Stability in Linear Optimization: applications

Transcription

1 Stability in Linear Optimization: applications Virginia N Vera de Serio Universidad Nacional de Cuyo Argentina ization Problems and Related Topics September 12-16, 2011, Borovets, Bulgaria

2 Outline of the presentation Some de nitions Two applications to duality theory : Stability of the primal-dual partition in linear semi-in nite programming (Qualitative perspective) Lipschitz-like property of the stability of the feasible sets of the primal and the dual problem in Banach spaces (Quantitative perspective) An application to convex geometry : Stability of the Voronoi cells (Qualitative perspective)

3 De nitions Z and Y topological spaces M : Z Y a set-valued mapping M is lower semicontinuous at z 0 2 Z (lsc, in brief) if, for each open set W Y such that W \ M(z 0 ) 6=, there exists an open set V Z, containing z 0, such that W \ M(z) 6= for each z 2 V

4 De nitions Z and Y topological spaces M : Z Y a set-valued mapping M is lower semicontinuous at z 0 2 Z (lsc, in brief) if, for each open set W Y such that W \ M(z 0 ) 6=, there exists an open set V Z, containing z 0, such that W \ M(z) 6= for each z 2 V M is upper semicontinuous at z 0 2 Z (usc, in brief) if, for each open set W Y such that M(z 0 ) W, there exists an open set V Z, containing z 0, such that M(z) W for each z 2 V

5 De nitions Z and Y topological spaces M : Z Y a set-valued mapping M is lower semicontinuous at z 0 2 Z (lsc, in brief) if, for each open set W Y such that W \ M(z 0 ) 6=, there exists an open set V Z, containing z 0, such that W \ M(z) 6= for each z 2 V M is upper semicontinuous at z 0 2 Z (usc, in brief) if, for each open set W Y such that M(z 0 ) W, there exists an open set V Z, containing z 0, such that M(z) W for each z 2 V M is closed at z 0 2 dom M if for all nets fz r g Z and fx r g Y n satisfying x r 2 M(z r ) for all r in some index set, z r! z 0 and x r! x 0, one has x 0 2 M(z 0 )

6 Painlevé-Kuratowski limit Z and Y metric spaces M : Z Y a set-valued mapping We consider the lower (inner) and upper (outer) limits of set-valued mappings lim inf M (z) and lim sup M (z) z!z 0 z!z 0 According to Rockafellar and Wets (1998), M closed and lsc at z 0 is equivalent to lim inf z!z0 M (z) = lim sup z!z0 M (z) = M (z 0 ) In this case we say that lim z!z0 M (z) = M (z 0 ) and call it the Painlevé-Kuratowski limit

7 Lipschitz properties Z and Y normed spaces M : Z Y a set-valued function M is Lipschitz-like around (bz, by) 2 gphm if there are exist ` 0, and neighborhood U of bz and V of by such that M(z) \ V M(u) + ` kz uk B, for all z, u 2 U

8 Lipschitz properties Z and Y normed spaces M : Z Y a set-valued function M is Lipschitz-like around (bz, by) 2 gphm if there are exist ` 0, and neighborhood U of bz and V of by such that M(z) \ V M(u) + ` kz uk B, for all z, u 2 U The in mum of such moduli `0s over all possible combinations f`, U, V g satisfying this is the exact Lipschitzian bound of M around (bz, by) lip M (bz, by) = lim sup (z,y )!(bz,by ) dist(y, M(z)) dist(z, M 1 (y))

9 This Lipschitz property is one of the various regularity concepts used in variational analysis Mordukhovich, Variational Analysis and Generalized Di erentiation (2005) Rockafellar & Wets, Variational Analysis (1998) Surveys: Io e (2000), Azè (2006) Bonnans, Borwein, Cánovas, Dontchev, Io e, Klatte, Lewis, López, Parra, Sekiguchi, Shapiro, Zhu (in the 2000 s), etc the metric regularity property

10 This Lipschitz property is one of the various regularity concepts used in variational analysis Mordukhovich, Variational Analysis and Generalized Di erentiation (2005) Rockafellar & Wets, Variational Analysis (1998) Surveys: Io e (2000), Azè (2006) Bonnans, Borwein, Cánovas, Dontchev, Io e, Klatte, Lewis, López, Parra, Sekiguchi, Shapiro, Zhu (in the 2000 s), etc the metric regularity property the open mapping property of the inverse mapping M 1 : Y Z

11 This Lipschitz property is one of the various regularity concepts used in variational analysis Mordukhovich, Variational Analysis and Generalized Di erentiation (2005) Rockafellar & Wets, Variational Analysis (1998) Surveys: Io e (2000), Azè (2006) Bonnans, Borwein, Cánovas, Dontchev, Io e, Klatte, Lewis, López, Parra, Sekiguchi, Shapiro, Zhu (in the 2000 s), etc the metric regularity property the open mapping property of the inverse mapping M 1 : Y Z lip M (bz, by) = sur M 1 (by, bz) 1 1 = lim inf (y,z,λ)!(by,bz,0 + ) λ sup r 0 : z + rb Z M 1 (y + λb Y sur M 1 (by, bz) is the rate of surjection (openness) of M 1 around (by, bz)

