What can be expressed via Conic Quadratic and Semidefinite Programming?

Transcription

1 What can be expressed via Conic Quadratic and Semidefinite Programming? A. Nemirovski Faculty of Industrial Engineering and Management Technion Israel Institute of Technology Abstract Tremendous recent progress in the performance of optimization techniques for Linear, Conic Quadratic and Semidefinite Programming makes it important to know how to recognize optimization problems lending themselves to these advanced techniques. To this end, we present in the talk a kind of simple, powerful and completely algorithmic calculus of problems reducible to CQP and SDP. We demonstrate also that problems representable via Conic Quadratic Programming admit polynomial time Linear Programming approximations.

2 . What can be expressed via Conic Quadratic and Semidefinite Programming? Arkadi Nemirovski Faculty of Industrial Engineering and Management Technion Israel Institute of Technology

3 What the story is about Consider the following optimization program: (a) (b) (c) minimize c T x subject to Ax = b x 0 ( ) 8 x i 3 1/3 1/7 x 2 x 2/7 3 x 3/ x 1/5 1 x 2/5 5 x 1/5 6 i=1 5x 2 1 x 1/2 1 x x 1/3 2 x 3 3 x5/8 4 x 2 x 1 x 1 x 4 x 3 x 3 x 6 x 3 x 3 x 8 5I x 1 x 2 x 1 x 3 x 2 x 4 x 3 x 2 x 1 x 2 x 3 x 2 x 4 x 3 x 3 x 2 x 3 x 2 x 3 x 4 x 3 x 4 x 3 x 4 x 3 x 4 x 3 x 4 0 x 1 + x 2 sin(φ) + x 3 sin(2φ) + x 4 sin(4φ) 0 φ [ 0, π 2 The problem can be converted, in a systematic way, into an equivalent semidefinite program d T z min P 0 + dim z i=1 z i P i 0. ] (SDP) Removing constraints (c), the resulting problem can be converted, in a systematic way, into an equivalent conic quadratic program d T z min P i z + p i 2 q T i z + r i, i = 1,..., m. (CQP) The resulting problem (CQP) can be approximated, in a polynomial time fashion, by a linear programming program d T z min P z + p 0. (LP) 3

4 (a) Ax = b x 0 x i minimize c T x subject to v i 0, 0, i = 1,..., 8, v i v 8+i x i v v i v 17 ; i=9 v 18 v 19 0, v 19 x 2 x 4 v 22 0, v 22 v 20 x 6 v 25 v , v i x i x 4 v 20 0, v 20 1 v 21 v 18 0, v 18 v 22 v 23 v 26 0, v 26 v 24 v 26 v 23 0, v 23 v 27 v 17 v v 23 ; v 19 v 21 0, v 21 x 3 x 1 v 24 0, v 24 v 23 x 5 v 27 v 27 v 25 0, (b) x 1 v 29 0, v 29 1 x 4 v 32 0, v 32 1 v 31 v 35 0, v 35 1 v 35 v 38 v 38 v 31 0, v 28 v 30 0, v , v 30 v 29 1 x 2 x 4 v 33 0, x 4 v 34 0, v 33 v 32 v 34 v 33 x 2 v 36 0, x 4 v 37 0, v 36 x 3 v 37 v 34 v 36 v 39 0, v 39 v 37 x 3 1 0, 1 v 40 v v 31 5x 2 ; x 2 x 1 x 1 x 4 x 3 x 3 x 6 x 3 x 3 x 8 4 v 38 v 40 v 40 v 39 5I 0,

