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Quasi-Relative Interior and Optimization

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Quasi-Relative Interior and Optimization Aim and framework Separation theorems Duality results in scalar optimization Quasi-relative interior and optimization Constantin Z˘alinescu University \Al. I. Cuza" Ia¸si,Faculty of Mathematics CRESWICK, 2016 C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Separation theorems Duality results in scalar optimization 1 Aim and framework Aim Framework and notation The quasi-relative interior 2 Separation theorems 3 Duality results in scalar optimization Sets of epigraph type and their associated functions Lagrange duality for minimization problems with constraints Duality results for (almost) convex minimization problems C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Aim Many results in Optimization Theory are obtained using interiority conditions. However, in infinite dimensional spaces there are many convex sets with empty interior (even empty algebraic interior). Such sets are the natural (usual) cones in Lp spaces (p 2 [1; 1)). A substitute for the interior is the so called \quasi (relative) interior", which coincides with the (relative) interior in the case of convex sets in finite dimensional spaces. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior The first sistematic use of the quasi relative interior in Convex Optimization was accomplished in Borwein{Lewis (1992). A turnaround happened in the last ten years when several papers were published making use of the quasi relative interior. Our aim is to point out some properties of the quasi relative interior, and, using them, to present in a unified manner several results, some of them under (even) weaker conditions. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Framework and notation X is a real separated locally convex space whose topological dual is X ∗ endowed with its weak∗ topology (so (X ∗)∗ = X ) For x 2 X and x∗ 2 X ∗ we set hx; x∗i := x∗(x) For ;= 6 A ⊂ X , conv A, cone A, lin A, aff A are the convex, conic, linear and affine hulls of A, respectively; lin0 A is the linear space parallel with aff A. The closure of the set conv A is denoted by convA, and similarly for the others. For x 2 A, the normal cone to A at x is defined by ∗ ∗ 0 ∗ 0 NA(x) := fx 2 X j hx − x; x i ≤ 0 8x 2 Ag: For ;= 6 A ⊂ X we set A+ := fx∗ 2 X ∗ j hx; x∗i ≥ 0 8 x 2 Ag; A− := −A+ A# := fx∗ 2 X ∗ j hx; x∗i > 0 8 x 2 A n f0gg C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior The quasi-relative interior All the sets in this section are assumed to be convex if not mentioned explicitly otherwise. Let C ⊂ X ; the quasi interior of C (cf. Borwein{Goebel 2003) is qi C := fx 2 C j cone(C − x) = X g ; the quasi-relative interior of C (cf. Borwein{Lewis 1992) is qri C := fx 2 C j cone(C − x) is a linear spaceg : We set qri ; := qi ; := ;. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior These notions extend those of algebraic interior (or core) and algebraic relative interior (or intrinsic core) as easily seen from x 2 icr C () cone(C − x) is a linear space, x 2 core C () cone(C − x) = X : Clearly icr C ⊂ qri C; core C ⊂ qi C: The inclusions become equalities if dim X < 1. For dim X = 1 the inclusions could be strict even if icr C 6= ; or core C 6= ;. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Example 1 (i) (Borwein-Lewis, 1992) Let C1 := fx 2 `2 j kxk1 ≤ 1g: C1 is closed (in `2), icr C1 = fx 2 `2 j kxk1 < 1g, and qri C1 = qi C1 = C1 n fx 2 `2 j kxk1 = 1; 9n0; 8n ≥ n0 : xn = 0g: −n For example (2 )n≥1 2 qri C1 n icr C1. (ii) Let ' : X ! R be linear but not continuous and C2 := [' ≥ 0] := fx 2 X j '(x) ≥ 0g. Then core C2 = [' > 0] 6= [' ≥ 0] = C2 = qi C2: C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior From the definition of qri C we get qri C = x 2 C j cone(C − x) = lin0C : Because affC = X () lin0C = X , qri C if affC = X ; qi C = ; otherwise. It follows that qi C 6= ; =)0 2 qi(C − C) () affC = X () lin0C = X =) qi C = qri C: C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Observe that [x 2 X ; cone(C − x) is a linear space] ) x 2 cl C; and so, qi C = C \ qi(cl C); qri C = C \ qri(cl C): This shows that we may concentrate on the case of closed convex sets when determining the quasi-relative interior. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Example 2 (empty quasi-relative interior) Consider + c00 := f(xn)n≥1 ⊂ R j 9n0; 8n ≥ n0 : xn = 0g; C := c00; clearly C ⊂ c00 ⊂ `2. We endow c00 with the norm k · k2. Then (i) qri`2 C = ;; (ii) C is closed in c00 and qric00 C = ;. + (i) Indeed, because cl`2 C = `2 , we get + qri`2 C = C \ qri`2 `2 = C \ f(xn)n≥1 2 `2 j 8n ≥ 1 : xn > 0g = ;. (ii) The closedness of C in c00 is rapid. For the conclusion one uses the fact that aff C = C − C = c00 and some more calculus. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Fact (Borwein{Lewis 1992) If X is separable and Fr´echetand C ⊂ X is closed and convex then qri C 6= ;. The preceding example shows that completeness of X and closedness of C are essential in the preceding result. Fact (Borwein{Lewis 1992) One has (1 − λ)C + λ qri C ⊂ qri C 8λ 2 (0; 1): From here we get rapidly that qri C 6= ;) [cl C = cl(qri C) ^ lin0C = lin0(qri C)]: C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Another important property is the following Fact (Borwein{Lewis 1992) Assume that T 2 L(X ; Y ), where Y is another separated lcs, and C ⊂ X is convex. Then T (qri C) ⊂ qri T (C) with equality if qri C 6= ; and dim Y < 1. From the very definition we have that qri(x + C) = x + qri C. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Having the convex sets A; B ⊂ X , it is clear that qi A ⊂ qi B provided A ⊂ B; moreover qri A ⊂ qri B ⊂ qri(cl A); A ⊂ B ⊂ cl A ) qri A = A \ qri B = A \ qri(cl A): From these we get rapidly qri(qri C) = qri C; qi(qi C) = qi C: C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Proposition (Z. 2015) Let C; D ⊂ X be convex sets; then C + qi D = qi(C + qi D) ⊂ qi(C + D); qri C + qri D = qri(qri C + qri D) ⊂ qri(C + D) Borwein and Goebel (2003) say \Can qri C + qri D be a proper subset of qri(C + D)? (Almost certainly such sets do exist.)", while Grad and Pop (2014) say: \we conjecture that in general when A; B ⊆ V are convex sets with qi B 6= ;, it holds A + qi B = qi(A + B)". The next example answers to both problems mentioned above in the sense that the answer to Borwein and Goebel question is affirmative and that Grad and Pop's conjecture is false. C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior Example 3 (Z. 2015) P 2 Take X := `2 := (xn)n≥1 ⊂ R j n≥1 xn < 1 endowed with its −1 usual norm, x := (n )n≥1 2 `2; C := [0; 1]x ⊂ `2 and + P D := `1 := (xn)n≥1 ⊂ R+ j n≥1 xn < 1 ⊂ `2. Clearly C and D are convex sets, qri C = (0; 1)x, qri D = qi D = (xn)n≥1 ⊂ `1 j xn > 0 8n ≥ 1 and x 2 qi(C + D) = qri(C + D), but x 2= C + qi D ⊃ qri C + qri D: C. Z˘alinescu Quasi-relative interior and optimization Aim and framework Aim Separation theorems Framework and notation Duality results in scalar optimization The quasi-relative interior It is worth observing that for x0 2 C we have that ∗ ∗ ∗ ∗ x0 2= qi C () 9x 2 X n f0g : inf x (C) = hx0; x i (that is x0 is a support point of C), or equivalently fx0g and C can be separated; so (Borwein{Lewis, 1992) x0 2 qi C () NC (x0) := @ιC (x0) = f0g: Similarly, for x0 2 C, ∗ ∗ ∗ ∗ ∗ x0 2= qri C () 9x 2 X : sup x (C) > inf x (C) = hx0; x i ; or equivalently fx0g and C can be properly separated; so (Borwein{Lewis, 1992) x0 2 qri C () NC (x0) is a linear space: C.
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