12 Part I: Stability of the primal-dual partition in LSIP Goberna & Todorov (2007, 2008, 2009) The continuous case Ochoa & V (2011) The general case, no continuity assumptions Goberna, López & Todorov (2001), Cánovas, López, Parra & Toledo (2006) Stability of consistency of the dual problem

13 π = (a, b, c) Stability of the primal-dual partition in LSIP 2 Primal problem: P : inf c 0 x, st a 0 tx b t, for all t 2 T, T is an arbitrary index set c and x are elements of R n a : T! R n and b : T! R are arbitrary functions Haar s dual problem of P : D : sup λ t b t, t2t st λ t a t = c, λ 2 R (T ) +, t2t P and D involve the same data a, b and c

14 Stability of the primal-dual partition in LSIP 3 Parameter space Π = (R n ) T R T R n (general case) The Banach space Π C = C (T ) n C (T ) R n (continuous case) Extended distance d : Π Π! [0, + ], n d(π 1, π 2 ) := max k c 1 c 2 k, sup t2t k(at 1, bt 1 ) (at 2, bt 2 )k o, π i = (a i, b i, c i ) 2 Π, i = 1, 2

15 Stability of the primal-dual partition in LSIP 4 Di erences with the continuous case in Π 3 = Π P IC \ ΠD UB and Π 4 = Π P IC \ ΠD IC

16 Stability of the primal-dual partition in LSIP 5 Some important sets: n a t C := conv The rst and the second moment cones of π: b t o : t 2 T n a t M := cone fa t : t 2 T g and N := cone b t o : t 2 T, The characteristic cone K : n 0 o n K := N + cone 1

17 Stability of the primal-dual partition in LSIP 6 π 2 int Π P B \ ΠD B, 0 n+1 /2 cl C c 2 int M π 2 int Π P UB \ ΠD IC 0 n /2 cl conv fa t : t 2 T g, (0 n ; 1) 0 /2 cl N c /2 cl M int Π P B \ ΠD B is dense in Π P B \ Π D B, if jt j n int Π P UB \ ΠD IC is dense in Π P UB \ Π D IC, for any T and any n

18 Stability of the primal-dual partition in LSIP 7 π 2 int Π P IC \ ΠD UB, M = R n, (0 n, 1) 0 2 int N 0 n /2 int convfa t : t 2 T g π 2 int Π P IC \ ΠD UB, M 6= R n, (0 n, 1) 0 2 O + (cl C ) c 2 int M int Π P IC \ ΠD UB is dense in Π P IC \ Π D UB, if jt j n + 1

19 Stability of the primal-dual partition in LSIP 8 π 2 int Π P IC \ ΠD IC 0 n /2 cl conv fa t : t 2 T g, (0 n, 1) 0 2 O + (cl C ) c /2 cl M int Π P B \ ΠD IC = int Π P IC \ ΠD B = Π P B \ ΠD IC bd ΠP B \ ΠD B \ bd Π P UB \ Π D IC Π P IC \ ΠD B bd ΠP IC \ ΠD IC

20 Part II: On coderivatives and Lipschitzian properties of the dual pair in optimization Characterizations of Lipschitzian stability of feasible solutions for the primal and the dual problem in an in nite-dimensional setting Cánovas, López, Mordukhovich & Parra I, II (2009, 2010) (primal problem) López, Ridol & V (2011) (primal and dual problem with a conic constraint) Du n (1956), Kretschmer (1961) Anderson & Nash (1987) Linear Programming and In nite-dimensional Spaces Dinh, Goberna & López (2010)

21 An in nite-dimensional linear optimization problem T X P : Sup hc, xi st ha t, xi b t, t 2 T, x 2 Q, an arbitrary index set, possibly in nite real Banach space (not necessarily re exive) Q closed convex cone in X c in X fa t, t 2 T g X bounded A : X! ` (T ) de ned as Ax := ha (), xi b t, t 2 T, real numbers

22 P can be reformulated P : Sup hc, xi st Ax b, x 2 Q b = b t t2t The associated dual problem D : D : Inf st µ, b A µ 2 c Q, µ 0, µ 2 ` (T ) = µ : 2 T! R: µ is bounded and additive A : ` (T )! X adjoint operator of A Q the dual cone of Q

23 Tools from variational analysis: coderivative and its norm

24 Tools from variational analysis: coderivative and its norm their relationship with the exact bound of the Lipschitzian moduli

25 M : Z Y (bz, by) 2 gph M, the coderivative of M at (bz, by) is D M (bz, by) : Y Z D M (bz, by) (y ) := fz 2 Z : (z, y ) 2 N((bz, by) ; gph M)g, y 2 Y N((bz, by) ; gph M) the limiting normal cone to gph M at (bz, by) (the normal cone when gph M is convex) The norm of this coderivative is kd M (bz, by)k := sup fkz k : z 2 D M (bz, by) (y ), ky k 1g