5 (c) x 1 x 2 x 1 x 3 x 2 x 4 x 3 x 2 x 1 x 2 x 3 x 2 x 4 x 3 x 3 x 2 x 3 x 2 x 3 x 4 x 3 x 4 x 3 x 4 x 3 x 4 x 3 x 4 0 x 1 0 v 41 v 42 v 43 v 44 v 45 v 46 v 47 0 v 48 v 42 v 49 v 50 v 51 v 52 v 53 v 54 v 41 v 42 v 55 v 56 v 57 v 58 v 59 v 60 v 61 v 42 v 49 v 56 v 62 v 63 v 64 v 65 v 66 v 67 v 43 v 50 v 57 v 63 v 68 v 69 v 70 v 71 v 72 v 44 v 51 v 58 v 64 v 69 v 73 v 74 v 75 v 76 v 45 v 52 v 59 v 65 v 70 v 74 v 77 v 76 v 78 v 46 v 53 v 60 v 66 v 71 v 75 v 76 v 79 0 v 47 v 54 v 61 v 67 v 72 v 76 v 78 0 v 80 0, 2v 41 + v 48 8x 1 2x 2 4x 3 8x 4 = 0 2v v 49 + v 55 32x 1 14x 2 28x 3 56x 4 = 0 v 44 + v 50 + v 56 = 0 2v v v 57 + v 62 80x 1 48x 2 88x 3 112x 4 = 0 v 46 + v 52 + v 58 + v 63 = 0 2v v v v 64 + v x 1 100x 2 160x 3 = 0 v 54 + v 60 + v 65 + v 69 = 0 2v v v 70 + v x 1 136x 2 176x x 4 = 0 v 67 + v 71 + v 74 = 0 2v v 75 + v x 1 120x 2 112x x 4 = 0 2v 78 + v 79 64x 1 64x 2 32x x 4 = 0 v 80 16x 1 16x 2 = 0 5

6 What can be expressed via CQP and SDP? Let us look at three generic families of convex programs: Linear Programming: c T x min Ax + b 0 (LP) Conic Quadratic Programming: c T x min A i x + b i 2 c T i x + d i, i = 1,..., m (CQP) Semidefinite Programming: c T x min n i=1 x i A i + B 0 (SDP) Geometrically, all these problems are of the form c T x min Ax + b K, (CP) where K is a closed pointed convex cone with a nonempty interior belonging to a specific for the generic problem in question family K of convex cones. 6

7 c T x min Ax + b K [K K] (CP) LP: K is comprised of direct products of rays R + CQP: K is comprised of direct products of the Lorentz cones L n = {(x, t) R n+1 : x 2 t} SDP: K is comprised of direct products of the semidefinite cones S n + = {A S n : A 0} Note: The above families of cones are closed w.r.t. (1) taking (finite) direct products of cones; (2) passing from a cone K to its dual cone K = {η η T x 0 x K}. 7

8 Assume that we are given a family K of finitedimensional convex cones (closed, pointed and with a nonempty interior) and know how to solve problems of the form c T y min Ay + b K ( ) with all possible data (c, A, b) and all K K. Question: What is the family of problems we can actually solve? When an optimization problem e T x min x X R n (P) can be equivalently reformulated in the form of ( )? An answer: This is the case when X is a K-representable set (K-r.s.), i.e., there exists a K-representation (K-r.) of X X = { x R n u R k : P x + Qu + b K } [K K] (R) Indeed, given the data P, Q, b, K of (R), we can rewrite (P) equivalently in the form of ( ): d T x min x X c T y d T x min Ay + b P x + Qu + b K y = x u 8

9 An example: Consider the optimization problem t min xyzt 1, A 1 xyz x y z min x, y, z 0, A + b 0, x, y, z 0 x y z + g 0. It turns out that the feasible set of this problem is semidefinite representable: (1) xyzt 1 & x, y, z 0 & A u 1, u 2 : (a) (b) x y z + b 0 x u 1 0, z u 2 0 y u 1 u 2 t }{{} say that xy u 2 1, zt u 2 2 and x, y, z, t 0 u u 2 }{{} says that u 1 u 2 1 and u 1, u 2 0 [(1) says exactly that xyzt 1 & x, y, z 0] x (2) A y + b 0 z Thus, (*) is equivalent to the semidefinite problem t min x u 1 u 1 y 0, z u 2 u 2 t 0, u u 2 0, A ( ) x y z +b 0. 9

10 We see that a natural interpretation of the question Given a possibility to solve problems c T x min Ax + b K K what can we actually solve? is What are K-representable sets? How to recognize K-representability? 10

