Some Contributions to Convex Infinite-Dimensional Optimization Duality

Transcription

1 Some Contributions to Convex Infinite-Dimensional Optimization Duality Marco A. López Alicante University King s College London Strand Campus June 2014

2 Introduction Consider the convex infinite programming problem (P) Min f (x) s.t. f t (x) 0, t 2 T, x 2 C, (1) where: T : index set (possibly infinite) C : convex subset of a lchtvs X (C 6= ) f ; f t, t 2 T : proper convex functions defined on X Feasible set of (P) : F \ C, where F := fx 2 X : f t (x) 0, t 2 Tg Optimal value: inf(p) 2 [, + ] Optimal set: S(P) = fx 2 F \ C : f (x) = inf(p)g (possibly, empty)

3 We consider two different associated duals: Lagrangian dual of (P) : (D) Max λ2r (T) + inf x2c ff (x) + t2t λ t f t (x)g (2) where o R (T) + nλ := (λ t ) t2t 2 R T + j only finitely many λ t > 0, and R (T) R T 3 hλ, f i := λ t f t (x) := t2t ( 0, if λ = 0T, t2supp λ λ t f t (x), if λ 6= 0 T.

4 We consider two different associated duals: Lagrangian dual of (P) : (D) Max λ2r (T) + inf x2c ff (x) + t2t λ t f t (x)g (2) where o R (T) + nλ := (λ t ) t2t 2 R T + j only finitely many λ t > 0, and R (T) R T 3 hλ, f i := λ t f t (x) := t2t Modified dual of (P) : ( ) Max λ2r (T) + 8f0 Tg Weak dual inequalities: ( 0, if λ = 0T, t2supp λ λ t f t (x), if λ 6= 0 T. inf x2c ff (x) + t2t λ t f t (x)g (3) sup( ) sup(d) inf (P) +. (4)

5 MAIN GOAL: To establish conditions for the converse strong Lagrangian duality (minsup duality) min(p) = sup(d), and also for strong Lagrangian duality (infmax duality) inf(p) = max(d).

6 MAIN GOAL: To establish conditions for the converse strong Lagrangian duality (minsup duality) min(p) = sup(d), and also for strong Lagrangian duality (infmax duality) inf(p) = max(d). Outline: Preliminaries Basic results for Infmax duality Minsup duality by assuming inf-compactness Minsup duality without inf-compactness Applications

7 Example 1 Let X = C = R 2, f (x) = exp (x 2 ), T = f1g, and f 1 (x) = x 1 + i RR+ (x). We compute sup ( ) = < max(d) = 0 (attained for λ = 0) < 1 = min (P) Slater condition does not guarantee even inf (P) = sup(d).

8 Example 1 Let X = C = R 2, f (x) = exp (x 2 ), T = f1g, and f 1 (x) = x 1 + i RR+ (x). We compute sup ( ) = < max(d) = 0 (attained for λ = 0) < 1 = min (P) Slater condition does not guarantee even inf (P) = sup(d). Example 2 Let X = C = R, f (x) = exp (x), T = f1g, and f 1 (x) = x. Then max ( ) = < max(d) = 0 = inf (P). In this case, Slater condition holds and, however, sup( ) 6= sup(d).

9 Example 1 Let X = C = R 2, f (x) = exp (x 2 ), T = f1g, and f 1 (x) = x 1 + i RR+ (x). We compute sup ( ) = < max(d) = 0 (attained for λ = 0) < 1 = min (P) Slater condition does not guarantee even inf (P) = sup(d). Example 2 Let X = C = R, f (x) = exp (x), T = f1g, and f 1 (x) = x. Then max ( ) = < max(d) = 0 = inf (P). In this case, Slater condition holds and, however, sup( ) 6= sup(d). Example 3 Let X = R, C = [ 1, 1], f (x) = x, T = f1g, and f 1 (x) = x if x 0, f 1 (x) = 0 if x < 0. Now max( ) = max(d) = min (P) = 0. However, Slater condition is not satisfied.

10 Preliminaries Given B X, we denote by co B, cone B, and aff B the convex hull of B, the smallest convex cone containing B [ f0 X g, and the smallest linear manifold containing B, respectively. The duality product of x 2 X (the topological dual of X) and x 2 X is represented by hx, xi. The positive and negative dual cones of a non-empty set C X are C + = fx 2 X : hx, xi 0 8x 2 Cg, and C = fx 2 X : hx, xi 0 8x 2 Cg. The recession cone of a non-empty convex set C X is C = fv 2 X : c + v 2 C 8c 2 Cg.