26 For a C 1 function f : R! R we have D f (z, f (z)) (y) = f 0 (z) y Let Z = R n, Y = R m For a C 1 function f : R n! R m, the coderivative D f (bz, by) : R m! R n is the adjoint operator of the di erential map df (bz, by) : R n! R m D f (bz, by) (y) = rf (bz) y

27 Io e and Sekiguchi (2009) de ned a convex set-valued mapping M 1 as being perfectly regular at (by, bz) 2 gph M 1 if and only if sur M 1 (by, bz) = inf ky k : y 2 D M 1 (by, bz) (z ), kz k = 1 If M 1 is perfectly regular at (by, bz) 2 gph M 1, the following equality holds: lip M (bz, by) = kd M (bz, by)k

28 We allow for perturbations c 2 X and b 2 ` (T ) of the xed c and b perturbed primal problem P (b, c ) : Sup hc + c, xi st hat, xi b t + b t, t 2 T, x 2 Q, perturbed dual problem D (b, c ) : Inf st µ, b + b A µ 2 c + c Q, µ 0

29 Feasible mappings F P : ` (T ) X F P (b) := fx 2 X : Ax b + b and x 2 Qg F D : X ` (T ) F D (c ) := fµ 2 ` (T ) : A µ 2 c + c Q and µ 0g

30 The corresponding inverse mappings F 1 P (x) := ( Ax b + ` (T ) +, if x 2 Q,, if x /2 Q, and F 1 D (µ) := ( A µ c + Q, if µ 0,, otherwise

31 Reformulations to apply known results to study their stability properties Choose a convenient closed bounded set eq not containing the null vector and spanning the cone Q Write P (b, c ) : Sup hc, xi + hc, xi st ha t, xi b t + b t, t 2 T, hq, xi 1, q 2 Q, D (b, c ) : Inf st µ, b + hµ, bi hµ, Aqi hc, qi + hc, qi, q 2 eq, hµ, pi 1, p 2 ` (T ) +,

32 F P satis es the strong Slater condition at b 2 ` (T ) if there is some bx 2 Q such that sup t2t ha t, bxi b t b t < 0 F D satis es the strong Slater condition at c 2 X if there is some bµ 2 ` (T ), bµ 0, such that inf q2 eq fhbµ, Aqi hc + c, qig > 0

33 Q has a compact base eq if there is some x 2 X, kx k = 1, such that the set eq = fq 2 Q : hx, qi = 1g, is weakly compact and spans Q Proposition If v P is nite, Q has a compact base, and there exists a strong Slater bµ point of F D at 0, then there is no-duality gap (ie v P = v D ) and P is solvable

34 Theorem Let F P (b) 6= If int Q 6=, then F P is Lipschitz-like around (b, x) for all x 2 F P (b) if and only if there exists some x 2 X such that (b, x) 2 int (gph F P ) Theorem Let bx 2 F P (0), then F P is Lipschitz-like around (0, bx) if and only if D F P (0, bx)(0) = f0g Theorem Let bx 2 F P (0), then lip F P (0, bx) = kd F P (0, bx)k

35 Characterization of the coderivative for the primal problem D F P (0, bx) Theorem Let bx 2 F P (0), µ 2 ` (T ), and x 2 X Then µ 2 D F P (0, bx) (x ) if and only if (µ, x, hx, bxi) 2 cl cone δ t, at, b t, t 2 T + f0g Q f0g

36 Norm of the coderivative for the primal problem Theorem Let bx 2 F P (0) Then, (i) If bx is a strong Slater point of F P at b = 0, then kd F P (0, bx)k = 0 (ii) If bx is not a strong Slater point of F P at b = 0, then kd F P (0, bx)k > 0 and it can be calculated as n o kd F P (0, bx)k = sup kx k 1 : (x, hx, bxi) 2 cl C P (0) The characteristic set of F P (b) is C P (b) := conv a t, b t + b t : t 2 T [ (Q f1g)

37 Consistency of the dual problem c 2 X and the linear system in ` (T ) ( hµ, Aqi hc σ D (c + c, qi, q 2 eq, ) := hµ, pi 1, p 2 ` (T ) + ), Theorem σ D (c ) is consistent () (0, 1) /2 cl H(c ), where H(c ) = f(ax, hc + c, xi) : x 2 Qg + ` (T ) + ( R + )

38 Characterization of stably consistent dual problems De nition A dual problem D (b, c ) is stably consistent if and only if c 2 int (dom F D ) This condition is equivalent to F D being Lipschitz-like around (c, µ) for all µ 2 F D (c ), because the graph of FD 1 : ` (T ) X is closed and convex, and ` (T ) and X are Banach spaces