11 Claim: Consider a family K of finite-dimensional closed pointed cones with a nonempty interior, and let this family be closed w.r.t. taking direct products passing from a cone to its dual. There exists a simple and powerful calculus of K- representable sets: essentially, Every standard convexity-preserving operation as applied to (finitely many) K-representable sets X 1,..., X k, yields a K-representable result X. Moreover, a K-representation of X is readily given by K-representations of X 1,..., X k. Applying the calculus machinery to (specific for a family K) collection of raw materials simple K- representable sets we get a possibility to recognize complicated K-representable sets and thus to pose various optimization problems in the K-form. 11

12 Calculus of K-representable sets I. The intersection of finitely many K-r.s. s is a K-r.s.: X i = {x u i : P i x + Q i u i + b i K i }, i = 1,..., k k i=1 X = { u u = u 1... : u k P 1 x + Q i u 1 + b 1... P k x + Q k u k + b k K 1... K k } II. The direct product of finitely many K-r.s. s is a K-r.s. III. The image of a K-r.s. under an affine mapping is a K-r.s. IV. The inverse image of a K-r.s. under an affine mapping is a K-r.s. V. The arithmetic sum of finitely many K-r.s. s is a K-r.s. 12

13 Calculus of K-representable sets (cont.) VI. A polyhedral set is a K-r.s. X = {x a T i x + b i 0, i = 1,..., k} Indeed, let K K and 0 e K. Then X = x [a T 1 x + b 1 ]e... [a T k x + b k ]e K... K Intersection of infinitely many half-spaces not necessarily is K-representable. It, however, is so when the half-spaces are well-organized.. 13

14 VIa. Assume that the data (α, β) defining a half-space vary in a K-r.s. U: X α,β = {x α T x + β 0} U = {(α, β) u : P α + βp + Qu + b K} and let the representation be strictly feasible: Then the set X = is a K-r.s.: ᾱ, β, ū : P ᾱ + βp + Qū + b int K. (α,β) U X α,β = { x α T x + β 0 (α, β) U } X = x η : η K P T η = x p T η = 1 Q T η = 0 b T η 0 Thus, X is the projection of the intersection of the K-r.s. K and a polyhedral set and thus is a K-r.s. 14

15 Calculus of K-representable sets (cont.) Several operations with sets nearly preserve K- representability. Definition. A set X R is called nearly K-representable, if there exists a K-r.s. X such that X X cl X. ( ) A nearly K-representation of X is a K-representation of a set X satisfying ( ). Note: Nearly K-representable closed set X is K- representable, and every nearly K-representation of such a set is its K-representation. VII. The conic hull Cone(X) = {0} { (x, t) R n R t > 0, t 1 x X } of a nonempty K-r.s. X is a nearly K-r.s. VIII. Convex hull of the union of finitely many K-r.s. s is a nearly K-r.s. IX. Let X = {x u : P x + Qu + b K} be a K-r.s., and let the representation ( ) be strictly feasible: ( x, ū) : P x + Qū + b int K. Then the polar cone of X the set ( ) is a K-r.s. X = {(η, s) η T x s x X} 15

16 K-representable functions In many applications, sets are represented by (systems of) inequalities f(x) 0 [f : R n R {+ }] ( ) Definition: f is called a K-representable function (Kr.f.), if the epigraph of f Epi(f) = {(x, t) t f(x)} is a K-r.s. A K-representation of Epi(f) is called a K-representation (K-r.) of f. Observation: If f is K-representable, then so are all level sets {x f(x) a} [a R] of f. Indeed, f(x) t u : P x + tp + Qu + b K {x f(x) a} = {x u : P x + Qu + [ap + b] K} 16

17 K-representable functions (cont.) Calculus of K-representable sets can be straightforwardly converted into a calculus of K-representable functions. The latter, essentially, can be summarized in the following statements: (i). [Conic combinations and taking maximum] A linear combination, with nonnegative coefficients, and the maximum of finitely many K-r.f. s are themselves K-r.f. s. (ii). [Adding affine form] The sum of a K-representable and an affine function is K-representable (iii). [Affine substitution of argument] The superposition of a K-r.f. and an affine mapping is a K-r.f. (iv). [Taking superpositions] Let functions and a function f 1,..., f k : R n R {+ } g : R k R {+ } be K-representable. Let also g be monotone: Then the superposition z z g(z) g(z ). is a K-r.f. h(x) = g(f 1 (x),..., f k (x)) 17