11 Infmax duality We are dealing with the problem (P) Min ff (x), s.t. f t (x) 0, t 2 T, x 2 Cg, assuming that C is closed and f, f t 2 Γ (X). The function h : X! R h := inf λ2r (T) + nf0 Tg! f + i C + λ t f t, t2t is crucial in our approach; h enjoys the following properties:

12 Infmax duality We are dealing with the problem (P) Min ff (x), s.t. f t (x) 0, t 2 T, x 2 Cg, assuming that C is closed and f, f t 2 Γ (X). The function h : X! R h := inf λ2r (T) + nf0 Tg! f + i C + λ t f t, t2t is crucial in our approach; h enjoys the following properties: (i) h is proper and convex, and h (0 X ) = sup( )

13 Infmax duality We are dealing with the problem (P) Min ff (x), s.t. f t (x) 0, t 2 T, x 2 Cg, assuming that C is closed and f, f t 2 Γ (X). The function h : X! R h := inf λ2r (T) + nf0 Tg! f + i C + λ t f t, t2t is crucial in our approach; h enjoys the following properties: (i) h is proper and convex, and h (0 X ) = sup( ) (ii) h = f + i C\F

14 Infmax duality We are dealing with the problem (P) Min ff (x), s.t. f t (x) 0, t 2 T, x 2 Cg, assuming that C is closed and f, f t 2 Γ (X). The function h : X! R h := inf λ2r (T) + nf0 Tg! f + i C + λ t f t, t2t is crucial in our approach; h enjoys the following properties: (i) h is proper and convex, and h (0 X ) = sup( ) (ii) h = f + i C\F (ii) h (0 X ) = inf (P) 2 R

15 Infmax duality We are dealing with the problem (P) Min ff (x), s.t. f t (x) 0, t 2 T, x 2 Cg, assuming that C is closed and f, f t 2 Γ (X). The function h : X! R h := inf λ2r (T) + nf0 Tg! f + i C + λ t f t, t2t is crucial in our approach; h enjoys the following properties: (i) h is proper and convex, and h (0 X ) = sup( ) (ii) h = f + i C\F (ii) h (0 X ) = inf (P) 2 R Γ (X ) : w -lsc proper convex functions.

16 Let us introduce the set A := S λ2r (T) + nf0 Tg epi! f + i C + λ t f t t2t

17 Let us introduce the set A := S λ2r (T) + nf0 Tg epi! f + i C + λ t f t t2t It holds that epi s h A epi h, (5) epi h = cl w A, (6) and h = (f + i C\M ) = h. (7)

18 Definition ([2]) Having two subsets A and B of a topological space (X, τ), A is said to be closed regarding to B if B \ cl A = B \ A.

19 Definition ([2]) Having two subsets A and B of a topological space (X, τ), A is said to be closed regarding to B if B \ cl A = B \ A. Remark Closedness of A regarding B is clearly stronger than the closedness of A in the topology induced by τ in B, as the last one only requires that B \ A be the intersection of a closed set in (X, τ) with B (not especifically cl A). We are now in a position to state the main result for infmax duality.

20 Definition ([2]) Having two subsets A and B of a topological space (X, τ), A is said to be closed regarding to B if B \ cl A = B \ A. Remark Closedness of A regarding B is clearly stronger than the closedness of A in the topology induced by τ in B, as the last one only requires that B \ A be the intersection of a closed set in (X, τ) with B (not especifically cl A). We are now in a position to state the main result for infmax duality. Theorem Assume that inf (P) < +. The following assertions are equivalent: (i) A is w -closed regarding to the set f0 X g R. (ii) inf (P) = max( ), including the value.

21 Minsup duality under inf-compactness Theorem x 2 h (0 X ), x 2 S (P) and min (P) = sup ( ).

22 Minsup duality under inf-compactness Theorem x 2 h (0 X ), x 2 S (P) and min (P) = sup ( ). g : X! R := R[ f g is said to be inf-compact (inf-locally-compact) when [g r] := fx 2 X : g (x) rg is a compact set (a locally compact set, respectively) for every r 2 R.

