M d ( g z )with rows and columns indexed by N nd , and whose entry ( α, β ) is just P γ g γ z α + β + γ ,for every α, β ∈ N nd . If z has a representing measure µ whose support is containedin the set { x : g ( x ) ≥ } then M d ( g z ) (cid:23) h f , M d ( g z ) f i = Z f g dµ ≥ , ∀ f ∈ R s ( d ) . Definition 2.1 (Archimedean property) . A set of polynomials q j ∈ R [ x ], j =0 , . . . , p (with q = 1), satisfy the Archimedean property if the quadratic polyno-mial x M − k x k can be written in the form: M − k x k = p X j =0 σ j ( x ) q j ( x ) , for some sums of squares polynomials ( σ j ) ⊂ Σ[ x ].Of course the Archimedean property implies that the set D := { x ∈ R n : q j ( x ) ≥ , j = 1 , . . . , p } is compact. For instance, it holds whenever the levelset { x : q k ( x ) ≥ } is compact for some k ∈ { , . . . , p } , or if the q j ’s are affineand D is compact (hence a polytope). On the other hand, if D is compact then M − k x | ≥ x ∈ D and some M sufficiently large. So if one adds theredundant quadratic constraint x q p +1 ( x ) = M − k x k ≥ D then the Archimedean property holds. Hence it is not a restrictive assumption. et D := { x ∈ R n : q j ( x ) ≥ , j = 1 , . . . , p } , and given a polynomial h ∈ R [ x ],consider the hierarchy of semidefinite programs:(2.1) ( ρ ℓ = min z L z ( h )s.t. M ℓ ( z ) , M ℓ − v j ( q j z ) (cid:23) , j = 1 , . . . , p, where z = ( z α ), α ∈ N n ℓ , and v j = ⌈ (deg q j ) / ⌉ , j = 1 , . . . , p . Theorem 2.2 ([6, 8]) . Let a family of polynomials ( q j ) ⊂ R [ x ] satisfy theArchimedean property. Then as ℓ → ∞ , ρ ℓ ↑ h ∗ = min x { h ( x ) : x ∈ D } .Moreover, if z ∗ is an optimal solution of (2.1) and (2.2) rank M ℓ ( z ∗ ) = rank M ℓ − v ( z ∗ ) (=: r ) (where v = max j v j ) then ρ ℓ = h ∗ and one may extract r global minimizers x ∗ k ∈ D , k = 1 , . . . , r . The size (resp. the number of variables) of the semidefinite program (2.1)grows as (cid:0) n + ℓn (cid:1) (resp. as (cid:0) n +2 ℓn (cid:1) ) and so becomes rapidly prohibitive, especiallyin view of the present status of available semidefinite solvers. Therefore, andeven though practice reveals that convergence is fast and often finite, so far, theabove methodology is limited to small to medium size problems (typically, anddepending on the degree of the polynomials appearing in the data, problems withup to n ∈ [10 ,
20] variables). However, for larger size problems with sparsity inthe data and/or symmetries, adhoc and tractable versions of (2.1) exist. See forinstance the sparse version of (2.1) proposed in [12], and whose convergence wasproved in [7] when the sparsity pattern satifies the so-called running intersectionproperty . In [12] this technique was shown to be successful on a sample of nonconvex problems with up to 1000 variables.3.