39 Proposition Let c 2 dom F D Suppose that 0 /2 cl conv eq, then the following statements are equivalent: (i) There is some bµ 0 that is strong Slater for F D at c (ii) (0, 0) /2 cl C D (c ) (iii) c 2 int (dom F D ) The characteristic set of F D (c ), relative to eq, is the following subset of ` (T ) R: n o! C D (c (Aq, hc ) := conv + c, qi) : q 2 eq [ (p, 1) : p 2 ` (T ) +

40 Remark The hypothesis 0 /2 cl conv eq is only needed for the implication (iii) ) (i) The existence of a strong Slater point implies the condition 0 /2 cl conv eq (ii) (0, 0) /2 cl C D (c ) is equivalent to (ii) 0 (0, 0) /2 cl C D (c ), (we always consider the w topology on ` (T ) )

41 Characterization of the normal cone for the dual problem N ((bc, bµ) ; gph F D ) Let (bc, bµ) 2 gph F D and let (x, b ) 2 X ` (T ) Then (x, b ) 2 N ((bc, bµ) ; gph F D ) if and only if (x, b, hbc, x i + hbµ, b i) belongs to cl f( q, Aq, hc, qi) : q 2 Qg + f0g ` (T ) + ( R + )

42 Lemma Let bµ 2 F D (0), x 2 X and b 2 ` (T ) If x 2 D F D (0, bµ)(b ) then there exists a net f(q ν, p ν g ν2n Q ` (T ) + such that x = w - lim ν2n q ν, b = w - lim ν2n (Aq ν + p ν ), hbµ, b i = lim ν2n hc, q ν i Moreover, if bµ is strong Slater for F D at 0, then x = 0

43 Characterization of the coderivative of the dual problem Theorem Let bµ 2 F D (0) If x 2 X and b 2 ` (T ), then x 2 D F D (0, bµ)(b ) if and only if (x, b, hbµ, b i) belongs to cl f(q, Aq, hc, qi) : q 2 Qg + f0g ` (T ) + f0g

44 Estimate of the norm of the coderivative (A): eq is a closed spanning subset of Q such that there are two positive real numbers r and R, and some x 2 X, kx k = 1, satisfying for all q 2 eq 0 < r hx, qi kqk R Theorem Suppose that Q satis es condition (A) and that bµ 2 F D (0) Then, (i) If bµ is a strong Slater point of F D at 0, then kd F D (0, bµ)k = 0 (ii) If bµ is not a strong Slater point of F D at 0, then kd F D (0, bµ)k > 0 and where r kd F D (0, bµ)k R n o := sup kb k 1 : (b, hbµ, b i) 2 cl C D (0)

45 Example (Constants r = R = 1) T = N, X = `1 Q = f(x n ) 2 `1 : x n 0 for all n 2 Ng c = 1 := (1n) 2 ` = X, an := en 2 `, n = 1, 2, eq := fq 2 Q : kqk 1 = 1g Condition (A) holds for eq with r = R = 1 µ = 0 is a strong Slater point of F D at 0, the Dirac measure bµ = δ 1 2 F D (0) is not The equality n o kd F D (0, bµ)k = sup kb k 1 : (b, hbµ, b i) 2 cl C D (0) takes place at (0, bµ) = (0, δ 1 )

46 Example Let X = R 2 T = N, c = (0, 1), a n := (1, 2) for all n, bµ = δ 1, Q = (x, y) 2 R 2 : y jxj, Take eq 1 = f(x, y) 2 Q : jxj + jyj = 1g The largest r and smallest R we can choose for eq 1 are r 1 = 1 2 and R 1 = 1 Another possibility is to take eq 2 = f(x, y) 2 Q : jxj 2, y = 2g Now r 2 = 2 (h(0, 1), (x, y)i = y 2) and R 2 = 2 p 2

47 We have, for i = 1, 2, CD i (0, 0) := n conv (x + 2y) n2n, y o : (x, y) 2 eq i [ (p, 1) : p 2 ` (N) + In particular, C 2 D (0, 0) = conv (x + 4) n2n, 2 : jxj 2 [ (p, 1) : p 2 ` (N) + Then: n o (i) sup kb k 1 : (b, hbµ, b i) 2 cl CD 1 (0, 0) = 2, n o (ii) sup kb k 1 : (b, hbµ, b i) 2 cl CD 2 (0, 0) = 1 2 n o and R 2 sup kb k 1 : (b, hbµ, b i) 2 cl CD 2 (0, 0) = p 2, (iii) kd F D (0, bµ)k = p 2, n o so kd F D (0, bµ)k = R 2 sup kb k 1 : (b, hbµ, b i) 2 cl CD 2 n o (0, 0) < R 1 sup kb k 1 : (b, hbµ, b i) 2 cl CD 1 (0, 0)

