Luận văn thạc sĩ Toán ứng dụng: Computing the robustly quasiconvex envelopes

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

NGUYEN HOANG HAI

COMPUTING THE ROBUSTLYQUASICONVEX ENVELOPES

TÍNH CÁC BAO TỰA LỒI VỮNG

Major: APPLIED MATHEMATICSMajor Code: 8460112

MASTER’S THESIS

HO CHI MINH CITY, January 2024

Advisor: Assoc Prof Phan Thanh An.

Examiner 1: PhD Le Xuan Dai

Examiner 2: Assoc Prof Nguyen Huy Tuan

This Master’s thesis is defended at Ho Chi Minh City University of Technology onJanuary 05, 2024.

Members of the Master’s Thesis Examination Committee1 Chairman: Assoc Prof Nguyen Dinh Huy

2 Secretary: PhD Phan Thi Huong3 Examiner 1: PhD Le Xuan Dai

4 Examiner 2: Assoc Prof Nguyen Huy Tuan5 Member: Assoc Prof Cao Thanh Tinh

Confirmation of the Chairman of the Master’s Thesis Examination Committee and theDean of the faculty after receiving the modified thesis (if any).

APPLIED SCIENCES

MASTER’S THESIS ASSIGNMENT

(TÍNH CÁC BAO TỰA LỒI VỮNG)

TASKS AND CONTENT

• Background on generalized convex functions

• Computing the robustness index of quasiconvex functions• Computing the quasiconvex envelope of a given function

Ho Chi Minh City, January 05, 2024

DEAN OF FACULTY OF APPLIED SCIENCES

PhD Le Xuan Dai

I would like to express my deepest gratitude and appreciation to all thosewho have supported and guided me throughout this journey of research andthesis writing.

First and foremost, I would like to express my heartfelt gratitude toAssoc Prof Phan Thanh An for his unwavering dedication and guidancethroughout the completion of this thesis I am also deeply appreciative of thefaculty members at the Faculty of Applied Sciences, Institute of Mathemat-ical and Computational Sciences, particularly those within the Departmentof Applied Mathematics, for creating an enriching academic environmentand providing valuable resources for my research Additionally, I extend mysincere thanks to Assoc Prof Nguyen Dinh Huy for not only introduc-ing me to opportunities but also encouraging and facilitating the successfulcompletion of this thesis.

This thesis is funded by the Type B project of Vietnam National versity Ho Chi Minh City (VNU-HCM) in 2024.

Uni-I am thankful to my family for their love, understanding, and constantencouragement Their belief in me has been my greatest motivation Inparticular, I would like to thank my wife and our lovely baby bear for beinga strong support system for me.

I am indebted to the authors of the works that I have cited and referencedin this thesis Their contributions to the field have been instrumental inshaping my research.

Finally, I would like to express my gratitude to all my friends who stoodby me during this journey and provided much-needed moral support.

This thesis would not have been possible without the collective effortsand support of all these individuals and institutions I am deeply thankfulto each and every one of you.

Ho Chi Minh City, January 01, 2024

Nguyen Hoang Hai

We presente the computation of the robustness indices of quasiconvex tions ([6]) The robustness index of a quasiconvex function on a convexand compat set D ⊂ Rn is equal to the infimum of its robustness indexon line segments contained in D An efficient approximation algorithm forcomputing the robustness index of quasiconvex functions on a compact andconvex setD ⊂ Rn using its robustness index on line segments contained inD is presented (Algorithm 3) An algorithm for computing the robustnessindex of a given continuously twice differentiable quasiconvex function on[a, b] ⊂ R is demonstrated (Algorithm 2) We implement Algorithms 4-5for computing the quasiconvex envelopes of the functions ([1]) Some exam-ples for computing the quasiconvex envelopes of the functions are presented.Applications of the robustness indices of quasiconvex functions in economics([4]) and healthcare ([6]) are presented.