18 K-representable functions (cont.) (v). [Legendre transformation] The Legendre transformation f (η) = sup x [η T x f(x)] of a K-r.f. f with strictly feasible K-r. is itself a K-r.f. Example: f(x) = Ax Bx c 4 2 f(x) = Ax Bx c 4 2 f (ξ) = sup x [ ξ T x f(x) ] =??? t f(x) τ f (ξ) s 1, s 2, s 3 : η 1, η 2, σ 1, σ 2, σ 3, σ 4, σ 5 : Ax 2 s 1 Bx c 2 s 2 2s 2 1 s s 3 2s 1 2s 3 1 t t η 1 2 σ 1 η 2 2 σ 2 σ 2 σ 3 2 σ 4 2σ 1 σ 3 + σ 4 4 2σ 5 2 2σ 5 2c T η 2 σ 3 + σ 4 + 2σ 5 2 2τ 18

19 What can be expressed via LP? K = LP {R n +} n=1 Of course, LP-representable sets are exactly the polyhedral sets, LP-representable functions are exactly convex piecewise linear functions. E.g., the function f(x 1,..., x n ) = n i=1 x i n i is not LP-representable. But... : int R n + R 19

20 What can be expressed via CQP? K = CQ L n i i L n = {(x, t) R n R x 2 t} Elementary CQ-representable functions/sets The following functions/sets are CQ-representable with explicit CQ-r. s: 1. An affine function f(x) = a T x + b 2. The Euclidean norm x 2 : R n R 3. The set {(x, s, t) R n R R : s, t 0, x T x st}. ( ) 4. The squared Euclidean norm f(x) = x T x : R n R Note: by Calculus it follows from 4. that every convex quadratic form is CQ-representable. 5. A simple fractional-quadratic function f(x, s) = x2 s : {(x, s) R2 s 0} R {+ } is CQ-representable. Extension: The set of triples (τ R, x R n, y R m +) for which τ x is CQ-representable. xt A T 0 Diag(y)A 20

21 6. Let π 1,..., π k be positive rationals with k i=1 π i 1. Then the function f(x) = k i=1 x π i i : R k + R admits an explicit CQ-r. The size of the representation is proportional to the magnitude of the common denominator of the fractions π 1,..., π k. Example: A CQ-r. of the set t x x x 3 7 3, x 0 is s, u 1, u 2, u 3, u 4 : 2u 1 x 1 x 3 L2 x 1 + x 3 2u 3 u 1 u 2 L2 u 1 + u 2 2s 2t u 3 u 4 u 3 + u 4 2u 2 s t 1 s t + 1 s 0 2u 4 x 2 x 3 x 2 + x 3 L2 L2 L2 21

22 Elementary CQ-representable functions/sets (cont.) 7. Let π 1,..., π k be positive rationals. Then the function f(x) = k i=1 x π i i : int R k + R admits an explicit CQ-r. The size of the representation is proportional to the magnitude of the common denominator of the fractions π 1,..., π k. Example: A CQ-r. of the set is u 1, u 2, u 3, u 4, u 5, u 6 : 2u 1 x 2 x 3 x 2 + x 3 2u 4 u 1 u 2 u 1 + u 2 L2 L2 t x x x u 2 t 1 t + 1 2u 5 u 3 u 4 u 3 + u 4 2 u 3 u 6 u 3 + u 6 L2 L2 L2 2u 3 x 2 x 3 x 2 + x 3 2u 6 u 2 t u 2 + t L2 L2 22

23 Elementary CQ-representable functions/sets (cont.) 8. Let π 1 be a rational. Then the function admits an explicit CQ-r. So does the function f + (x) = (max[x, 0]) π g(x) = x π [ f(x) + f( x)] 9. For rational π 1, the π-norm f(x) = x π : R n R is CQ-representable. 10. The cone of symmetric 3-diagonal positive semidefinite matrices is CQ-representable. In particular, the maximum eigenvalue λ max (X) of a 3-diagonal symmetric matrix X is a CQ-representable function of X. 23