48 Theorem Let bµ 2 F D (0) and suppose that 0 /2 cl conv eq Then, F D is Lipschitz-like around (0, bµ) if and only if D F D (0, bµ)(0) = f0g Remark If the condition 0 /2 cl conv eq does not hold, we may have D F D (0, bµ)(0) = f0g and F D not Lipschitz-like around (0, bµ)

49 Example T = ft 0 g X = c 0 the Banach space of bounded real sequences converging to 0, with the supremum norm X = `1 = `1 (N) and X = ` = ` (N) Q = fq 2 c 0 : q n 0, n 2 Ng and eq = fq 2 Q : kqk = 1g a t 0 = c = 1 n! n=1 2 `1 Observing that each e n = (0,, 0, 1, 0, ), is in eq, it is easy to see that bµ = 1 2 F D (0) D F D (0, bµ)(0) = f0g c = 0 /2 int (dom F D ) F D is not Lipschitz-like around (0, 1)

50 Estimate of the exact Lipschitzian bound for the dual problem lip F D (0, bµ) = Extended Ascoli distance formula dist (µ, F D (c )) = lim sup (c,µ)!(0,bµ) dist (µ, F D (c )) dist c, FD 1(µ) [α hb, µi] sup + (b,α)2cl C D (c ) kb k holds true when F D satis es the strong Slater condition at c If kqk R for all q 2 eq Let c 2 X and µ 2 ` (T ) be such that (c, µ) /2 gph F D and FD 1 (µ) 6= Then dist c, FD 1(µ) R 1 sup [ hµ, Aqi + hc + c, qi] + > 0 q2 eq

51 Theorem Assume condition (A) and let bµ 2 F D (0) Then: (i) If bµ is a strong Slater point of F D at 0, then lip F D (0, bµ) = 0 (ii) If bµ is not a strong Slater point of F D at 0, then r lip F D (0, bµ) R where n o := sup kb k 1 : (b, hbµ, b i) 2 cl C D (0)

52 Example Let X = R 2 T = N, c = (0, 1), a n := (1, 2) for all n, bµ = δ 1, Q = (x, y) 2 R 2 : y jxj, eq = eq 2 = f(x, y) 2 Q : jxj 2, y = 2g Now r 2 = 2 and R 2 = 2 p 2, gives lip F D (0, bµ) = p 2 because p 2 = kd F D (0, bµ)k lip F D (0, bµ) n o R 2 max kb k 1 : (b, hbµ, b i) 2 cl CD 2 = p 2

53 Theorem In relation to the dual feasible set mapping F D the following two statements hold: (i) Assume condition (A) and suppose that F D satis es the strong Slater condition at c = 0 Let bµ 2 F D (0), then there are constants r and R, 0 < r R, which only depends on the cone Q, such that 0 lip F D (0, bµ) kd F D (0, bµ)k n o (R r) max kb k 1 : (b, hbµ, b i) 2 cl C D (0) (ii) If FD 1(` (T ) +) has nonempty interior for the norm topology in X and fq 2 Q : kqk = 1g is w -closed in X, then lip F D (0, bµ) = kd F D (0, bµ)k

54 The equality lip F D (0, bµ) = kd F D (0, bµ)k may hold even when r 6= R and fq 2 Q : kqk = 1g is not w -closed in X Example X = c 0 the Banach space of bounded real sequences converging to 0, with the supremum norm X = `1 = `1 (N) and X = ` = ` (N) Q = fq 2 c 0 : 2q 1 q n 0, n 2 Ng and eq = fq 2 Q : kqk = 1g ; 1 = (1) n=1 2 cl eq, so this set eq is not w -closed in ` T = ft 0 g and put at 0 := a 1 = 2 n 1 n=1 2 `1, also x c = a r = 1 2 is the largest r we can choose to satisfy hz, qi r for some z 2 `1, kz k 1 = 1, and for all q 2 eq Here R = 1 Then µ = 0 is a strong Slater point for F D at c = 0, while bµ = 1 2 F D (0) is not strong Slater n o! (ha C D (0) = conv, qi, ha, qi) : q 2 eq R 2 ; [ f(p, 1) : p 2 R, p 0g

55 n o kd F D (0, bµ)k sup kb k 1 : (b, hbµ, b i) = (b, b ) 2 cl C D (0) Also lip F D (0, bµ) 2 On the other hand, for each k 2 N, k 6= 1, q k de ned by q k 1 1 = 1 2 k 1, qk k+1 = 2q k 1, and qn k = 0 otherwise, belongs to D F D (0, bµ) b k, where b k := a, q k Since b k = a, q k 1 = 1 < 1, we obtain that 2 k(k 2) kd F D (0, bµ)k = sup fkz k : z 2 D F D (0, bµ) (b ), kb k 1g q k q = k 1 = k 1 By letting k! kd F D (0, bµ)k 2 It follows that kd F D (0, bµ)k = 2 Finally, since kd F D (0, bµ)k lip F D (0, bµ) 2, we obtain the equalities kd F D (0, bµ)k = lip F D (0, bµ) = 2

