Lecture 13 Introduction to quasiconvex analysis Didier Aussel Univ. de Perpignan – p.1/48 Outline of lecture 13 I- Introduction II- Normal approach a- First definitions b- Adjusted sublevel sets and normal operator III- Quasiconvex optimization a- Optimality conditions b- Convex constraint case c- Nonconvex constraint case – p.2/48 Quasiconvexity A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if, for all x, y ∈ K and all t ∈ [0, 1], f(tx + (1 − t)y) ≤ max{f(x),f(y)}. – p.3/48 Quasiconvexity A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if, for all λ ∈ IR, the sublevel set Sλ = {x ∈ X : f(x) ≤ λ} is convex. – p.3/48 Quasiconvexity A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if, for all λ ∈ IR, the sublevel set Sλ = {x ∈ X : f(x) ≤ λ} is convex. A function f : X → IR ∪ {+∞} is said to be semistrictly quasiconvex on K if, f is quasiconvex and for any x,y ∈ K, f(x) <f(y) ⇒ f(z) <f(y), ∀ z ∈ [x,y[. – p.3/48 I Introduction – p.4/48 f differentiable f is quasiconvex iff df is quasimonotone iff df(x)(y − x) > 0 ⇒ df(y)(y − x) ≥ 0 f is quasiconvex iff ∂f is quasimonotone iff ∃ x∗ ∈ ∂f(x) : hx∗,y − xi > 0 ⇒∀ y∗ ∈ ∂f(y), hy∗,y − xi≥ 0 – p.5/48 Why not a subdifferential for quasiconvex programming? – p.6/48 Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of ∂f if f is not supposed to be Lipschitz – p.6/48 Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of ∂f if f is not supposed to be Lipschitz No sufficient optimality condition HH¨ x¯ ∈ Sstr(∂f,C)=¨¨⇒H x¯ ∈ arg min f C – p.6/48 II Normal approach of quasiconvex analysis – p.7/48 II Normal approach a- First definitions – p.8/48 A first approach Sublevel set: Sλ = {x ∈ X : f(x) ≤ λ} > Sλ = {x ∈ X : f(x) < λ} Normal operator: X∗ Define Nf (x) : X → 2 by Nf (x) = N(Sf(x),x) ∗ ∗ ∗ = {x ∈ X : hx ,y − xi≤ 0, ∀ y ∈ Sf(x)}. > With the corresponding definition for Nf (x) – p.9/48 But ... Nf (x) = N(Sf(x),x) has no upper-semicontinuity properties > > Nf (x) = N(Sf(x),x) has no quasimonotonicity properties – p.10/48 But ... Nf (x) = N(Sf(x),x) has no upper-semicontinuity properties > > Nf (x) = N(Sf(x),x) has no quasimonotonicity properties Example Define f : R2 → R by 8 |a| + |b| , if |a| + |b|≤ 1 f (a,b)= < . 1, if |a| + |b| > 1 : Then f is quasiconvex. Consider x = (10, 0), x∗ = (1, 2), y = (0, 10) and y∗ = (2, 1). We see that x∗ ∈ N <(x) and y∗ ∈ N < (y) (since |a| + |b| < 1 implies (1, 2) · (a − 10,b) ≤ 0 and (2, 1) · (a,b − 10) ≤ 0) while hx∗, y − xi > 0 and hy∗, y − xi < 0. Hence N < is not quasimonotone. – p.10/48 But ... Nf (x) = N(Sf(x),x) has no upper-semicontinuity properties > > Nf (x) = N(Sf(x),x) has no quasimonotonicity properties These two operators are essentially adapted to the class of semi- strictly quasiconvex functions. Indeed this case, for each x ∈ < dom f \ arg min f, the sets Sf(x) and Sf(x) have the same closure < and Nf (x) = Nf (x). – p.10/48 II Normal approach b- Adjusted sublevel sets and normal operator – p.11/48 Definition Adjusted sublevel set For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. – p.12/48 Definition Adjusted sublevel set For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. a > Sf (x) coincides with Sf(x) if cl(Sf(x)) = Sf(x) e.g. f is semistrictly quasiconvex – p.12/48 Definition Adjusted sublevel set For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. a > Sf (x) coincides with Sf(x) if cl(Sf(x)) = Sf(x) Proposition 1 Let f : X → IR ∪ {+∞} be any function, with domain dom f. Then a f is quasiconvex ⇐⇒ Sf (x) is convex , ∀ x ∈ dom f. – p.12/48 Proof a Let us suppose that Sf (u) is convex for every u ∈ dom f. We will show that for any x ∈ dom f, Sf(x) is convex. a If x ∈ arg min f then Sf(x) = Sf (x) is convex by assumption. Assume now that x∈ / arg min f and take y, z ∈ Sf(x). If both y and z belong to