Robustly Quasiconvex Function

Robustly quasiconvex function

Bui Thi Hoa

Centre for Informatics and Applied Optimization, Federation University Australia

May 21, 2018

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 1 / 15 Convex functions

1 All lower level sets are convex.

2 Each local minimum is a global minimum.

3 Each stationary point is a global minimizer.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 2 / 15 Deﬁnition f is called explicitly quasiconvex if it is quasiconvex and for all λ ∈ (0, 1)

f (λx + (1 − λ)y) < max{f (x), f (y)}, with f (x) 6= f (y).

Example

1 f1 : R → R; f1(x) = 0, x 6= 0; f1(0) = 1.

2 f2 : R → R; f2(x) = 1, x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex.

4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex.

Generalised Convexity

Deﬁnition

A function f : X → R, with a convex domf , is called quasiconvex if for all x, y ∈ domf , and λ ∈ [0, 1] we have f (λx + (1 − λ)y) ≤ max{f (x), f (y)}.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 3 / 15 Example

1 f1 : R → R; f1(x) = 0, x 6= 0; f1(0) = 1.

2 f2 : R → R; f2(x) = 1, x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex.

4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex.

Generalised Convexity

Deﬁnition

A function f : X → R, with a convex domf , is called quasiconvex if for all x, y ∈ domf , and λ ∈ [0, 1] we have f (λx + (1 − λ)y) ≤ max{f (x), f (y)}.

Deﬁnition f is called explicitly quasiconvex if it is quasiconvex and for all λ ∈ (0, 1)

f (λx + (1 − λ)y) < max{f (x), f (y)}, with f (x) 6= f (y).

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 3 / 15 Generalised Convexity

Deﬁnition

A function f : X → R, with a convex domf , is called quasiconvex if for all x, y ∈ domf , and λ ∈ [0, 1] we have f (λx + (1 − λ)y) ≤ max{f (x), f (y)}.

Deﬁnition f is called explicitly quasiconvex if it is quasiconvex and for all λ ∈ (0, 1)

f (λx + (1 − λ)y) < max{f (x), f (y)}, with f (x) 6= f (y).

Example

1 f1 : R → R; f1(x) = 0, x 6= 0; f1(0) = 1.

2 f2 : R → R; f2(x) = 1, x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex.

4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 3 / 15 Example x 2 f : R → R, f (x) = t sin(1/t)dt + x . f is pseudoconvex but not stable. ˆ−1

x x f (x) = t sin(1/t)dt + x2. f (x) = t sin(1/t)dt + x2 − x/π. ˆ−1 ˆ−1

Pseudoconvex function

Deﬁnition

A Gˆateauxdiﬀerentiable function f : X → R, with a convex domf , is called pseudoconvex if for x, y ∈ domf , f (x) < f (y) ⇒ h∇f (y), x − yi < 0.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 4 / 15 x x f (x) = t sin(1/t)dt + x2. f (x) = t sin(1/t)dt + x2 − x/π. ˆ−1 ˆ−1

Pseudoconvex function

Deﬁnition

A Gˆateauxdiﬀerentiable function f : X → R, with a convex domf , is called pseudoconvex if for x, y ∈ domf , f (x) < f (y) ⇒ h∇f (y), x − yi < 0.

Example x 2 f : R → R, f (x) = t sin(1/t)dt + x . f is pseudoconvex but not stable. ˆ−1

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 4 / 15 Pseudoconvex function

Deﬁnition

A Gˆateauxdiﬀerentiable function f : X → R, with a convex domf , is called pseudoconvex if for x, y ∈ domf , f (x) < f (y) ⇒ h∇f (y), x − yi < 0.

Example x 2 f : R → R, f (x) = t sin(1/t)dt + x . f is pseudoconvex but not stable. ˆ−1

x x f (x) = t sin(1/t)dt + x2. f (x) = t sin(1/t)dt + x2 − x/π. ˆ−1 ˆ−1 Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 4 / 15 Generalized Convexity

Quasiconvexity satisfies (1), not (2), (3), and explicitly quasiconvexity satisfies (1), (2), not (3). Pseudoconvexity satisfies (3). These three properties are not stable w/r to three generalized convex properties.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 5 / 15 A function is call s–quasiconvex if there is α > 0 such that f is α−robustly quasiconvex.

1 s–quasiconvex functions satisfy (1), (2), (3).

2 Consequently, these three properties are stable under a small linear perturbation.

3 A lower semicontinuous α–robustly quasiconvex function are not necessary continuous in its intdom .

Theorem (J.-P. Crouzeix, 1977) ∗ ∗ ∗ A function f : X → R¯ is convex provided all its linear perturbations f + x , x ∈ X are quasiconvex.