56 Part III: Stability of Voronoi cells 1 Goberna & V (2011) Nature provides beautiful examples:

57 The world s largest salt ats: Salar de Uyuni (Bolivia)

58 A salty Voronoi cell

59 Voronoi Cells Let T R n be the Voronoi sites (T with more than one element) The Voronoi cell of s 2 T is the set V T (s) = fx 2 R n : d (x, s) d (x, T n fsg)g Descartes (1644), Dirichlet (1850), Voronoi (1908) V T (s) = o nx 2 R n : (t s) 0 x ktk2 ksk 2 2, t 2 T Voigt (2008), Voigt & Weis (2010), Goberna, Rodríguez & V (2011, submitted)

60 Voronoi Diagram Vor (T ) := fv T (s) : s 2 T g (Wikipedia)

61 An unbounded Voronoi cell V T (s) is bounded if and only if s 2 int conv T T = f0 2 g [ (f 1g Z), s = 0 2

62 Unbounded set of sites Let T = 1 k 1, 1 k, k 1, k, k 2 N

63 k = 2, 10

64 Perturbations of the sites In many real applications, the position of some elements of T is uncertain What are the e ect on the Voronoi cell of small changes in s or in a given non empty set P T n fsg? Di erent types of perturbations can be considered: only the point s is allowed to be perturbed

65 Perturbations of the sites In many real applications, the position of some elements of T is uncertain What are the e ect on the Voronoi cell of small changes in s or in a given non empty set P T n fsg? Di erent types of perturbations can be considered: only the point s is allowed to be perturbed global perturbations of the subset of sites P

66 Perturbations of the sites In many real applications, the position of some elements of T is uncertain What are the e ect on the Voronoi cell of small changes in s or in a given non empty set P T n fsg? Di erent types of perturbations can be considered: only the point s is allowed to be perturbed global perturbations of the subset of sites P pointwise perturbations of P

67 Perturbations of the sites In many real applications, the position of some elements of T is uncertain What are the e ect on the Voronoi cell of small changes in s or in a given non empty set P T n fsg? Di erent types of perturbations can be considered: only the point s is allowed to be perturbed global perturbations of the subset of sites P pointwise perturbations of P simultaneous perturbations in s and P

68 Stability of Voronoi cells 1 Strong Slater condition: Given a set P with 6= P T n fsg, an element x of the set n o XP s := x 2 R n : (t s) 0 x ktk2 ksk 2 2, t 2 T np is called a strong Slater point for the pair (P, s) with associated scalar δ > 0 whenever (t s) 0 x ktk2 ksk 2 2 δ, for all t 2 P When there exists a strong Slater point for (P, s) we say that P satis es the strong Slater condition for s

69 Stability of Voronoi cells 2 Proposition Given a set P such that 6= P T n fsg, the following statements are equivalent: (i) P satis es the strong Slater condition for s (ii) s /2 cl Q, ie, d (s, P) > 0 (iii) s is a strong Slater point for (P, s) Proposition Assume that P is bounded If x is a strong Slater point for (P, s), then there exists ε > 0 such that x is a strong Slater point for any pair (P 1, s) such that P 1 P + εb n

70 Stability of Voronoi cells Case 1 Case 1: only s can be perturbed Parameter space Ω 1 = R n V : R n R n is the mapping ( ) V (s 1 ) = x 2 R n : (t s 1 ) 0 x ktk2 ks 1 k 2, t 2 T n fsg 2 for any s 1 2 R n Theorem The following statements are true for any s 2 T : (i) V is closed at s (ii) V is lsc at s if and only if V T (s) = fsg or s is an isolated point of T (iii) V is usc at s if V T (s) is bounded

71 Stability of Voronoi cells Case 2 Case 2: P is a given nonempty compact subset of sites, s /2 P Global perturbations of P that are also nonempty compact sets not containing s are allowed s is kept xed Parameter space Ω : the family of nonempty compact subsets of R n equipped with the Hausdor distance d H The set-valued mapping to be analyzed is V : Ω R n ( ) V (P 1 ) = x 2 X : (t s) 0 x ktk2 ksk 2, t 2 P 1, 2 for P 1 2 Ω, where X := X s P = will remain xed o nx 2 R n : (t s) 0 x ktk2 ksk 2 2, t 2 T np V (P) = V T (s)

72 Stability of Voronoi cells Case 2 Theorem The following statements are true for any P T, P 2 Ω : (i) V is closed at P (ii) V is lsc at P (iii) V is usc at P if V T (s) is bounded

73 Stability of Voronoi cells Case 3 Case 3: P is a given nonempty bounded subset of sites, s /2 P Pointwise perturbations on P, s /2 P remains xed Pointwise perturbations of P: The result of perturbing P is represented by the (bounded) range, f (P), of certain f 2 l n (P) The set P is represented by the identity mapping on it, say i P The parameter space is the Banach space l n (P) of all bounded functions from P R n to R n equipped with the norm kf k := sup kf (t)k, for all f 2 l n (P) t2p

