SDP Hierarchies for Approximate Optimal Designs

We introduce a new approach aiming at computing approximate optimal
designs for multivariate polynomial regressions on compact (semialgebraic)
design spaces. We use the moment-sum-of-squares hierarchy of semidefinite
programming problems to solve numerically the approximate optimal design
problem. The geometry of the design is recovered via semidefinite programming
duality theory. This article shows that the hierarchy converges to the
approximate optimal design as the order of the hierarchy increases. Furthermore,
we provide a dual certificate ensuring finite convergence of the hierarchy
and showing that the approximate optimal design can be computed
numerically with our method. As a byproduct, we revisit the equivalence theorem
of the experimental design theory: it is linked to the Christoffel polynomial
and it characterizes finite convergence of the moment-sum-of-square
hierarchies.

Transcript

1. Yohann DE CASTRO (CERMICS) SDP Hierarchies for Approximate Optimal Designs SPO, 14th January, 2019 SDP Hierarchies for Approximate Optimal Designs
2. Yohann DE CASTRO Based on a joint work with… 2 F. Gamboa R. Hess D. Henrion J.-B. Lasserre
3. Yohann DE CASTRO Based on a joint work with… 2 F. Gamboa R. Hess D. Henrion J.-B. Lasserre
4. Yohann DE CASTRO Outline 3
5. Yohann DE CASTRO Outline 3 1. Optimal Design Theory
6. Yohann DE CASTRO Outline 3 1. Optimal Design Theory 2. Problem Reformulation as Convex Program
7. Yohann DE CASTRO Outline 3 1. Optimal Design Theory 2. Problem Reformulation as Convex Program 3. SDP Hierarchies
8. Yohann DE CASTRO Outline 3 1. Optimal Design Theory 2. Problem Reformulation as Convex Program 3. SDP Hierarchies 4. Equivalence Theorems
9. Yohann DE CASTRO Outline 3 1. Optimal Design Theory 2. Problem Reformulation as Convex Program 3. SDP Hierarchies 4. Equivalence Theorems 5. Christoffel Polynomial
10. Yohann DE CASTRO Optimal Design Theory 4 zi = p X j=1 ✓jfj(ti) + "i <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> z = f?(t) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Select inputs t , observe outputs z , and estimate f? s.t. <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> In a standard regression scheme, one may consider Design is given by ⇠ := ✓ x1 · · · x` n1 N · · · n` N ◆ <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where nk denote the number of times <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> the particular point xk occurs among t1, . . . , tN . <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
11. Yohann DE CASTRO Optimal Design Theory 4 zi = p X j=1 ✓jfj(ti) + "i <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> z = f?(t) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Select inputs t , observe outputs z , and estimate f? s.t. <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> In a standard regression scheme, one may consider Design is given by ⇠ := ✓ x1 · · · x` n1 N · · · n` N ◆ <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where nk denote the number of times <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> the particular point xk occurs among t1, . . . , tN . <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
12. Yohann DE CASTRO Optimal Design Theory 5 M(⇠) := ` X i=1 wiF(xi) F>(xi) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> The Fisher Information Matrix where F := (f1, . . . , fp) are regression functions <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> and wi := ni/N is the weight corresponding to the point xi. <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Objective function q(M) := 8 < : (1 p trace(Mq))1/q if q 6= 1, 0 det(M)1/p if q = 0 min(M) if q = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> which is positively homogeneous, upper semi-continuous, isotonic (with respect to the Loewner ordering) and concave functions.
13. Yohann DE CASTRO Optimal Design Theory 5 M(⇠) := ` X i=1 wiF(xi) F>(xi) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> The Fisher Information Matrix where F := (f1, . . . , fp) are regression functions <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> and wi := ni/N is the weight corresponding to the point xi. <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Objective function q(M) := 8 < : (1 p trace(Mq))1/q if q 6= 1, 0 det(M)1/p if q = 0 min(M) if q = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> which is positively homogeneous, upper semi-continuous, isotonic (with respect to the Loewner ordering) and concave functions.