func-Tóm tắt

Chúng tôi trình bày việc tính toán các chỉ số vững của các hàm tựa lồi ([6]).Chỉ số vững của một hàm tựa lồi trên một tập lồi và compact D ⊂ Rn bằng vớiinfimum của chỉ số vững của nó trên các đoạn thẳng nằm trong D Chúng tôitrình bày một thuật toán hiệu quả để tính xấp xỉ chỉ số vững của các hàm tựalồi trên một tập hợp lồi và compact D ⊂ Rn bằng cách sử dụng chỉ số vững trêncác đoạn thẳng nằm trong D (Thuật toán 3) Chúng tôi trình bày một thuậttoán để tính toán chỉ số vững của một hàm tựa lồi liên tục hai lần trên [a, b] ⊂ R(Thuật toán 2) Chúng tôi thực thi Thuật toán 4-5 để tính toán các bao tựa lồicủa các hàm cho trước ([1]) Một số ví dụ về việc tính toán các bao tựa lồi củacác hàm cho trước được trình bày Ứng dụng của các chỉ số vững của các hàmtựa lồi trong lĩnh vực kinh tế ([4]) và y tế ([6]) được trình bày.

I, Nguyen Hoang Hai, Student ID: 2170955, am a graduate student izing in Applied Mathematics, Class of 2021 - 2023, at Ho Chi Minh CityUniversity of Technology, Vietnam National University Ho Chi Minh City.

special-I hereby solemnly declare that everything presented in this thesis is myown work under the direct guidance of Assoc Prof Phan Thanh An,Institute of Mathematical and Computational Sciences, Ho Chi Minh CityUniversity of Technology, Vietnam National University Ho Chi Minh City.

The main results presented in this thesis are sourced from [1] and [6].Throughout this thesis, whenever I have collected, selected, cited, or referredto research results from the scientific works of other authors, I have clearlydocumented the sources for readers to reference.

I solemnly affirm that everything mentioned above is true, and I take fullresponsibility for any copyright violations, if any, in this thesis.

Ho Chi Minh City, January 01, 2024Thesis Author

Nguyen Hoang Hai

Acknowledgments i

1.1 Some Basic Definitions and Properties 3

1.2 Convex Functions 4

1.3 Quasiconvex Functions 5

1.4 Robustly Quasiconvex Functions 8

2 Computing the Robustness Index of Quasiconvex Functions 122.1 Definition and Properties 12

2.2 Approximating Robustness Index of Quasiconvex Functions 172.3 Algorithms for Computing the Robustness Index 17

2.4 Implementation 21

2.5 Final Remarks 24

3 Computing the Quasiconvex Envelopes of Functions 253.1 Definition and Properties 25

3.2 QCE in One Dimensional 26

3.3 QCE in Higher Dimensional 29

3.4 Algorithms for Computing the QCE 30

3.5 Implementation 31

3.6 Final Remarks 34

4 Applications 354.1 The Stability Index of Excess Demand Functions 35

4.2 Estimating the Growth of Acne 36

4.3 Final Remarks 40

References 41

Code Execution for Robustness Index Computation 44Code Execution for QCE Computation 48

1.1 Epigraph of functions 31.2 The α-sublevel set of functions 41.3 f (x) = ln(x2+ 1) (left) and fξ(x) (right) on [−2, 2] , where

ξ(x) =−0.9x 61.4 The functions f (x) + px on [0, 1] for some p∈ [−2, 2], where

f (x) = x3+ x 81.5 The non-quasiconvex (a), quasiconvex (b) and robustly qua-

siconvex functions (c) 91.6 The function f (x, y) = ln(x2 + 2y2) and f

u + t v− u∥v − u∥

,where u = (1, 1), v = (2, 2) 112.1 The functions f2(x) + px on D2 for some p in [−0.5, 0.5] 222.2 s(m)f7 (D7) are computed by Algorithm 3 for m ∈ {1, 2, 3, 4} 232.3 s(m1,m2)

f8 (D8) are computed by Algorithm 3 for m1, m2 ∈ {1, 2, 3} 243.1 A function g and QCEg 253.2 The function u on [a, b] and the symbols in Definition 3.2.1 273.3 g(x) and QCEg(x) onD1 = [−1, 1] and D2 = [−1.2, 1], where

g(x) = x2(x + 1) 283.4 g(x) and QCEg(x) onD = [−2, 2] where g(x) = ln(x4−2x2+2) 293.5 The quasiconvex envelopes of some continuous functions on

[a, b] ⊂ R are computed by Algorithm 4 323.6 The QCD of the some function are computed by Algorithm 5 334.1 Left: Robustness indices of some random data (colored in

red) Right: 3D images of acne lesions from some random data 384.2 Left: Robustness indices of some acne lesions (colored in

white) Right: Contour plots of some acne lesions 39

2.1 The robustness index of fi on D ⊂ R, where i ∈ {1, , 6} 212.2 s(m)f7 (D7) are computed by Algorithm 3 for m ∈ {1, 2, , 7} 22