Main result
Let B ⊂ R n be a simple set like a box or an ellipsoid. Let p s ∈ R [ x ], s =1 , . . . , sx , and h j ∈ R [ x , y ], j = 1 , . . . , m , be given polynomials and let X ⊂ R n be the basic semi-algebraic set X := { x ∈ R n : p s ( x ) ≥ , s = 1 , . . . , sx } . Next, for every x ∈ R n , let Y x ⊂ R p be the basic semi-algebraic set described by:(3.1) Y x = { y ∈ R p : h j ( x , y ) ≥ , j = 1 , . . . , m } , and with B ⊇ X , let K ⊂ R n × R p be the set(3.2) K := { ( x , y ) ∈ R n + p : x ∈ B ; h j ( x , y ) ≥ , j = 1 , . . . , m } . Observe that problem P in (1.1) is equivalent to: P : f ∗ = min x ∈ X { f ( x ) : Φ( x ) ≤ } (3.3) where Φ( x ) = max y { g ( x , y ) : y ∈ Y x } , x ∈ B . (3.4) emma 3.1. Let K ⊂ R n + p in (3.2) be compact and assume that for every x ∈ B ⊂ R n , the set Y x defined in (3.1) is nonempty. Then Φ is upper semicon-tinuous (u.s.c.) on B . Moreover, if there is some compact set Y ⊂ R p such that Y x = Y for every x ∈ B , then Φ is continuous on B .Proof. Let x ∈ B be fixed, arbitrary, and let ( x k ) k ∈ N ⊂ B be a sequence thatconverges to x and such thatlim sup x → x Φ( x ) = lim k →∞ Φ( x k ) . As K is compact then so is Y x for every x ∈ B . Therefore, as Y x = ∅ forall x ∈ B and g is continuous, there exists an optimal solution y ∗ k ∈ Y x k forevery k . By compactness there exist a subsequence ( k ℓ ) and y ∗ ∈ R p such that( x k ℓ , y ∗ k ℓ ) → ( x , y ∗ ) ∈ K , as ℓ → ∞ . Hencelim sup x → x Φ( x ) = lim k →∞ Φ( x k )= lim k →∞ g ( x k , y ∗ k ) = lim ℓ →∞ g ( x k ℓ , y ∗ k ℓ )= g ( x , y ∗ ) ≤ Φ( x ) , which proves that Φ is u.s.c. at x . As x ∈ B was arbitrary, Φ is u.s.c. on B .Next, assume that there is some compact set Y ⊂ R p such that Y x = Y forevery x ∈ B . Let x ∈ B be fixed arbitrary with Φ( x ) = g ( x , y ∗ ) for some y ∗ ∈ Y . Let ( x n ) ⊂ B , n ∈ N , be a sequence such that x n → x as n → ∞ ,and Φ( x ) ≥ lim inf x → x Φ( x ) = lim n →∞ Φ( x n ). Again, let y ∗ n ∈ Y be such that Φ( x n ) = g ( x n , y ∗ n ), n ∈ N . By compactness, consider an arbitrary converging subsequence( n ℓ ) ⊂ N , i.e., such that ( x n ℓ , y ∗ n ℓ ) → ( x , y ∗ ) ∈ K as ℓ → ∞ , for some y ∗ ∈ Y .Suppose that Φ( x ) (= g ( x , y ∗ )) > g ( x , y ∗ ), say Φ( x ) > g ( x , y ∗ ) + δ for some δ >
0. By continuity of g , g ( x n ℓ , y ∗ n ℓ ) < g ( x , y ∗ ) + δ/ ℓ > ℓ (for some ℓ ). But again, by continuity, | g ( x n ℓ , y ∗ ) − g ( x , y ∗ ) | < δ/ ℓ > ℓ (forsome ℓ ). And so we obtain the contradictionΦ( x n ℓ ) ≥ g ( x n ℓ , y ∗ ) > Φ( x ) − δ/ x n ℓ ) = g ( x n ℓ , y ∗ n ℓ ) < Φ( x ) − δ/ , whenever ℓ > max[ ℓ , ℓ ]. Therefore, g ( x , y ∗ ) = g ( x , y ∗ ) and so, g ( x , y ∗ ) = Φ( x ) = g ( x , y ∗ ) = lim ℓ →∞ Φ( x n ℓ ) = lim inf x → x Φ( x ) ≤ Φ( x ) , which combined with Φ being u.s.c., yields that Φ is continuous at x . (cid:3) We next explain how to • approximate the function x Φ( x ) on B by a polynomial, and • evaluate (or at least approximate) Φ( x ) for some given x ∈ B , to checkwhether Φ( x ) ≤ .1. Certificate of Φ( x ) ≤ . For every x ∈ X fixed, let g x , h x j ∈ R [ y ] be thepolynomials y g x ( y ) = g ( x , y ) and y h x j ( y ) := h j ( x , y ), j = 1 , . . . , m , andconsider the hierarchy of semidefinite programs:(3.5) Q ℓ ( x ) : ( ρ ℓ ( x ) = max z L z ( g x )s.t. M ℓ ( z ) , M ℓ − v j ( h x j z ) (cid:23) , j = 1 , . . . , m, where z = ( z β ), β ∈ N p ℓ , and v j = ⌈ (deg h x j ) / ⌉ , j = 1 , . . . , m . Obviously one has ρ ℓ ( x ) ≥ Φ( x ) for every ℓ , and Corollary 3.2.
