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 2 (0; 1) f (λx + (1 − λ)y) < maxff (x); f (y)g; 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 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: 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 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Definition f is called explicitly quasiconvex if it is quasiconvex and for all λ 2 (0; 1) f (λx + (1 − λ)y) < maxff (x); f (y)g; 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 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Definition f is called explicitly quasiconvex if it is quasiconvex and for all λ 2 (0; 1) f (λx + (1 − λ)y) < maxff (x); f (y)g; 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 G^ateauxdifferentiable function f : X ! R, with a convex domf , is called pseudoconvex if for x; y 2 domf , f (x) < f (y) ) hrf (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 G^ateauxdifferentiable function f : X ! R, with a convex domf , is called pseudoconvex if for x; y 2 domf , f (x) < f (y) ) hrf (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 G^ateauxdifferentiable function f : X ! R, with a convex domf , is called pseudoconvex if for x; y 2 domf , f (x) < f (y) ) hrf (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 2 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 2 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 2 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 2 αB , the ∗ function fv ∗ : v 7! f (x) + hv ; xi is quasiconvex. 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 2 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 2 αB , the ∗ function fv ∗ : v 7! f (x) + hv ; xi is quasiconvex. 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 X {Banach space. Definition ∗ ∗ For α > 0, a function f : X ! R is called α{robustly quasiconvex if, for every v 2 αB , the ∗ function fv ∗ : v 7! f (x) + hv ; xi is quasiconvex. 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 2 X are quasiconvex. Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 6 / 15 Fr´echetsubdifferential set of ' atx ¯ '(x) − '(x) − hx∗; x − xi @F '(x) := x∗ 2 X ∗ : lim inf ≥ 0 : x!x kx − xk Theorem 1 ' is convex, then @F '(x) = @'(x); x 2 X : ∗ F 2 x 2 @ '(x) if and only if there exists a function g : X ! R such that 1 g(x) = '(x), and g(y) ≤ '(y) for all y 2 X . ∗ 2 g is Fr´echetdifferentiable at x, and rg(x) = x . 3 x¯ is a local minimum of ', then 0 2 @F '(¯x): Dual Characterization: subdiferentials X − normed vector space, ' : X ! R¯; x¯ 2 X Convex subdifferential set of ' atx ¯ @'(¯x) := fx∗ 2 X ∗ : '(x) − '(¯x) ≥ hx∗; x − x¯i; 8x 2 X g : Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 7 / 15 Theorem 1 ' is convex, then @F '(x) = @'(x); x 2 X : ∗ F 2 x 2 @ '(x) if and only if there exists a function g : X ! R such that 1 g(x) = '(x), and g(y) ≤ '(y) for all y 2 X . ∗ 2 g is Fr´echetdifferentiable at x, and rg(x) = x . 3 x¯ is a local minimum of ', then 0 2 @F '(¯x): Dual Characterization: subdiferentials X − normed vector space, ' : X ! R¯; x¯ 2 X Convex subdifferential set of ' atx ¯ @'(¯x) := fx∗ 2 X ∗ : '(x) − '(¯x) ≥ hx∗; x − x¯i; 8x 2 X g : Fr´echetsubdifferential set of ' atx ¯ '(x) − '(x) − hx∗; x − xi @F '(x) := x∗ 2 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, ' : X ! R¯; x¯ 2 X Convex subdifferential set of ' atx ¯ @'(¯x) := fx∗ 2 X ∗ : '(x) − '(¯x) ≥ hx∗; x − x¯i; 8x 2 X g : Fr´echetsubdifferential set of ' atx ¯ '(x) − '(x) − hx∗; x − xi @F '(x) := x∗ 2 X ∗ : lim inf ≥ 0 : x!x kx − xk Theorem 1 ' is convex, then @F '(x) = @'(x); x 2 X : ∗ F 2 x 2 @ '(x) if and only if there exists a function g : X ! R such that 1 g(x) = '(x), and g(y) ≤ '(y) for all y 2 X .