74 Stability of Voronoi cells Case 3 The set-valued mapping to be analyzed is V : l n (P) R n de ned by ( ) V (f ) = x 2 X : (f (t) s) 0 x kf (t)k2 ksk 2, t 2 P, 2 for all f 2 l n (P), where X := X s P In this framework we have: V (i P ) = V T (s) If T is nite, then V (f ) = V (T np )[f (P ) (s) is always a polyhedral convex set because (T np) [ f (P) is nite If T is discrete, (T np) [ f (P) is discrete too because it is the union of discrete sets, so that V (f ) is quasipolyhedral for all f 2 l n (P) The boundedness of V T (s) is maintained in some neighborhood of i P

75 Stability of Voronoi cells Case 3 The number of extreme points (or facets) of a Voronoi cell can change even for arbitrarily small pointwise perturbations Let T = f(0, 0), (1, 0), (2, 0)g, P = f(2, 0)g, and s = 0 2 l n (P) = R 2 and V (i P ) = x 2 R 2 : x has a unique facet and no extreme point If f (2, 0) = (a, b) 2 R 2, we have V (f ) = x 2 R 2 : x 1 12, ax 1 + bx 2 a2 + b 2 2 and this polyhedral convex set has two facets and one extreme point whenever b 6= 0,

76 Stability of Voronoi cells Case 3 Theorem Given s 2 T np, the following statements are true: (i) V is closed at i P (ii) V is lsc at i P if and only if V T (s) = fsg or P satis es the strong Slater condition for s (iii) V is usc at i P if V T (s) is bounded

77 Stability of Voronoi cells Case 3 The assumption V (i P ) 6= fsg gives that s /2 cl P is equivalent to V being lsc at i P Since V is closed, the Painlevé-Kuratowski characterization applies: Corollary If V (i P ) 6= fsg, then s /2 cl P if and only if for any sequence ff k g l n (P), lim inf k! V (f k ) = lim sup k! V (f k ) = V (i P ) Corollary If V T (s) = V (i P ) 6= fsg, then s /2 cl P if and only if V T (s) is the set formed by all the possible cluster points of sequences fy k g with y k 2 V (f k ), for all the sequences ff k g l n (P), f k! i P, and for any x 2 V T (s) there exists a convergent sequence fx k g such that x k 2 V (f k ) and lim k! x k = x

78 Stability of Voronoi cells Simultaneous perturbations in s and P : the nominal set P will be replaced by a new set P 1, the nominal site s by a new site s 1, new set of sites (T n (P [ fsg)) [ P 1 [ fs 1 g

79 Stability of Voronoi cells Case 4 Case 4: P is a given nonempty compact subset of sites, s /2 P Global perturbations of P that are also nonempty compact sets not containing s are allowed The parameter space Ω 2 is the family of pairs (P 1, s 1 ) such that P 1 is a nonempty compact set, s 1 2 R n, and s 1 /2 P 1, equipped with the distance d ((P 1, s 1 ), (P 2, s 2 )) := max fd H (P 1, P 2 ), d (s 1, s 2 )g The corresponding set-valued mapping is V : Ω 2 R n such that ( ) V (P 1, s 1 ) = x 2 X (s 1 ) : (t s 1 ) 0 x ktk2 ks 1 k 2, t 2 P 1, 2 where X (s 1 ) := ( ) x 2 R n : (t s 1 ) 0 x ktk2 ks 1 k 2, t 2 T np, t 6= s, t 6= s 1 2

80 Stability of Voronoi cells Case 4 Theorem Global perturbations: For V : Ω 2 R n and any xed (P, s) 2 Ω 2 : (i) V is closed at (P, s) (ii) V is lsc at (P, s) (iii) If V T (s) is bounded, then V is locally bounded at (P, s) (iv) V is usc at (P, s) if V T (s) is bounded

81 Stability of Voronoi cells Case 5 Case 5: P is a given nonempty bounded subset of sites, s /2 P Pointwise perturbations of P: The result of perturbing P is represented by the (bounded) range, f (P), of certain f 2 l n (P) The set P is represented by the identity mapping on it, say i P The parameter space Ω 3 is the Banach space l n (P) R n with the product topology V : Ω 3 R n is de ned by ( ) V (f, s 1 ) = x 2 X (s 1 ) : (f (t) s 1 ) 0 x kf (t)k2 ks 1 k 2, t 2 P 2

82 Stability of Voronoi cells Case 5 Theorem Pointwise perturbations: For V : Ω 2 R n, given s 2 T np, the following statements are true: (i) V is closed at (i P, s) (ii) V is lsc at (i P, s) if and only if V T (s) = fsg or s is isolated in T (iii) If V T (s) is bounded, then V is locally bounded at (i P, s) (iv) V is usc at (i P, s) if V T (s) is bounded T nite set of sites: in all the cases V is closed and lower semicontinuous In particular we always have the Painlevé Kuratowski characterization of V T (s)