Let x ∈ X and assume that the polynomials ( h x j ) ⊂ R [ y ] satisfythe Archimedean property. Then: (a) As ℓ → ∞ , ρ ℓ ( x ) ↓ Φ( x ) = max { g ( x , y ) : y ∈ Y x } . In particular, if ρ ℓ ( x ) ≤ for some ℓ , then Φ( x ) ≤ . (b) Moreover, if z ∗ is an optimal solution of (3.5) that satisfies rank M ℓ ( z ∗ ) = rank M ℓ − v ( z ∗ ) (=: r ) , (where v := max j v j ), then ρ ℓ ( x ) = Φ( x ) and there are r global maximizers y ( k ) ∈ Y x , k = 1 , . . . , r . Corollary 3.2 is a direct consequence of Theorem 2.2.3.2.
Approximating the function Φ . Recall that B ⊇ X is a simple set likee.g., a simplex, a box or an ellipsoid and let µ be the finite Borel probabilitymeasure uniformly distributed on B . Therefore, the vector γ = ( γ α ), α ∈ N n , ofmoments of µ , i.e., γ α := Z B x α dµ ( x ) , α ∈ N n , can be computed easily. For instance, in the sequel we assume that B = [ − , n = { x : θ i ( x ) ≥ , i = 1 , . . . n } with θ i ∈ R [ x , y ] being the polynomial ( x , y ) θ i ( x , y ) := 1 − x i , i = 1 , . . . , n .Observe that the function Φ is defined in (3.4) via a parametric polynomialoptimization problem (with x being the parameter vector). Therefore, following[9], let r j = ⌈ (deg h j ) / ⌉ , j = 1 , . . . , m , and consider the hierarchy of semidefiniterelaxations indexed by d ∈ N :(3.6) ρ d = max z L z ( g )s.t. M d ( z ) , M d − r j ( h j z ) (cid:23) , j = 1 , . . . , m M d − ( θ i z ) (cid:23) , i = 1 , . . . , nL z ( x α ) = γ α , α ∈ N n d , here the sequence z is now indexed in N n + p d , i.e., z = ( z αβ ), ( α, β ) ∈ N n + p d .Writing g ≡
1, the dual of the semidefinite program (3.6) reads(3.7) ρ ∗ d = min q,σ j ,θ i Z B q ( x ) dµ ( x )s.t. q ( x ) − g ( x , y ) = m X j =0 σ j ( x , y ) h j ( x , y ) + n X i =1 ψ i ( x , y ) θ i ( x , y ) q ∈ R [ x ] d , σ j , ψ i ∈ Σ[ x , y ]deg σ j h j ≤ d, j = 0 , . . . , m. deg ψ i θ i ≤ d, i = 1 , . . . , n. It turns out that any optimal solution of the semidefinite program (3.7) permitsto approximate Φ in a strong sense.
Theorem 3.3 ([9]) . Let K ⊂ R n + p in (3.2) be compact. Assume that the polyno-mials h j , θ i ∈ R [ x , y ] satisfy the Archimedean property and assume that for every x ∈ B , the set Y x defined in (3.1) is nonempty. Let Φ d ∈ R [ x ] d be an optimalsolution of (3.7). Then : (a) Φ d ≥ Φ and as d → ∞ , (3.8) Z B (Φ d ( x ) − Φ( x )) dµ ( x ) = Z B | Φ d ( x ) − Φ( x ) | dµ ( x ) → , that is, Φ d → Φ for the L ( B , µ ) -norm . (b) There is a subsequence ( d ℓ ) , ℓ ∈ N , such that Φ d ℓ → Φ , µ -almost uniformly in B , as ℓ → ∞ . The proof of (a) can be found in [9], whereas (b) follows from (a) and [1, The-orem 2.5.3].3.3.
An algorithm.