Symbols and

A convex function is the function for which the line segment connecting anytwo points on its graph lies above or touches the graph itself This con-cept first appeared and was studied in the early 20th century in papers byJensen; see [13], [14] An important property of convex functions is thattheir lower level sets are convex sets, where a convex set is a set for whichthe line segment connecting any two points in the set lies within the set.However, a function for which the lower level sets are convex is not neces-sarily convex itself In 1949, Finetti [12] based on this property of convexfunctions introduced a new definition, now known as a quasiconvex function.Similarly, based on other characteristic properties of convex functions, var-ious extensions have emerged, such as explicitly quasiconvex functions andpseudoconvex functions.

In 1996, Phu and An originally introduced the definition of the robustlyquasiconvex function under the name s-quasiconvex (“s”stands for “stable”)in [7] These are quasiconvex functions that remain quasiconvex when sub-jected to sufficiently small linear perturbations In 2018, An introduced theconcept of the stability index of s-quasiconvex functions in [3] The stabilityindex of an s-quasiconvex function is defined as the largest number of lin-ear maps causing perturbations as mentioned above Subsequently, variousproperties of s-quasiconvex functions have been studied in a series of recentpapers (see [3], [6], [9], [10], [11], [15]) under the name robustly quasiconvexfunctions.

Convex functions and their extensions have numerous applications in ious fields Therefore, finding the convex envelope of non-convex functionsor finding extensions (such as quasiconvex envelopes, robust quasiconvexenvelopes) of non-convex functions (such as quasiconvex functions, robustquasiconvex functions) is of great interest.

var-In 2018, Abbasi and Oberman introduced the concept of the quasiconvexenvelope (QCE) of a given function in [1] The QCE of a given function isdefined as the maximal quasiconvex (QC) function below it The authorspresented an algorithm to find the QCE of a given function over a closedinterval in R Finding the QCE of a function defined on D ⊂ Rn (n > 1) isreduced to finding its QCE in a finite number of directions Building uponthe method for finding the quasiconvex envelope (QCE) of a given function,they also introduced an algorithm to find the robust quasiconvex envelope(robust QCE).

To measure such robustness, the robustness index of a quasiconvex

func-tion on a nonempty and convexD was firstly introduced in [3] In [6], N N.Hai, P T An and N H Hai proved that the robustness index of a quasi-convex function onD is equal to the infimum of its robustness index on linesegments contained inD Efficient approximation algorithms for computingthe robustness index of quasiconvex functions on compact and convex setD ⊂ Rn using its robustness index on line segments contained inD are pre-sented Additionally, an algorithm for computing the robustness index of agiven continuously twice differentiable quasiconvex function on [a, b]⊂ R isdemonstrated.

In this thesis, we present the properties of (robustly) quasiconvex tions and their robustness index as introduced in [3] and [6] Furthermore,we provide illustrative examples for their definitions and properties We in-troduce examples of computing the robustness index for quasiconvex func-tions and computing the quasiconvex envelopes of given functions Someapplications of the robustness index of quasiconvex functions are presented.Specifically, the thesis consists of three chapters, outlined as follows

func-• Chapter 1: Background on Generalized Convex Functions

• Chapter 2: Computing the Robustness Index of Quasiconvex Functions• Chapter 3: Computing the Quasiconvex Envelopes of Functions

• Chapter 4: Applications

In Chapter 1, we present the definitions and some properties of generalconvex functions (see [3], [7], [18]) In Chapter 2, we compute the robustnessindex of quasiconvex functions [6] In Chapter 3, we compute the quasicon-vex envelope (QCE) of a given function [1] Some examples of robustnessindices for QCEs are computed in this chapter.

The content of Chapter 2 was presented at the 21st Workshop on timization and Scientific Computing, held on April 20-22, 2023, in Ba Vi,Vietnam This content was also presented at the International Symposiumon Applied Science 2023, held from October 13 to October 15, 2023, at HoChi Minh City University of Technology, Ho Chi Minh City, Vietnam.