24 There are, of course, nice convex functions which are not CQ-representable, e.g., the exponent But: for every p 1, f(x) = exp{x}. ( x exp{x} = r lim r 2 ( x ) 2 1 ( x ) p ) (2 r ) r p! 2 r }{{} h p( x 2 r) It follows that if p is such that h p ( ) is CQ-representable and nonnegative, then exp{x} can be approximated by CQ-r.f. ( ) g p,r (x) = h (2r ) x p. For p fixed and for a given bounded range x T of values of x, the quality of this approximation grows rapidly with r: r O ( ln ( ) ) 1 ε + T, x T (1 ε) exp{x} g p,r (x) (1 + ε) exp{x}. 2 r 24

25 E.g., the system of 24 conic quadratic constraints with 23 variables: u 1,..., u 21 : 2 + x u u 1 L x u u 2 L2 2u 1 1 u u 3 L u u 3 u 4 2u l 1 1 u l 1 + u l L2, l = 5,..., 21 2u 21 1 t 1 + t L2 512 x 512 is a conic representation of the univariate function ( ) g(x) = min{t u 1,..., u 21 : (t, x, u) satisfies ( )}. The function g approximates exp{x} in the segment = {x x 512} = {x 4.38e-223 exp{x} 2.28e222} within relative accuracy 1.e-10. For any practical purpose exp{x} is CQ-representable! 25

26 Whether Conic Quadratic Programming does exist? Surprisingly, Lorentz cones (and thus all CQrepresentable sets!) are nearly polyhedrally representable. Theorem [BT-N, 98] For every n > 0 and every ε (0, 1/2), there exist (and can be explicitly written down) matrices P n,ε, Q n,ε and vector b n,ε such that (i) The row and the column sizes of P n,ε, Q n,ε do not exceed O ( n ln 1 ) ε (ii) If (x, t) L n = {(x, t) R n R x 2 t}, then there exists a vector u such that P n,ε x + tb n,ε + Q n,ε u 0, and nearly vice versa : if (x, t) can be extended by some u to a solution of ( ), then (x, t) L n ε = {(x, t) R n R x 2 (1 + ε)t}. ( ) Thus, L n can be approximated within accuracy ε by a cone M n ε: L n N n ε L n ε which is the projection onto the (x, t)-space of a simple polyhedral cone N n ε = {(x, t) u : P n,ε x + tb n,ε + Q n,ε u 0}. given by just O ( n ln 1 ) ( ) ε linear inequalities involving O n ln 1 ε variables. 26

27 Corollary. Consider a conic quadratic program c T x min Ax+b 0, P x+q K = L n 1... L n k (CQP) and let the problem be both r-strictly feasible for some r > 0: x : A x + b 0 & (ξ : ξ 2 r) : P x + q + ξ K R-semibounded: Ax + b 0, P x + q K P x + q 2 R. For every ε (0, 1), there exists an LP program c T x min Bx + Cv + d 0 (LP) which is an ε-relaxation of (CQP) of polynomial complexity: If x is feasible for (CQP), then x can be extended to a feasible solution (x, v) of (LP); If (x, v) is feasible for (LP), then the vector is feasible for (CQP). (1 ε)x + ε x The size dim x+dim v+dim d of (LP) does not exceed dim x + dim b + O (dim q) ln 2R εr and the data (c, B, C, d) of (LP) is readily given by the data (c, A, b, P, q) of (CQP). 27

28 Fast polyhedral approximations of the Lorentz cone are based on the fact that if P ν is the polytope defined as u 0 x 1 ; u 1 x 2 ; (x, u) R 2 R 2ν+4 : u 2j = cos ( π ) 2 j+1 u2j 2 ) u2j 1, + sin ( π 2 j+1 u 2j+1 sin ( π + cos ( π 2 j+1 j = 1,..., ν; ) 2 j+1 u2j 2 ) u2j 1, j = 1,..., ν; u 2ν+3 1; u 2ν+4 tan ( π 2 ν+1 ) u2ν+3 then the projection P ν x of the polytope on the x-plane approximates the unit 2D disk D 2 within accuracy 1 cos ( π 2 ν+1 ) 1 π 2 2 2ν+3. 28

