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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 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 .
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