Op-Chapter 1

Background on GeneralizedConvex Functions

Before we begin the analysis of Generalized Convex Functions that will beused in the following sections, we present some basic definitions and prop-erties, see [18, p 10-11] For x, y∈ Rn, we denote

[x, y] := {(1 − λ)x + λy : 0 ≤ λ ≤ 1} and (x, y) := [x, y] \ {x, y}.We say that [x, y] is a line segment in Rn Let ξ : Rn → R be a linearfunction We set

∥ξ∥ := sup

where∥x∥ is the norm of x ∈ Rn By Riesz Theorem, there exists a uniquea∈ Rnsuch that ξ(x) = xTa and∥ξ∥ = ∥a∥, where xT denotes the transposeof x We denote the following sets

R∗n:={ξ : Rn → R such that ξ is a linear mapping},

B :={ξ : Rn → R such that ξ is a linear mapping and ∥ξ∥ < 1},

ϵB :={ξ : Rn → R such that ξ is a linear mapping and ∥ξ∥ < ϵ}, for ϵ > 0.

Figure 1.1: Epigraph of functions

Given a function f : D ⊂ Rn → R, set epif := {(x, α) | x ∈ D, α ∈R, α ≥ f(x)} and is said to be the epigraph of f (see Figure 1.1) A set

D ⊂ Rn is called a convex set if the line segment connecting any two pointsx, y inD lies within D, meaning [x, y] ⊂ D for all x, y ∈ D Figure 1.1 showsthat the set epif1 is convex, but epif2 is not convex.

We denote Sα(f ) := {x ∈ D | f(x) ≤ α} as the α-sublevel set of afunction f defined on a set D ⊂ Rn, where α∈ R (see Figure 1.2).

Theorem 1.2.1 (See [18, p 25]) Let f : D ⊂ Rn → R and D be a convexset Then, f is a convex function on D if and only if

f (xλ)≤ (1 − λ)f(x0) + λf (x1),for all x0, x1 ∈ D, λ ∈ [0, 1], and xλ = (1− λ)x0+ λx1.

Theorem 1.2.2(See [18, p 26]) Let f be a twice continuously differentiablereal-valued function on an open interval (a, b) throughout R Then, f isconvex if and only if its second derivative f′′ is non-negative on (a, b).Example 1.2.1 Consider the function f (x) = x3 + x We have f′′(x) =6x > 0 for all x > 0 However, if x < 0, then f′′(x) < 0 By Theorem 1.2.2,the function f (x) is convex on (0, +∞) but not convex on R.

Theorem 1.2.3(See [18, p 27]) Let f be a twice continuously differentiablereal-valued function on an open convex set D in Rn Then, f is convex onD if and only if its Hessian matrix

H(x) =

∂xi∂xj (x1, , xn)

,is positive semi-definite for every x = (x1, , xn)∈ D.

Noted that A symmetric square matrix Q is called positive semi-definiteif for every column vector v, then vTQv ≥ 0.

Example 1.2.2 Consider the function f (x, y) = x2 + 2xy + y2 on R2 Todetermine if f is a convex function on R2 or not, we need to examine itsHessian matrix

H(x, y) =

2 2

Since vTHv = 16 > 0 for all v as column vectors of H, the Hessian matrixof f is positive semi-definite Therefore, by Theorem 1.2.3, we conclude thatthe given function f (x, y) is a convex function on R2.

Theorem 1.2.4 (See [18, p 28]) Let f be a convex function defined onD ⊂ Rn, and α∈ R Then, Sα(f ) is convex set.

The converse of Theorem 1.2.4 is not true For example, consider thefunction f (x) = x3+ x onR in Example 1.4 We observe that the α-sublevelsets of this function are always convex sets; however, this function is notconvex on R Finetti introduced the definition of quasiconvex functionsbased on this property in [12].

By Definition 1.3.1 and Theorem 1.2.4, we observe that every convexfunction is a quasiconvex function The following proposition provides bothnecessary and sufficient conditions for a quasiconvex function to be convex.Proposition 1.3.1 ([7, p 313]) A function f defined on a non-empty con-vex set D ⊂ Rn is convex on D if and only if f + ξ is quasiconvex on D forall ξ∈ R∗n.

Next, we present some properties of quasiconvex functions, which arereferenced from [16] and [1] Note that if subsequent sections do not specifythe properties of the setD, we assume that D is a nonempty and convex setinRn.