23 Minsup duality under inf-compactness Theorem x 2 h (0 X ), x 2 S (P) and min (P) = sup ( ). g : X! R := R[ f g is said to be inf-compact (inf-locally-compact) when [g r] := fx 2 X : g (x) rg is a compact set (a locally compact set, respectively) for every r 2 R. Additionally h (0 X ) 6=, h is finite and τ (X, X) -cont. at 0 X h2γ(x ) () h is w-inf-compact, i.e., the sublevel sets of h are σ (X, X )-compact.

24 We are now in position to state our first result.

25 We are now in position to state our first result. Theorem Assume that one of the following conditions holds: (a) 9λ 2 R (T) + such that the function f + i C + t2t λ t f t is w-inf-compact. (b) x 2 cone([ t2t dom ft ) such that f + i C+ x is w-inf-compact.

26 We are now in position to state our first result. Theorem Assume that one of the following conditions holds: (a) 9λ 2 R (T) + such that the function f + i C + t2t λ t f t is w-inf-compact. (b) x 2 cone([ t2t dom ft ) such that f + i C+ x is w-inf-compact. Then min(p) = sup(d) and S (P) is a non-empty weakly compact set.

27 We are now in position to state our first result. Theorem Assume that one of the following conditions holds: (a) 9λ 2 R (T) + such that the function f + i C + t2t λ t f t is w-inf-compact. (b) x 2 cone([ t2t dom ft ) such that f + i C+ x is w-inf-compact. Then min(p) = sup(d) and S (P) is a non-empty weakly compact set. For the linear SIP (X = X = R n ) problem (P) Min fhc, xi, s.t. hat, xi b, t 2 T, x 2 Cg, with c, at 2 R n, b t 2 R, the theorem (under (b)) asserts that, if there exists a 2 cone fat, t 2 Tg such that C \ [c + a 0] = f0 R ng, then min(p) = sup(d) holds and S (P) is compact.

28 We are now in position to state our first result. Theorem Assume that one of the following conditions holds: (a) 9λ 2 R (T) + such that the function f + i C + t2t λ t f t is w-inf-compact. (b) x 2 cone([ t2t dom ft ) such that f + i C+ x is w-inf-compact. Then min(p) = sup(d) and S (P) is a non-empty weakly compact set. For the linear SIP (X = X = R n ) problem (P) Min fhc, xi, s.t. hat, xi b, t 2 T, x 2 Cg, with c, at 2 R n, b t 2 R, the theorem (under (b)) asserts that, if there exists a 2 cone fat, t 2 Tg such that C \ [c + a 0] = f0 R ng, then min(p) = sup(d) holds and S (P) is compact. If C = R n corollary does not apply since [c + a 0] is either a halfspace or R n. So, inf-compactness is useless in linear SIP.

29 Minsup duality without inf-compactness Definition The function h 2 Γ(Y), with Y being a l.c.h.t.v.s., is quasicontinuous (Def in Laurent 72) if: a) aff dom h is closed and of finite codimension; b) ri dom h 6= and the restriction of h to aff dom h is continuous on ri dom h.

30 Minsup duality without inf-compactness Definition The function h 2 Γ(Y), with Y being a l.c.h.t.v.s., is quasicontinuous (Def in Laurent 72) if: a) aff dom h is closed and of finite codimension; b) ri dom h 6= and the restriction of h to aff dom h is continuous on ri dom h. Lemma If g 2 Γ (X), then g is w-inf-locally-compact if and only if g is τ (X, X)-quasicontinuous.

31 Minsup duality without inf-compactness Definition The function h 2 Γ(Y), with Y being a l.c.h.t.v.s., is quasicontinuous (Def in Laurent 72) if: a) aff dom h is closed and of finite codimension; b) ri dom h 6= and the restriction of h to aff dom h is continuous on ri dom h. Lemma If g 2 Γ (X), then g is w-inf-locally-compact if and only if g is τ (X, X)-quasicontinuous. Lemma Let h : Y! R be convex and quasicontinuous, and let y 0 2 Y be such that h (y 0 ) > and cl cone (dom h y 0 ) is a linear subspace. Then h (y 0 ) is the sum of a non-empty w -compact convex set and a finite dimensional linear subspace.

32 Minsup duality without inf-compactness Definition The function h 2 Γ(Y), with Y being a l.c.h.t.v.s., is quasicontinuous (Def in Laurent 72) if: a) aff dom h is closed and of finite codimension; b) ri dom h 6= and the restriction of h to aff dom h is continuous on ri dom h. Lemma If g 2 Γ (X), then g is w-inf-locally-compact if and only if g is τ (X, X)-quasicontinuous. Lemma Let h : Y! R be convex and quasicontinuous, and let y 0 2 Y be such that h (y 0 ) > and cl cone (dom h y 0 ) is a linear subspace. Then h (y 0 ) is the sum of a non-empty w -compact convex set and a finite dimensional linear subspace. If cl cone (dom h y 0 ) = Y, h is continuous at y 0, and h (y 0 ) is a non-empty w -compact convex set.