Robust quasiconvexity

X –Banach space. Deﬁnition ∗ ∗ For α > 0, a function f : X → R is called α–robustly quasiconvex if, for every v ∈ αB , the ∗ function fv ∗ : v 7→ f (x) + hv , xi is quasiconvex.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 6 / 15 1 s–quasiconvex functions satisfy (1), (2), (3).

2 Consequently, these three properties are stable under a small linear perturbation.

3 A lower semicontinuous α–robustly quasiconvex function are not necessary continuous in its intdom .

Theorem (J.-P. Crouzeix, 1977) ∗ ∗ ∗ A function f : X → R¯ is convex provided all its linear perturbations f + x , x ∈ X are quasiconvex.

Robust quasiconvexity

A function is call s–quasiconvex if there is α > 0 such that f is α−robustly quasiconvex.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 6 / 15 Theorem (J.-P. Crouzeix, 1977) ∗ ∗ ∗ A function f : X → R¯ is convex provided all its linear perturbations f + x , x ∈ X are quasiconvex.

Robust quasiconvexity

A function is call s–quasiconvex if there is α > 0 such that f is α−robustly quasiconvex.

1 s–quasiconvex functions satisfy (1), (2), (3).

2 Consequently, these three properties are stable under a small linear perturbation.

3 A lower semicontinuous α–robustly quasiconvex function are not necessary continuous in its intdom .

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 6 / 15 Robust quasiconvexity

A function is call s–quasiconvex if there is α > 0 such that f is α−robustly quasiconvex.

1 s–quasiconvex functions satisfy (1), (2), (3).

2 Consequently, these three properties are stable under a small linear perturbation.

3 A lower semicontinuous α–robustly quasiconvex function are not necessary continuous in its intdom .

Theorem (J.-P. Crouzeix, 1977) ∗ ∗ ∗ A function f : X → R¯ is convex provided all its linear perturbations f + x , x ∈ X are quasiconvex.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 6 / 15 Fr´echetsubdiﬀerential set of ϕ atx ¯

ϕ(x) − ϕ(x) − hx∗, x − xi ∂F ϕ(x) := x∗ ∈ X ∗ : lim inf ≥ 0 . x→x kx − xk

Theorem 1 ϕ is convex, then ∂F ϕ(x) = ∂ϕ(x), x ∈ X . ∗ F 2 x ∈ ∂ ϕ(x) if and only if there exists a function g : X → R such that 1 g(x) = ϕ(x), and g(y) ≤ ϕ(y) for all y ∈ X . ∗ 2 g is Fr´echetdiﬀerentiable at x, and ∇g(x) = x .

3 x¯ is a local minimum of ϕ, then 0 ∈ ∂F ϕ(¯x).

Dual Characterization: subdiferentials

X − normed vector space, ϕ : X → R¯, x¯ ∈ X Convex subdiﬀerential set of ϕ atx ¯

∂ϕ(¯x) := {x∗ ∈ X ∗ : ϕ(x) − ϕ(¯x) ≥ hx∗, x − x¯i, ∀x ∈ X } .

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 7 / 15 Theorem 1 ϕ is convex, then ∂F ϕ(x) = ∂ϕ(x), x ∈ X . ∗ F 2 x ∈ ∂ ϕ(x) if and only if there exists a function g : X → R such that 1 g(x) = ϕ(x), and g(y) ≤ ϕ(y) for all y ∈ X . ∗ 2 g is Fr´echetdiﬀerentiable at x, and ∇g(x) = x .

3 x¯ is a local minimum of ϕ, then 0 ∈ ∂F ϕ(¯x).

Dual Characterization: subdiferentials

X − normed vector space, ϕ : X → R¯, x¯ ∈ X Convex subdiﬀerential set of ϕ atx ¯

∂ϕ(¯x) := {x∗ ∈ X ∗ : ϕ(x) − ϕ(¯x) ≥ hx∗, x − x¯i, ∀x ∈ X } .

Fr´echetsubdiﬀerential set of ϕ atx ¯

ϕ(x) − ϕ(x) − hx∗, x − xi ∂F ϕ(x) := x∗ ∈ X ∗ : lim inf ≥ 0 . x→x kx − xk

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 7 / 15 Dual Characterization: subdiferentials

X − normed vector space, ϕ : X → R¯, x¯ ∈ X Convex subdiﬀerential set of ϕ atx ¯

∂ϕ(¯x) := {x∗ ∈ X ∗ : ϕ(x) − ϕ(¯x) ≥ hx∗, x − x¯i, ∀x ∈ X } .

Fr´echetsubdiﬀerential set of ϕ atx ¯

ϕ(x) − ϕ(x) − hx∗, x − xi ∂F ϕ(x) := x∗ ∈ X ∗ : lim inf ≥ 0 . x→x kx − xk

3 x¯ is a local minimum of ϕ, then 0 ∈ ∂F ϕ(¯x).