29 P 3 lives in R 9 and is given by 12 linear inequalities. P 3 x approximates D 2 within accuracy 5.e-3 (as good as the 16-side perfect polygon) P 6 lives in R 12 and is given by 18 linear inequalities. P 6 x approximates D 2 within accuracy 3.e-4 (as good as the 127-side perfect polygon) P 12 lives in R 18 and is given by 30 linear inequalities. P 12 x approximates D 2 within accuracy 7.e-8 (as good as the 8,192-side perfect polygon) P 24 lives in R 30 and is given by 54 linear inequalities. P 24 x approximates D 2 within accuracy 4.e-15 (as good as the 34,200,933-side perfect polygon) 29

30 K = SD What can be expressed via SDP? Direct products of the cones of positive semidefinite matrices Examples of SD-representable functions/sets Every CQ-r. function/set is SD-r. as well. Indeed, the Lorentz cone L n is SD-r.: (x, t) L n Θ n (x) x 1 x 2. x n t x 1 x 2... x n t t... t Consequently, a CQ-r. of a set X: 0 X = {x u : P i x + Q i u + b i L n i, i = 1,..., k} yields an SD-r. of X: X = {x u : Θ ni (P i x + Q i u + b i ) 0, i = 1,..., k}. 30

31 Elementary SD-representable functions/sets (cont.) Functions of eigenvalues of symmetric matrices, I For a symmetric n n matrix X, let λ 1 (X) λ 2 (X)... λ n (X) be the eigenvalues of X taken with their multiplicities in the non-ascending order. 11. Maximum eigenvalue λ max (X) of a symmetric matrix X is SD-representable: t λ max (X) ti X The sum S k (X) = λ 1 (X) λ k (X) of k largest eigenvalues of a symmetric matrix X is SD-representable: S k (X) t s, Z : t ks Tr(Z) 0 Z 0 Z X + si 0 31

32 Elementary SD-representable functions/sets (cont.) Functions of eigenvalues of symmetric matrices, II 13. Let a function f on R n SD-representable and symmetric (i.e., invariant w.r.t. permutations of coordinates). Then the function F (X) = f(λ(x)) of a symmetric n n matrix X also is SD-representable. E.g., the following functions of a symmetric n n matrix X are SD-representable explicit SD-r. s: (Det X) π, X 0, 0 < π 1 n [f(x) = ( n x i) π, x 0] i=1 X π λ(x) π, π 1 is rational [f(x) = x π ] is rational X + π {max[λ i (X), 0]} n i=1 π, π 1 is rational [f(x) = {max[x i, 0]} n i=1 π ] 32

33 Elementary SD-representable functions/sets (cont.) Functions of singular values of rectangular matrices For a n m matrix X [n m], let σ(x) = λ ( XX T ) R n be the vector of singular values of X taken in the nonascending order. 14. Let a function f on R n + be symmetric, monotone and SD-representable. Then the function F (X) = f(σ(x)) : R n m R {+ } is SD-representable. E.g., the following functions of rectangular n m matrix X are SD-representable with explicit SD-r. s: X = σ max (X) X t ti n X X T 0 ti m Σ k (X) = k i=1 σ i(x), 1 k n X π = σ(x) π, π 1 is rational 33

34 Elementary SD-representable functions/sets (cont.) Convex hulls of some sets of matrices 15. Let S {x R n x 1 x 2... x n } be an SDrepresentable set. The convex hull of the set of symmetric matrices X such that λ(x) S is SD-representable. Let, in addition, S be monotone: (y 0, y 1 y 2... y n, x S : y x) y S. Then the convex hull of the set of rectangular n m matrices X such that σ(x) S is SD-representable. 34