83 Stability of Voronoi cells Case 5 The lsc and the usc properties may fail simultaneously in cases where P is a non closed set with s 2 bd P, and s /2 int conv T Example Let T = t 2 R 2 : kt e 1 k = 1, s = 0 2 and and P = T n f0 2 g (a) For any ε, 0 < ε < 1, let f ε : P! R 2 be such that f ε (t 1, t 2 ) = (t 1 ε, t 2 ) Then, V is not lsc at (i P, s) Indeed, V (i P, s) is not bounded and V (f ε, s) is bounded

84 Stability of Voronoi cells Case 5 Example (b) For any ε, 0 < ε < 1, let f ε : P! R 2 be such that f ε (t) = (1 cos ε, sin ε), if t 1 < 1 cos ε, and 0 < t 2 < sin ε, and f ε (t) = t, otherwise Then kf ε i P k < p 2 (1 cos ε)! 0 as ε # 0 whereas n V (f ε, s) = x 2 R 2 : x 1 + cot ε o x 2 1, x is not contained in V (i P, s) + B 2 for any ε Therefore V is not usc at (i P, s)

85 References Anderson E, Nash P (1987) Linear Programming and In nite-dimensional Spaces, John Wiley and Sons, Chichester Azè D, A uni ed theory for metric regularity of multifunctions, J Convex Analysis 13 (2006), Borwein JM (1983) Semi-in nite programming duality: How special is it?, in AV Fiacco and KO Kortanek (eds), Semi-In nite Programming and Applications, Springer-Verlag, Berlin Bonnans JF, Shapiro A (2000) Perturbation Analysis of Optimization Problems, Springer-Verlag, New York Borwein JM, and Zhu J (2005) Techniques of Variational Analysis, Springer-Verlag Cánovas, MJ, Dontchev, AL, López, MA and Parra, J (2005), Metric regularity of semi-in nite constraint systems, Math Program, Ser B 104,

86 Cánovas MJ, Klatte D, López MA, Parra J (2007) Metric regularity in convex semi-in nite optimization under canonical perturbations SIAM J Optim 18: Cánovas MJ, López MA, Mordukhovich BS, Parra J (2009) Variational Analysis in Semi-in nite and In nite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions SIAM J Optim 20, no 3, Cánovas MJ, López MA, Mordukhovich BS, Parra J (2010) Variational Analysis in Semi-in nite and In nite Programming, II: Necessary optimality conditions SIAM J Optim 20, no 3, Dinh, N, Goberna, MA, López, MA, Son, TQ: (2007) New Farkas-type constraint quali cations in convex in nite programming ESAIM: Control, Optim Cal Variations, 13, Dinh N, Goberna MA, López MA (2010) On the stability of the feasible set in optimization problems SIAM J Optim 20,

87 Dontchev AL and Rockafellar, RT, Regularity and Conditioning of Solution Mappings in Variational Analysis, Set-Valued Analysis, 12, (2004) Dontchev AL, Lewis AS and Rockafellar, RT, The radius of metric regularity, Trans Amer Math Soc, 355, (2003) Du n RJ (1956) In nite Programs Linear inequalities and related systems, Annals of Mathematics Studies 38, Goberna, MA and López, MA (1998), Linear Semi-In nite Optimization, J Wiley, Chichester Goberna MA, Rodríguez MML, Vera de Serio VN (2011), Voronoi cells of arbitrary sets via linear inequality systems, submitted

88 Goberna MA and Vera de Serio VN (2011), On stability of Voronoi cells, to apper in TOP DOI: /s Gruber PM (2007) Convex and discrete geometry Springer-Verlag, Berlin Io e AD (2000), Metric regularity and subdi erential calculus, Uspehi Mat Nauk 55:3, (in Russian), English translation: Russian Math Surveys, 55:3, Io e AD, Sekiguchi Y (2009) Regularity estimates for convex multifunctions Math Program, Ser B 117, Io e AD (2010) Io e AD (2010) On stability of solutions to systems of convex inequalities CRM Preprints (984) Kretschmer KS (1961) Programmes in paired spaces Canad J Math 13,

89 López MA, Ridol AB, and Vera de Serio VN (2011), On coderivatives and Lipschitzian properties of the dual pair in optimization, to appear in J Nonlinear Analysis DOI: /jna Mordukhovich BS (2006) Variational Analysis and Generalized Di erentiation I:Basic Theory, Springer, Berlin Ochoa PD and Vera de Serio VN (2011), Stability of the primal-dual partition in linear semi-in nite programming, to appear in Optimization DOI: / Rockafellar RT and Wets RB, Variational Analysis, Springer-Verlag, New York (1998) Voigt I (2008) Voronoizellen diskreter Punktmengen (German) Dissertation, TU Dortmund University Voigt I, Weis S (2010) Polyhedral Voronoi cells Contr Beiträge Algebra Geom 51:

90 The end!