The idea behind the algorithm is to approximate P in (1.1)with the polynomial optimization problem: ( P ǫd ):(3.9) P ǫd : f ǫd = min x ∈ X { f ( x ) : Φ d ( x ) ≤ ǫ } , d = 1 , , . . . with d ∈ N , ǫ > d as in Theorem 3.3, for every d = 1 , . . . .Obviously, for ǫ = 0 one has f d ≥ f ∗ for all d because by definition Φ d ≥ Φfor every d ∈ N . However, it may happen that P d has no solution. Next, if x ∗ is an optimal solution of P and Φ( x ∗ ) <
0, it may also happen that Φ d ( x ∗ ) > d is not large enough. This is why one needs to relax the constraint Φ ≤ d ≤ ǫ for some ǫ >
0. However, in view of Theorem 3.3, one expects that f ǫd ≈ f ∗ provided that d and ǫ are sufficiently large and small, respectively. Andindeed: L ( B , µ ) is the Banach space of µ -integrable functions on B , with norm k f k = R B | f | dµ . If one fixes ǫ > A ∈ B ( B ) such that µ ( A ) < ǫ and Φ d ℓ → Φuniformly on B \ A , as ℓ → ∞ . heorem 3.4. Assume that X is the closure of an open set. Let ǫ ≥ be fixed,arbitrary and with f ǫd be as in (3.9), let x ǫd ∈ X be any optimal solution of (3.9)(including the case where ǫ = 0 ), and let ˜ f ǫd := min { f ǫℓ : ℓ = 1 , . . . , d } = f ( x ǫℓ ( d ) ) for some ℓ ( d ) ∈ { , . . . , d } . (a) If ǫ > there exists d ǫ ∈ N such that for every d ≥ d ǫ , f ( x ǫℓ ( d ) ) < f ∗ + ǫ . (b) If there is an optimal solution x ∗ ∈ X of (1.1) such that Φ( x ∗ ) < , thenthere exists d ∈ N such that for every d ≥ d , f ∗ ≤ f ( x ℓ ( d ) ) < f ∗ + ǫ .Proof. (a) With ǫ > x ∗ ǫ ∈ X be such that Φ( x ∗ ǫ ) ≤ f ( x ∗ ǫ ) < f ∗ + ǫ/
2. We may assume that x ∗ ǫ is not on the boundary of X . Let O ǫ := { x ∈ int X : Φ( x ) < ǫ/ } which is an open set because Φ is u.s.c. (byLemma 3.1), and so µ ( O ǫ ) >
0. Next, as f is continuous, there exists ρ > f < f ∗ + ǫ whenever x ∈ O ǫ := { x ∈ int X : k x − x ∗ ǫ k < ρ } . Observethat ρ := µ ( O ǫ ∩ O ǫ ) > O ǫ ∩ O ǫ is an open set (with x ∗ ǫ ∈ O ǫ ∩ O ǫ ).Next, by Theorem 3.3(b), there is a subsequence ( d ℓ ), ℓ ∈ N , such that Φ d ℓ → Φ, µ -almost uniformly on B . Hence, there is some Borel set A ǫ ⊂ B , and integer ℓ ǫ ∈ N , such that µ ( A ǫ ) < ρ/ x ∈ X \ A ǫ | Φ( x ) − Φ d ℓ ( x ) | < ǫ/ ℓ ≥ ℓ ǫ . Inparticular, as µ ( A ǫ ) < ρ/ < µ ( O ǫ ∩ O ǫ ), the set ∆ ǫ := ( O ǫ ∩ O ǫ ) \ A ǫ has positive µ -measure. Therefore, f ( x ) < f ∗ + ǫ and Φ d ℓ ( x ) < ǫ whenever ℓ ≥ ℓ ǫ and x ∈ ∆ ǫ ,which in turn implies f ǫd ℓ < f ∗ + ǫ , and consequently, ˜ f ǫd = f ( x ǫℓ ( d ) ) < f ∗ + ǫ , thedesired result.(b) Let ǫ ′ := − Φ( x ∗ ), and let O ǫ ′ := { x ∈ int X : Φ( x ) < − ǫ ′ / } which is anonempty open set because it contains x ∗ and Φ is u.s.c.. Let O ǫ ′ be as O ǫ inthe proof of (a), but now with x ∗ ǫ ′ = x ∗ ∈ X . Both O ǫ ′ and O ǫ ′ are open andnonempty because they contain x ∗ . The rest of the proof is like for the proof of(a), but noticing that now for every x ∈ ∆ ǫ ′ one has Φ d ℓ ( x ) < − ǫ ′ / ǫ ′ / x is feasible for (3.9) with ǫ = 0. Next, by feasiblity f ( x ) ≥ f ∗ since theresulting feasible set in (3.9) is smaller than that of (1.2) because Φ d ≥ Φ, forall d . And so f ∗ ≤ f ( x ) < f ∗ + ǫ whenever x ∈ ∆ ǫ , and ℓ ≥ ℓ ǫ , from which (b)follows. (cid:3) Theorem 3.4 provides a rationale behind the algorithm that we present below.In solving (3.9) with d sufficiently large and small ǫ (or even ǫ = 0), f ǫd wouldprovide a good approximation of f ∗ . But in principle, computing the globaloptimum f ǫd is still a difficult problem. However, P ǫd is a polynomial optimizationproblem. Therefore, by Theorem 2.2, if the polynomials ( p s ) ⊂ R [ x ] that define X satisfy the Archimedean property (see Definition 2.1) we can approximate f ǫd from below, as closely as desired, by a monotone sequence ( f ǫdt ), t ∈ N , obtained y solving the hierarchy of semidefinite relaxations (2.1), which here read:(3.10) f ǫdt = min z L z ( f )s.t. M t ( z ) , M t − d ( ǫ − Φ d z ) (cid:23) M t − t s ( p s z ) (cid:23) , s = 1 , . . . , sx, where t s = ⌈ (deg p s ) / ⌉ , s = 1 , . . . , sx . Corollary 3.5.
Assume that the polynomials ( p s ) ⊂ R [ x ] satisfy the Archimedeanproperty. Then f ǫdt ↑ f ǫd as t → ∞ . Moreover, if z ∗ is an optimal solution of (3.10)and (3.11) rank M t ( z ∗ ) = rank M t − t ( z ∗ ) (=: r ) (where t := max[ d, max s [ t s ]] ) then f ǫdt = f ǫd and one may extract r global mini-mizers x ∗ d ( k ) ∈ X , k = 1 , . . . , r . That is, for every k = 1 , . . . , r , f ( x ∗ d ( k )) = f ǫd and Φ d ( x ∗ d ( k )) ≤ ǫ . However, given a minimizer x ∗ d ∈ X , if on the one hand Φ d ( x ∗ d ) ≤ ǫ , on the otherhand it may not satisfy Φ( x ∗ d ) ≤
0. (Recall that checking whether Φ( x ∗ d ) ≤ Q ℓ ( x ) in (3.5) with x := x ∗ d .) Ifthis happens then one solves again (3.10) for a smaller value of ǫ , etc., until oneobtains some x ∗ d ∈ X with Φ( x ∗ d ) ≤ d is relatively large, the size of semidefiniterelaxations (3.10) to compute f ǫdt becomes too large for practical implementation(as one must have t ≥ d ). So in practice one let d be fixed at a small value,typically the smallest possible value of d , i.e., 1 (Φ d is quadratic) or 2 (Φ d isquartic)), and one updates ǫ as indicated above. So the resulting algorithmreads: Algorithm.Input: ℓ, d, k ∗ ∈ N , ǫ > ǫ := 10 − ), d ∈ N , ˜ x := ⋆ , f ( ⋆ ) = + ∞ . Output: f ( x ∗ d ) with x ∗ d ∈ X and Φ( x ∗ d ) ≤ k = 1 and ǫ ( k ) = 1.Step 2: While k ≤ k ∗ , solve P ǫ ( k ) d in (3.9) → x ∗ k ∈ X .Step 3: Solve Q ℓ ( x ∗ k ) in (3.5) → ρ ℓ ( x ∗ k ).If − ǫ ≤ ρ ℓ ( x ∗ k ) ≤ x ∗ d := x ∗ k and STOP.If ρ ℓ ( x ∗ k ) < − ǫ then: • if f (˜ x ) > f ( x ∗ k ) then set ˜ x := x ∗ k . If k = k ∗ then x ∗ d := x ∗ k . • set ǫ ( k + 1) := 2 ǫ ( k ), k := k + 1 and go to Step 2.If ρ ℓ ( x ∗ k ) > • If k < k ∗ set ǫ ( k + 1) := ǫ ( k ) / k := k + 1 and go to Step 2. • If k = k ∗ then set set x ∗ d = ˜ x .Observe that in Step 2 of the above algorithm, one assumes that by solving P ǫ ( k ) d one obtains x ∗ k ∈ X . .4. Numerical experiments.