Theorem 1.3.1 (See [16, p 115-116]) Let D ⊂ Rn be a non-empty convexset, and f is a function defined on D Then the following statements areequivalent:

i) f is a quasiconvex function on D.

ii) f (xλ)≤ max{f(x0), f (x1)} for all x0, x1 ∈ D, λ ∈ [0, 1].

iii) If f (x0)≤ f(x1), then f (xλ)≤ f(x1) for all x0, x1 ∈ D, λ ∈ [0, 1].Let I = [a, b] be a bounded interval in R Write C(I) for the set ofcontinuous functions on the interval I Suppose f ∈ C(I) We say f is(nonstrictly) increasing, and write,

f ∈ CI+(I), if x < y implies f (x)≤ f(y).We say u is (nonstrictly) decreasing, and write

f ∈ CI−(I), if x < y implies f (x)≥ f(y).

We say f is down-up if there exists a global minimizer x∗ of f and if therestriction of f to [a, x∗] is decreasing and the restriction of f to [x∗, b] isincreasing.

Proposition 1.3.2(See [1, p 4]) Suppose f : [a, b] ⊂ R → R is continuous.Let x∗ be a global minimizer of f Then f is quasiconvex if and only if f isdown-up.

Example 1.3.1 Consider the function f (x) = ln(x2+1) on [−2, 2] We havex∗ = 0 as the global minimum point of f on [−2, 2], meaning x∗

∈ [−2, 2]satisfies f (x∗) = min{f(x) : x ∈ [−2, 2]} Furthermore, f is decreasing on[−2, 0] and increasing on [0, 2] According to Proposition 1.3.2, we concludethat f is a quasiconvex function on [−2, 2].

Consider the function fξ(x) := f (x)+ξ(x) on [−2, 2], where ξ(x) = −0.9x(see Figure 1.3.1) In this case, we have x∗ = 10−√19

point that satisfies fξ(x∗) = min{fξ(x)(x) : x ∈ [−2, 2]} Since fξ is notincreasing on [x∗, 2], by Proposition 1.3.2, fξ(x) is not quasiconvex on [−2, 2].According to Proposition 1.3.1, the function f is not convex on [−2, 2].

Figure 1.3: f (x) = ln(x2 + 1) (left) and fξ(x) (right) on [−2, 2] , whereξ(x) =−0.9x.

However, if we consider the function fξ(x) := f (x) + ξ(x) on [−2, 2],with ξ ∈ R∗n such that ∥ξ∥ ≤ 0.5, we have x∗

∈ [−2, 2] satisfying fξ(x∗) =

min{fξ(x)(x) : x ∈ [−2, 2]} Furthermore, fξ is decreasing on [−2, x∗] andincreasing on [x∗, 2] According to Proposition 1.3.2, fξ is quasiconvex on[−2, 2] for all ξ ∈ R∗n with ∥ξ∥ ≤ 0.5 This means that the function f isquasiconvex, and when perturbed by linear maps ξ with sufficiently smallnorms, it remains quasiconvex This functions were studied in [7, p 312-313].

Proposition 1.3.3 ([6, p 3]) Let f be a function on D ⊂ Rn The setΛ :={ξ ∈ R∗n : f + ξ is quasiconvex on D}

(ξm− ξ)(xi)≤ ∥(ξm− ξ)(xi)∥ ≤ ∥ξm− ξ∥ ∥xi∥ ≤ β − α, for m ≥ N.Therefore,

(f + ξm)(xi) = (f + ξ)(xi) + (ξm− ξ)(xi)≤ β, for m ≥ N.Since f + ξm is quasiconvex for n∈ N,

(f + ξm)(xλ)≤ β, for m ≥ N.Letting m→ ∞ gives

(f + ξ)(xλ)≤ β.As β > α is arbitrary, we get

(f + ξ)(xλ)≤ α.

Therefore, Sα is convex and fξ is quasiconvex on D Hence, ξ ∈ Λ.

Example 1.3.2 We consider the function f (x) = x3+ x onR in Example1.4 Set fp(x) := f (x) + px for p∈ R (see Figure 1.4) We have

p(x) = f′(x) + p = 3x2+ 1 + pTherefore,

• If p ≥ −1, then fp(x) is increasing on R By Proposition 1.3.2, fp isquasiconvex on R.

p =−2p =−1.8p =−1.6p =−1.4p =−1.2

p =−1p =−0.8p =−0.6p =−0.4p =−0.2p = 0p = 0.2p = 0.4p = 0.6p = 0.8p = 1p = 1.2p = 1.4p = 1.6p = 1.8p = 2

Figure 1.4: The functions f (x) + px on [0, 1] for some p ∈ [−2, 2], wheref (x) = x3+ x

• If p < −1, then fp is not down-up By Proposition 1.3.2, fp is notquasiconvex on R.