14. Yohann DE CASTRO Optimal Design Theory 6 Optimal Design max q(M(⇠)) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where the maximum is taken on all designs ⇠ := ✓ x1 · · · x` n1 N · · · n` N ◆ <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
15. Yohann DE CASTRO Optimal Design Theory 6 Optimal Design max q(M(⇠)) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where the maximum is taken on all designs ⇠ := ✓ x1 · · · x` n1 N · · · n` N ◆ <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
16. Yohann DE CASTRO Approximate Optimal Designs: A Convex Program 7 Feasible set n M(⇠) : ⇠ s.t. ⇠ := ✓ x1 · · · x` n1 N · · · n` N ◆ o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Convexifies to n M(⇠) : ⇠ s.t. ⇠ := ✓ x1 · · · x` w1 · · · w` ◆ o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> = n Z X F(x) F>(x)dµ : µ 2 M0(X) o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Leading to the notion of « Approximate Optimal Design »
17. Yohann DE CASTRO Approximate Optimal Designs: A Convex Program 7 Feasible set n M(⇠) : ⇠ s.t. ⇠ := ✓ x1 · · · x` n1 N · · · n` N ◆ o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Convexifies to n M(⇠) : ⇠ s.t. ⇠ := ✓ x1 · · · x` w1 · · · w` ◆ o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> = n Z X F(x) F>(x)dµ : µ 2 M0(X) o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Leading to the notion of « Approximate Optimal Design »
18. Yohann DE CASTRO SDP Hierarchies in a Nutshell 8 A Multivariate Polynomial Setting X := {x 2 Rm : gj(x) > 0, j = 1, . . . , m} <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> F = (f1, . . . , fp) 2 (R[x]d)p and gj 2 R[x]dj <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Based on an SDP Representation of Nonnegative Measures Md(X) := n y 2 R(n+d n ) : 9 µ 0 s.t. y↵ = Z X x↵ dµ, |↵|  d o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
19. Yohann DE CASTRO SDP Hierarchies in a Nutshell 8 A Multivariate Polynomial Setting X := {x 2 Rm : gj(x) > 0, j = 1, . . . , m} <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> F = (f1, . . . , fp) 2 (R[x]d)p and gj 2 R[x]dj <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Based on an SDP Representation of Nonnegative Measures Md(X) := n y 2 R(n+d n ) : 9 µ 0 s.t. y↵ = Z X x↵ dµ, |↵|  d o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
20. Yohann DE CASTRO SDP Hierarchies in a Nutshell 8 A Multivariate Polynomial Setting X := {x 2 Rm : gj(x) > 0, j = 1, . . . , m} <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> F = (f1, . . . , fp) 2 (R[x]d)p and gj 2 R[x]dj <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Based on an SDP Representation of Nonnegative Measures M2d(X) ✓ · · · ✓ MSDP 2(d+2) (X) ✓ MSDP 2(d+1) (X) ✓ MSDP 2d (X) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Md(X) := n y 2 R(n+d n ) : 9 µ 0 s.t. y↵ = Z X x↵ dµ, |↵|  d o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
21. Yohann DE CASTRO SDP Hierarchies in a Nutshell 9 M2d(X) ✓ · · · ✓ MSDP 2(d+2) (X) ✓ MSDP 2(d+1) (X) ✓ MSDP 2d (X) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> M <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFlw40asYB9Yi0ym03YwL5KJUKpu/QG3+lviH+hfeGdMQS2iE5KcOfeeM3Pv9WJfpspxXgvWzOzc/EJxsbS0vLK6Vl7faKZRlnDR4JEfJW2PpcKXoWgoqXzRjhPBAs8XLe/6SMdbNyJJZRSeq1EsugEbhLIvOVNEXVwGTA058+2Tq3LFqTpm2dPAzUEF+apH5RdcoocIHBkCCIRQhH0wpPR04MJBTFwXY+ISQtLEBe5QIm1GWYIyGLHX9B3QrpOzIe21Z2rUnE7x6U1IaWOHNBHlJYT1abaJZ8ZZs795j42nvtuI/l7uFRCrMCT2L90k8786XYtCH4emBkk1xYbR1fHcJTNd0Te3v1SlyCEmTuMexRPC3CgnfbaNJjW1694yE38zmZrVe57nZnjXt6QBuz/HOQ2ae1XXqbpn+5VaLR91EVvYxi7N8wA1HKOOBnmHeMQTnq1TK7NurfvPVKuQazbxbVkPH1bjkwA=</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> Sequence of nested OUTER
 SDP approximations
 of the cone of moments SDPs Non SDP
22. Yohann DE CASTRO SDP Hierarchies in a Nutshell 10 Idea: Replace test functions by an SDP representable subset Test functions are nonnegative polynomials Replaced by « Sum-Of-Squares » polynomials y 2 Md(X) , 8f 2 f 2 R[x] , X ↵2Nn f↵y↵ 0 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> h(x) = 0(x) + m X j=1 gj(x) j(x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> = vd+ (x)>Q0vd+ (x) + m X j=1 gj(x) vd+ ⌫j (x)>Qjvd+ ⌫j (x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> SOS Polynomials
23. Yohann DE CASTRO SDP Hierarchies in a Nutshell 10 Idea: Replace test functions by an SDP representable subset Test functions are nonnegative polynomials Replaced by « Sum-Of-Squares » polynomials y 2 Md(X) , 8f 2 f 2 R[x] , X ↵2Nn f↵y↵ 0 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> h(x) = 0(x) + m X j=1 gj(x) j(x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> = vd+ (x)>Q0vd+ (x) + m X j=1 gj(x) vd+ ⌫j (x)>Qjvd+ ⌫j (x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> SOS Polynomials