We have taken Examples 2 , ,
9, K, M, N, allfrom Bhattacharjee et al. [2, Appendix A] and whose data are polynomials, ex-cept for problem L. For the latter problem, the non-polynomial function x min[0 , ( x − x )] is semi-algebraic and can be generated by introducing an addi-tional variable x , with the polynomial constraints: x = ( x − x ) ; x ≥ . Indeed, 2 min[0 , ( x − x )] = x − x − x .Although these examples are quite small, they are still non trivial (and evendifficult) to solve, and we wanted to test the above methodology with smallrelaxation order d . In fact we have even considered the smallest possible d , i.e., d = 1 (Φ d is quadratic). Results in Table 1 are quite good since by using thesemidefinite relaxation of minimal order “ d ” one obtains an optimal value f ∗ d quite close to f ∗ , at the price of updating ǫ several times.Next, for Problem L, if we now increase d to d = 2, we improve the opti-mal value which becomes f ∗ d = 0 . ǫ = 2 .
2. However, for Problem M,increasing d does not improve the optimal value. est known value f ∗ d final value of ǫ problem 2 0.194 0.198 1.895problem 7 1.0 1.41 5problem 9 -12.0 -14.47 ∗ − Table 1.
Examples of [2, Table 6.1] with minimal d Conclusion
We have presented an algorithm for semi-infinite (global) polynomial optimiza-tion whose novelty with respect to previous works is to not rely on a discretizationscheme. Instead, it uses a polynomial approximation Φ d of the function Φ, ob-tained by solving some semidefinite relaxation attached to the “joint+marginal”approach developed in [9] for parametric optimization, which guarantees (strong)convergence Φ d → Φ in L -norm. Then for fixed d , one has to solve a polynomialoptimization problem, which can be done by solving an appropriate hierarchyof semidefinite relaxations. Of course, as already mentioned and especially inview of the present status of semidefinite solvers, so far the present methodologyis limited to small to medium size problems, unless sparsity in the data and/orsymmetries are taken into account appropriately, as described in e.g. [7, 12].Preliminary results on non trivial (but small size) examples are encouraging. References [1] R.B. Ash.
Real Analysis and Probability . Academic Press Inc., Boston (1972)[2] B. Bhattacharjee, W.H. Green Jr., P.I. Barton. Interval methods for semi-infiniteprograms, Comput. Optim. and Appl. , 63–93 (2005).[3] A. Ben-Tal, S. Boyd, A. Nemirovski. Extending Scope of Robust Optimization: Com-prehensive Robust Counterparts of Uncertain Problems, Math. Program. S´er. B ,63–89 (2006).[4] G.C. Calafiore, F. Dabbene. A probabilistic analytic center cutting plane method forfeasibility of uncertain LMIs, Automatica , 2022–2033 (2007)[5] D. Henrion, J.B. Lasserre, J. Lofberg. GloptiPoly 3: moments, optimization andsemidefinite programming, Optim. Methods and Softwares , 761–779 (2009).[6] J.B. Lasserre. Global optimization with polynomials and the problem of moments.SIAM J. Optim. , 796–817 (2001)[7] J.B. Lasserre. Convergent SDP-relaxations in polynomial optimization with sparsity,SIAM J. Optim. , 822–843 (2006)[8] J.B. Lasserre. Moments, Positive Polynomials and Their Applications , Imperial Col-lege Press, London (2009)[9] J.B. Lasserre. A “joint+marginal” approach to parametric polynomial optimization.SIAM J. Optim. , 1995–2022 (2010)
10] A. Mitsos, P. Lemonidis, Cha Kun Lee, P.I. Barton. Relaxation-based bounds forsemi-infinite programs, SIAM J. Optim. , 77–113 (2008)[11] P. Parpas, B. Rustem. An algorithm for the global optimization of a class of contin-uous minimax problems, J. Optim. Theor. Appl. , 461–473 (2009)[12] H. Waki, S. Kim, M. Kojima, M. Maramatsu. Sums of squares and semidefinite pro-gramming relaxations for polynomial optimization problems with structured sparsity,SIAM J. Optim. , 218–242 (2006). J.B. Lasserre: LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 ToulouseC´edex 4, France.
E-mail address : [email protected]@laas.fr