35 Elementary SD-representable functions/sets (cont.) -epigraphs of matrix-valued functions, I The following sets are SD-representable: 16. The set of solutions of the -convex quadratic matrix inequality (AXB)(AXB) T + CXD + (CXD) T + E Y in variable matrices X R k l and Y S m ; 17. The set of solutions of the fractional-quadratic matrix inequality F (X, Y ) XY 1 X T Z in variable matrices X R n m, 0 Y S m, Z S n ; 18. The set of solutions of the double fractional matrix inequality Y (A T X 1 A) 1 in symmetric matrices Y S n and 0 X S m ; 19. The -hypograph of the matrix square root, i.e., the set of solutions of the matrix inequality Y X 1/2 in symmetric n n matrices X 0 and Y. Note: Solution sets of matrix inequalities X 1/2 Y X 1/2 Y 2 X (a) (b) differ from each other! (And both are SD-r...) 35

36 Elementary SD-representable functions/sets (cont.) Univariate polynomials, I Let be a convex subset of the real axis and k be a positive integer. 20. [Yu. Nesterov, 97] The following sets are SDrepresentable with explicit SD-representations: The set of coefficients of all nonnegative on algebraic polynomials of degree k; p(x) = k l=0 p l x l Example: The cone of all nonnegative on the entire axis algebraic polynomials of degree not exceeding 2m is the image of S m+1 + under the linear mapping Z p Z (x) m+1 i,j=1 Z ij x i+j 2. The set of coefficients of all nonnegative on trigonometric polynomials of degree k. p(x) = p 0 + k [p 2l 1 cos(lx) + p 2l sin(lx)] l=1 36

37 Elementary SD-representable functions/sets (cont.) Univariate polynomials, II Corollary of 20: The functions f(p 0, p 1,..., p k ) = sup x g(p 0, p 1,..., p 2k ) = sup x are SD-representable. k p lx l l=1 [ p 0 + k [p 2l 1 cos(lx) + p 2l sin(lx)] l=1 21. Let p(x) = k p lx l be an algebraic polynomial. l=0 The closed convex hull of the epigraph of p is SDrepresentable. In particular, if p is convex on, then the epigraph of p is SD-representable. ] 37

38 Elementary SD-representable functions/sets (cont.) Families of ellipsoids Consider two parameterizations of ellipsoids in R n V (X, x) = {u R n u = Xv + x, v T v 1} [X R n n ] ellipsoid is an affine image of Euclidean ball W (Y, y) = {u R n u T Y T Y u 2u T Y T y + y T y 1} [Y R n n ] ellipsoid is a level set of a below bounded quadratic function Proposition [Boyd et al.] V (X, x) W (Y, y) if and only if there exists λ such that I Y x y Y X x T Y T y T 1 λ 0 X T Y T λi Observation: When one of the ellipsoids is fixed, ( ) is a linear matrix inequality in λ and in the (parameters of) the other ellipsoid. Thus, the set of (parameters of) all ellipsoids contained in/containing a fixed ellipsoid is SD-r. Corollary. The problems of (1) finding the largest volume ellipsoid contained in the intersection of finitely many given ellipsoids, (2) finding the smallest volume ellipsoid containing the union of finitely many given ellipsoids can be posed as semidefinite programs. ( ) 38

39 Plant: 5 material points linked by elastic springs and sliding without friction along given axes Input: the forces u i applied at green points Output: velocities ẋ i of the points Control law: linear feedback which brings the nominal plant to the origin while minimizing the cost 0 5 i=1 (ẋ i ) 2 (t) + u 2 1(t) + u 2 5(t) dt, Uncertainty: ρ%-perturbations in rigidities of springs, masses of points, and coefficients of the feedback matrix What is the largest level of (dynamical) perturbations ρ preserving the stability of the closed loop system? 4 39

40 The computable upper bound on ρ turns out to be 4.1%. Thus, one can be sure that the closed loop system remains stable at least up to 4.1% perturbations in the parameters of the plant and the coefficients in the feedback matrix. Simulation demonstrates that there exist dynamical 6.5% perturbations which make the closed loop system unstable. Thus, we have specified the stability margin of the system up to factor