24. Yohann DE CASTRO SDP Hierarchies in a Nutshell 10 Idea: Replace test functions by an SDP representable subset Test functions are nonnegative polynomials Replaced by « Sum-Of-Squares » polynomials y 2 Md(X) , 8f 2 f 2 R[x] , X ↵2Nn f↵y↵ 0 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> h(x) = 0(x) + m X j=1 gj(x) j(x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> = vd+ (x)>Q0vd+ (x) + m X j=1 gj(x) vd+ ⌫j (x)>Qjvd+ ⌫j (x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> SOS Polynomials
25. Yohann DE CASTRO Equivalence Theorems 11 Note that the Approximate Optimal Design Problem reads max y q(Md(y)) s.t. y 2 M2d(X), y0 = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> vd(x) = ( 1 |{z} deg. 0 , x1, . . . , xn | {z } degree 1 , x2 1 , x1x2, . . . , x2 n | {z } degree 2 , . . . , xd 1 , . . . , xd n | {z } degree d )> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> x 7! p? d (x) := vd(x)>Md(y?)q 1vd(x) = ||Md(y?)q 1 2 vd(x)||2 2 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Introduce Supp(µ?) ✓ n x 2 X : trace(Md(y?)q) p? d (x) = 0 o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> trace(Md(y?)q) p? d (x) 0 , x 2 X <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
26. Yohann DE CASTRO Equivalence Theorems 11 Note that the Approximate Optimal Design Problem reads max y q(Md(y)) s.t. y 2 M2d(X), y0 = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> vd(x) = ( 1 |{z} deg. 0 , x1, . . . , xn | {z } degree 1 , x2 1 , x1x2, . . . , x2 n | {z } degree 2 , . . . , xd 1 , . . . , xd n | {z } degree d )> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> x 7! p? d (x) := vd(x)>Md(y?)q 1vd(x) = ||Md(y?)q 1 2 vd(x)||2 2 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Introduce Supp(µ?) ✓ n x 2 X : trace(Md(y?)q) p? d (x) = 0 o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> trace(Md(y?)q) p? d (x) 0 , x 2 X <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
27. Yohann DE CASTRO Equivalence Theorems 11 Note that the Approximate Optimal Design Problem reads max y q(Md(y)) s.t. y 2 M2d(X), y0 = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> vd(x) = ( 1 |{z} deg. 0 , x1, . . . , xn | {z } degree 1 , x2 1 , x1x2, . . . , x2 n | {z } degree 2 , . . . , xd 1 , . . . , xd n | {z } degree d )> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> x 7! p? d (x) := vd(x)>Md(y?)q 1vd(x) = ||Md(y?)q 1 2 vd(x)||2 2 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Introduce Supp(µ?) ✓ n x 2 X : trace(Md(y?)q) p? d (x) = 0 o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> trace(Md(y?)q) p? d (x) 0 , x 2 X <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
28. Yohann DE CASTRO -5 0.8 -4 0.6 0.4 -3 x 2 0.2 -2 0.5 0 x 1 -0.2 0 -1 -0.4 -0.5 0 Equivalence Theorems 12 Supp(µ?) ✓ n x 2 X : trace(Md(y?)q) p? d (x) = 0 o <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> trace(Md(y?)q) p? d (x) 0 , x 2 X <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
29. Yohann DE CASTRO Equivalence Theorems 13 Relaxation of the Approximate Optimal Design Problem reads x 7! p? d (x) := vd(x)>Md(y?)q 1vd(x) = ||Md(y?)q 1 2 vd(x)||2 2 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Recall max y q(Md(y)) s.t. y 2 MSDP 2(d+ ) (X), y0 = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> y? 2 MSDP 2(d+ ) (X) unique <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> p? = trace(Md(y?)q) p? d 2 PSOS 2(d+ ) (X) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> hy?, p?i = 0 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
30. Yohann DE CASTRO Equivalence Theorems 13 Relaxation of the Approximate Optimal Design Problem reads x 7! p? d (x) := vd(x)>Md(y?)q 1vd(x) = ||Md(y?)q 1 2 vd(x)||2 2 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Recall max y q(Md(y)) s.t. y 2 MSDP 2(d+ ) (X), y0 = 1 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> y? 2 MSDP 2(d+ ) (X) unique <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> p? = trace(Md(y?)q) p? d 2 PSOS 2(d+ ) (X) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> hy?, p?i = 0 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
31. Yohann DE CASTRO Christoffel Polynomial 14 For q=0 one has the «Christoffel Polynomial» x 7! p? d (x) := vd(x)>Md(y?) 1vd(x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Which is known to satisfy 1 p? d (t) = min P 2R[x]d n Z P(x)2 dµ?(x) : P(t) = 1 o , 8t 2 Rn <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> p? d (t) = X |↵|d P↵(t)2, t 2 Rn <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where (P↵)|↵|d ✓ R[x]d are orthonormal polynomials <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
32. Yohann DE CASTRO Christoffel Polynomial 14 For q=0 one has the «Christoffel Polynomial» x 7! p? d (x) := vd(x)>Md(y?) 1vd(x) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Which is known to satisfy 1 p? d (t) = min P 2R[x]d n Z P(x)2 dµ?(x) : P(t) = 1 o , 8t 2 Rn <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> p? d (t) = X |↵|d P↵(t)2, t 2 Rn <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where (P↵)|↵|d ✓ R[x]d are orthonormal polynomials <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>
33. Yohann DE CASTRO References viewed today 15