33 This is a minsup duality theorem without inf-compactness: Theorem Assume that sup( ) < + and that 9λ 2 R (T) + n f0 T g : f + i C + t2t λ t f t is w-inf-locally-compact, holds and that \ [f 0] \ C \ [(f t2t t) 0] is a linear subspace, (8) where [f 0] and [(f t ) 0] are the recession cones of f and f t. Then min(p) = sup(d), and S (P) is the sum of a non-empty w-compact convex set and a finite dimensional linear subspace.

34 Consider now the (primal) ordinary convex program (P) Min ff (x), s.t. f i (x) 0, i = 1,..., m, x 2 C R n g, and its dual (D) Max λ2r m + inf x2c f (x) + m i=1 λ if i (x).

35 Consider now the (primal) ordinary convex program (P) Min ff (x), s.t. f i (x) 0, i = 1,..., m, x 2 C R n g, and its dual (D) Max λ2r m + inf x2c f (x) + m i=1 λ if i (x). Applying the last theorem we get a very general form of Clark-Duffin Theorem (Duffin 78):

36 Consider now the (primal) ordinary convex program (P) Min ff (x), s.t. f i (x) 0, i = 1,..., m, x 2 C R n g, and its dual (D) Max λ2r m + inf x2c f (x) + m i=1 λ if i (x). Applying the last theorem we get a very general form of Clark-Duffin Theorem (Duffin 78): Corollary (Generalized Clark-Duffin Theorem) Let f, f 1,..., f m 2 Γ (R n ) and C convex closed in R n be such that! \ [f 0] \ C \ [(f i ) 0] i=1,...,m is a linear subspace. Then min(p) = sup(d) 2 R and S (P) is the sum of a non-empty compact convex set and a linear subspace.

37 If we deal with the linear SIP problem (P) Min fhc, xi, s.t. hat, xi b, t 2 Tg, with X = C = R n = X, \ [c 0] \ t2t [a t 0] is a linear subspace if and only if c 2 ri conefa t, t 2 Tg.

38 If we deal with the linear SIP problem (P) Min fhc, xi, s.t. hat, xi b, t 2 Tg, with X = C = R n = X, \ [c 0] \ t2t [a t 0] is a linear subspace if and only if c 2 ri conefa t, t 2 Tg. This condition guarantees min(p) = sup(d), and that S (P) is the sum of a non-empty compact convex set and a linear subspace.

39 If we deal with the linear SIP problem (P) Min fhc, xi, s.t. hat, xi b, t 2 Tg, with X = C = R n = X, \ [c 0] \ t2t [a t 0] is a linear subspace if and only if c 2 ri conefa t, t 2 Tg. This condition guarantees min(p) = sup(d), and that S (P) is the sum of a non-empty compact convex set and a linear subspace. If, additionally, conefat, t 2 Tg is full-dimensional, S (P) is a non-empty compact convex set.

40 Definition A family (C t ) t2t of sets of a topological space is said to have the finite intersection property if every finite subfamily has non-empty intersection. Proposition Let (C t ) t2t be a family of closed convex subsets of a lchtvs having the finite intersection property. Moreover, assume the existence of t 1,..., t m 2 T such that T m i=1 C ti is w locally-compact and T t2t (C t ) is a linear space. Then T t2t C t is the sum of a non-empty w compact convex set and a finite dimensional linear space.

41 Definition A family (C t ) t2t of sets of a topological space is said to have the finite intersection property if every finite subfamily has non-empty intersection. Proposition Let (C t ) t2t be a family of closed convex subsets of a lchtvs having the finite intersection property. Moreover, assume the existence of t 1,..., t m 2 T such that T m i=1 C ti is w locally-compact and T t2t (C t ) is a linear space. Then T t2t C t is the sum of a non-empty w compact convex set and a finite dimensional linear space. Proof: If we apply Theorem 7 with C = X, f 0, and f t = i Ct, t 2 T, we observe the following: S (P) = T t2t C t, rec (P) = T t2t (C t ), and sup( ) < + amounts to say that the family (C t ) t2t has the finite intersection property.

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