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 7 / 15 ∂F ϕ(0) = {0}; but ϕ is not diﬀerentiable at 0.

Examples

 max{0, x sin(1/x)} if x 6= 0; ϕ(x) = 0 if x = 0.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 8 / 15 Examples

 max{0, x sin(1/x)} if x 6= 0; ϕ(x) = 0 if x = 0.

∂F ϕ(0) = {0}; but ϕ is not diﬀerentiable at 0.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 8 / 15  {(1, −1)} if x1 > 0, x2 > 0,   {(−1, −1)} if x1 < 0, x2 > 0,  {(−1, 1)} if x < 0, x < 0,  1 2 F  ∂ ϕ(x1, x2) = {(1, 1)} if x1 > 0, x2 < 0,   {(v, −1) : |v| ≤ 1} if x1 = 0, x2 > 0,  {(v, 1) : |v| ≤ 1} if x = 0, x < 0,  1 2  ∅ if x2 = 0.

Examples

2 X = R and ϕ : X → R 2 ϕ(x) = |x1| − |x2|, x = (x1, x2) ∈ R .

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 9 / 15 Examples

2 X = R and ϕ : X → R 2 ϕ(x) = |x1| − |x2|, x = (x1, x2) ∈ R .

 {(1, −1)} if x1 > 0, x2 > 0,   {(−1, −1)} if x1 < 0, x2 > 0,  {(−1, 1)} if x < 0, x < 0,  1 2 F  ∂ ϕ(x1, x2) = {(1, 1)} if x1 > 0, x2 < 0,   {(v, −1) : |v| ≤ 1} if x1 = 0, x2 > 0,  {(v, 1) : |v| ≤ 1} if x = 0, x < 0,  1 2  ∅ if x2 = 0.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 9 / 15 Theorem (Rockafellar (1970))

If ϕ is convex, then ∂F ϕ is maximal monotone.

Convex function–Monotone operator

Deﬁnition ∗ An operator F : X ⇒ X is called monotone if

hv ∗ − u∗, v − ui ≥ 0, for all u, v ∈ X and u∗ ∈ F (u), v ∗ ∈ F (v). F is maximal monotone if there is no monotone operator that properly contains it.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 10 / 15 Convex function–Monotone operator

Deﬁnition ∗ An operator F : X ⇒ X is called monotone if

hv ∗ − u∗, v − ui ≥ 0, for all u, v ∈ X and u∗ ∈ F (u), v ∗ ∈ F (v). F is maximal monotone if there is no monotone operator that properly contains it.

Theorem (Rockafellar (1970))

If ϕ is convex, then ∂F ϕ is maximal monotone.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 10 / 15 Theorem F X is Asplund, f : X → R¯. Then f is convex if and only if ∂ f is monotone.

X is NOT Asplund if and only if there is a proper l.s.c function ϕ : X → X¯ with ∂F ϕ is monotone, but not convex. Lemma (Lemma of three points)

Let ϕ : X → R be a proper, l.s.c on Asplund space X . Let u, v, w ∈ X with v ∈ [u, w], ϕ ∗ F ϕ(v) > ϕ(u). Then, there exist c ∈ [u, v), and two sequences xk → c, and xk ∈ ∂ ϕ(xk ) such that ∗ hxk , w − xk i > 0 ∀k ∈ N.

(Ausel,1988) ∂−smooth renorm space.

Asplund space

Definition A Banach space X is said to be Apslund if every continuous convex function defined on a nonempty open convex subset D of X is Fréchetdifferentiable at each point of some dense set Gδ subset of D.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 11 / 15 X is NOT Asplund if and only if there is a proper l.s.c function ϕ : X → X¯ with ∂F ϕ is monotone, but not convex. Lemma (Lemma of three points)

(Ausel,1988) ∂−smooth renorm space.

Asplund space

Theorem F X is Asplund, f : X → R¯. Then f is convex if and only if ∂ f is monotone.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 11 / 15 Lemma (Lemma of three points)

(Ausel,1988) ∂−smooth renorm space.

Asplund space

Theorem F X is Asplund, f : X → R¯. Then f is convex if and only if ∂ f is monotone.

X is NOT Asplund if and only if there is a proper l.s.c function ϕ : X → X¯ with ∂F ϕ is monotone, but not convex.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 11 / 15 Asplund space

Theorem F X is Asplund, f : X → R¯. Then f is convex if and only if ∂ f is monotone.