B S< ,ρ , then y, z ∈ Sa (x) thus [y, z] ⊆ Sa (x) ⊆ S . “ f(x) x” f f f(x) If both y and z do not belong to B S< ,ρ , then “ f(x) x” < < < f(x)= f(y)= f(z), Sf(z) = Sf(y) = Sf(x) and ρ ,ρ are positive. If, say, ρ ≥ ρ then y, z ∈ B S< ,ρ thus y z y z “ f(y) y” a a y, z ∈ Sf (y) and [y, z] ⊆ Sf (y) ⊆ Sf(y) = Sf(x). – p.13/48 Proof < < Finally, suppose that only one of y, z, say z, belongs to B(Sf(x),ρx) while y∈ / B(Sf(x),ρx). Then < < f(x)= f(y), Sf(y) = Sf(x) and ρy >ρx so we have z ∈ B S< ,ρ ⊆ B S< ,ρ and we deduce as before that “ f(x) x” “ f(y) y” a [y, z] ⊆ Sf (y) ⊆ Sf(y) = Sf(x). The other implication is straightforward. – p.14/48 Adjusted normal operator Adjusted sublevel set: For any x ∈ dom f, we define a < Sf (x) = Sf(x) ∩ B(Sf(x), ρx) < < where ρx = dist(x, Sf(x)), if Sf(x) =6 ∅. Ajusted normal operator: a ∗ ∗ ∗ a Nf (x) = {x ∈ X : hx ,y − xi≤ 0, ∀ y ∈ Sf (x)} – p.15/48 Example x © – p.16/48 Example x © < B(Sf(x), ρx) a < Sf (x)= Sf (x) ∩ B(Sf(x), ρx) – p.16/48 Example a < Sf (x)= Sf (x) ∩ B(Sf(x), ρx) a ∗ ∗ ∗ a Nf (x)= {x ∈ X : hx , y − xi≤ 0, ∀ y ∈ Sf (x)} – p.16/48 Subdifferential vs normal operator One can have a a Nf (x) ⊆/ cone(∂f(x)) or cone(∂f(x)) ⊆/ Nf (x) Proposition 2 f is quasiconvex and x ∈ dom f If there exists δ > 0 such that 0 6∈ ∂Lf(B(x,δ)) then L ∞ a cone(∂ f(x)) ∪ ∂ f(x) ⊂ Nf (x). – p.17/48 a Basic properties of Nf Nonemptyness: Proposition 3 Let f : X → IR ∪ {+∞} be lsc. Assume that rad. continuous on dom f or dom f is convex and intSλ =6 ∅, ∀ λ > infX f. Then a If f is quasiconvex , Nf (x) \ {0}= 6 ∅, ∀ x ∈ dom f \ arg min f. f quasiconvex a ⇐⇒ dom Nf \ {0} dense in dom f \ arg min f. Quasimonotonicity: a The normal operator Nf is always quasimonotone – p.18/48 Upper sign-continuity ∗ • T : X → 2X is said to be upper sign-continuous on K iff for any x,y ∈ K, one have : ∀ t ∈ ]0, 1[, inf hx∗,y − xi≥ 0 ∗ x ∈T (xt) =⇒ sup hx∗,y − xi≥ 0 x∗∈T (x) where xt = (1 − t)x + ty. – p.19/48 Upper sign-continuity ∗ • T : X → 2X is said to be upper sign-continuous on K iff for any x,y ∈ K, one have : ∀ t ∈ ]0, 1[, inf hx∗,y − xi≥ 0 ∗ x ∈T (xt) =⇒ sup hx∗,y − xi≥ 0 x∗∈T (x) where xt = (1 − t)x + ty. upper semi-continuous ⇓ upper hemicontinuous ⇓ upper sign-continuous – p.19/48 locally upper sign continuity ∗ Definition 5 Let T : K → 2X be a set-valued map. T est called locally upper sign-continuous on K if, for any x ∈ K there exist a neigh. Vx of x and a upper sign-continuous set-valued X∗ ∗ map Φx(·) : Vx → 2 with nonempty convex w -compact values such that Φx(y) ⊆ T (y) \ {0}, ∀ y ∈ Vx – p.20/48 locally upper sign continuity ∗ Definition 6 Let T : K → 2X be a set-valued map. T est called locally upper sign-continuous on K if, for any x ∈ K there exist a neigh. Vx of x and a upper sign-continuous set-valued X∗ ∗ map Φx(·) : Vx → 2 with nonempty convex w -compact values such that Φx(y) ⊆ T (y) \ {0}, ∀ y ∈ Vx Continuity of normal operator Proposition 7 Let f be lsc quasiconvex function such that int(Sλ) =6 ∅, ∀ λ > inf f. Then Nf is locally upper sign-continuous on dom f \ arg min f. – p.20/48 Proposition 8 If f is quasiconvex such that int(Sλ) =6 ∅, ∀ λ> inf f and f is lsc at x ∈ dom f \ arg min f, a ∗ Then Nf is norm-to-w cone-usc at x. ∗ A multivalued map with conical valued T : X → 2X is said to be cone-usc at x ∈ dom T if there exists a neighbourhood U of x and a base C (u) of T (u), u ∈ U, such that u → C (u) is usc at x. – p.21/48 a Integration of Nf Let f : X → IR ∪ {+∞} quasiconvex Question: Is it possible to characterize the functions a a g : X → IR ∪ {+∞} quasiconvex such that Nf = Ng ? – p.22/48 a Integration of Nf Let f : X → IR ∪ {+∞} quasiconvex Question: Is it possible to characterize the functions a a g : X → IR ∪ {+∞} quasiconvex such that Nf = Ng ? A first answer: Let C = {g : X → IR ∪ {+∞} cont.