X is NOT Asplund if and only if there is a proper l.s.c function ϕ : X → X¯ with ∂F ϕ is monotone, but not convex. Lemma (Lemma of three points)

(Ausel,1988) ∂−smooth renorm space.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 11 / 15 Theorem

Let ϕ : X → R be a proper, l.s.c function on a Banach space X . Consider the following statements (a) ϕ is quasiconvex; (b) ϕ(y) ≤ ϕ(x) =⇒ hx∗, y − xi ≤ 0 ∀x∗ ∈ ∂F ϕ(x). (c) ∂F ϕ is quasimonotone. Then, (a)⇒(b) ⇔ (c). Moreover, if X is an Asplund space then (c)⇒(a).

If X is NOT Asplund, we can always ﬁnd a proper l.s.c function that satisﬁes (c),(b); but not (a).

Quasiconvex function-Quasimonotone operator

Deﬁnition ∗ An operator F : X ⇒ X is quasimonotone if

hu∗, v − ui > 0 ⇒ hv ∗, v − ui ≥ 0 for all u, v ∈ X and u∗ ∈ F (u), v ∗ ∈ F (v).

Monotone =⇒ Quasimonotone.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 12 / 15 Moreover, if X is an Asplund space then (c)⇒(a).

If X is NOT Asplund, we can always ﬁnd a proper l.s.c function that satisﬁes (c),(b); but not (a).

Quasiconvex function-Quasimonotone operator

Deﬁnition ∗ An operator F : X ⇒ X is quasimonotone if

hu∗, v − ui > 0 ⇒ hv ∗, v − ui ≥ 0 for all u, v ∈ X and u∗ ∈ F (u), v ∗ ∈ F (v).

Monotone =⇒ Quasimonotone. Theorem

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 12 / 15 If X is NOT Asplund, we can always ﬁnd a proper l.s.c function that satisﬁes (c),(b); but not (a).

Quasiconvex function-Quasimonotone operator

Deﬁnition ∗ An operator F : X ⇒ X is quasimonotone if

hu∗, v − ui > 0 ⇒ hv ∗, v − ui ≥ 0 for all u, v ∈ X and u∗ ∈ F (u), v ∗ ∈ F (v).

Monotone =⇒ Quasimonotone. Theorem

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 12 / 15 Quasiconvex function-Quasimonotone operator

Deﬁnition ∗ An operator F : X ⇒ X is quasimonotone if

hu∗, v − ui > 0 ⇒ hv ∗, v − ui ≥ 0 for all u, v ∈ X and u∗ ∈ F (u), v ∗ ∈ F (v).

Monotone =⇒ Quasimonotone. Theorem

If X is NOT Asplund, we can always ﬁnd a proper l.s.c function that satisﬁes (c),(b); but not (a).

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 12 / 15 Moreover, if X is an Asplund space then (b)⇒(a).

Theorem

Let ϕ : X → R be proper, l.s.c on an Asplund space X and α ≥ 0. Then, ϕ is α−robustly quasiconvex if and only if for any x, y ∈ X and x∗ ∈ ∂F ϕ(x), y ∗ ∈ ∂F ϕ(y), we have

min{hx∗, y − xi, hy ∗, x − yi} > −αky − xk =⇒ hx∗ − y ∗, x − yi ≥ 0.

First Order Characterization

Theorem

Let ϕ : X → R be a proper, l.s.c function on a Banach space X , and α > 0. Consider the following statements (a) ϕ is α−robustly quasiconvex; (b) For every x, y ∈ X

ϕ(y) ≤ ϕ(x) =⇒ hx∗, y − xi ≤ − min {αky − xk, ϕ(x) − ϕ(y)} , ∀x∗ ∈ ∂F ϕ(x).

Then, (a)⇒(b).

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 13 / 15 First Order Characterization

Theorem

Let ϕ : X → R be a proper, l.s.c function on a Banach space X , and α > 0. Consider the following statements (a) ϕ is α−robustly quasiconvex; (b) For every x, y ∈ X

ϕ(y) ≤ ϕ(x) =⇒ hx∗, y − xi ≤ − min {αky − xk, ϕ(x) − ϕ(y)} , ∀x∗ ∈ ∂F ϕ(x).

Then, (a)⇒(b). Moreover, if X is an Asplund space then (b)⇒(a).

Theorem

Let ϕ : X → R be proper, l.s.c on an Asplund space X and α ≥ 0. Then, ϕ is α−robustly quasiconvex if and only if for any x, y ∈ X and x∗ ∈ ∂F ϕ(x), y ∗ ∈ ∂F ϕ(y), we have

min{hx∗, y − xi, hy ∗, x − yi} > −αky − xk =⇒ hx∗ − y ∗, x − yi ≥ 0.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 13 / 15 Robustly quasiconvex functions

1 All lower level sets are convex.

2 Each local minimum is a global minimum.

3 Each stationary point is a global minimizer.

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 14 / 15 THANK YOU!

Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 15 / 15