Λ ={p ∈ R : f(x) + px is quasiconvex on D} = [−1, +∞).

It shows that Λ is closed Figure 1.4 illustrates the quasiconvex functions fpin solid blue for some p∈ Λ, as well as the non-quasiconvex functions fp indashed red for some p̸∈ Λ.

Definition 1.4.1 ([7, p 311], [9, p 1091]) A function f : X ⊂ Rn → Ris said to be robustly quasiconvex on a nonempty and convex set D ⊂ X ifthere exists ϵ > 0 such that f + ξ is quasiconvex on D for all ξ ∈ ϵB.Proposition 1.4.1 ([8, p 197], [3, p 195]).

i) Every convex function on D is robustly quasiconvex on D.ii) Every robustly quasiconvex function on D is quasiconvex on D.

Figure 1.5 illustrates the robustly quasiconvex function The followingexample shows that the robustly quasiconvex function is also quasiconvex(by Proposition 1.4.1).

f (x0)− f (x1)

f (xλ)− f (x1)∥xλ− x1∥ ≤ δ

for |δ| < σ, x0, x1 ∈ D, x0 ̸= x1, xλ = (1− λ)x0+ λx1 and λ∈ (0, 1).

Proposition 1.4.2 ([6, p 7]) A function f : D ⊂ Rn → R is quasiconvex(robustly quasiconvex, respectively) on D if and only if f is quasiconvex(robustly quasiconvex, respectively) on every line segment contained in D.Proof Suppose [u, v]⊂ D is a line segment in D It is clear that [u, v] is aconvex set and f is quasiconvex on [u, v] Conversely, for x0, x1 ∈ D, if fis quasiconvex on [x0, x1], then f (xλ) ≤ max{f(x0), f (x1)} Therefore, f isquasiconvex onD.

If f is robustly quasiconvex on D, there exists ϵ > 0 such that f + ξis quasiconvex on D for ξ ∈ ϵB Therefore, f + ξ is quasiconvex on I, forall I = [u, v] ⊂ D, ξ ∈ ϵB Hence, f is robustly quasiconvex on I, for allI = [u, v]⊂ D.

Conversely, suppose that f is not robustly quasiconvex on D, then forall ϵ > 0, there exists ξ ∈ ϵB such that f + ξ is not quasiconvex So, thereexist x0, x1 ∈ D and λ ∈ (0, 1) such that

(f + ξ)(xλ) > max{(f + ξ)(x0), (f + ξ)(x1)}

where xλ = (1− λ)x0+ λx1 Thus, f + ξ is not quasiconvex on I = [x0, x1]for ϵ > 0, ξ ∈ ϵB, a contradiction with f is robustly quasiconvex on linesegments I ⊂ D.

Proposition 1.4.3 ([6, p 7]) Let f be a function on [u, v] ⊂ Rn Definefunction

g(t) := f

u + t v− u∥v − u∥

, t ∈ [0, ∥v − u∥]

The function f is robustly quasiconvex on [u, v] if and only if g is robustlyquasiconvex on [0,∥v − u∥].

Proof If f is a robustly quasiconvex function on [u, v], there exists σ > 0such that

′for |δ′

| < σ′

; t0, t1 ∈ [0, ∥v − u∥], t0 ̸= t1.We set

x0 = u + t0 v− u

∥v − u∥ ∈ [u, v] and x1 = u + t1

v− u

∥v − u∥ ∈ [u, v].Then,

g(t0)− g(t1)|t0− t1| =

f (x0)− f(x1)

∥x0− x1∥ ≤ δ with δ = δ′

f (xλ)− f(x1)

xλ = (1− λ)x0+ λx1 for λ∈ [0, 1)= u + tλ v− u

∥v − u∥ with tλ = (1− λ)t0+ λt1, λ∈ (0, 1).So,

f (xλ)− f(x1)

g(tλ)− g(t1)|tλ− t1| ≤ δ

′.Hence, g is the robustly quasiconvex function.

Conversely, if g is a robustly quasiconvex function on [0,|v − u|], then fis a robustly quasiconvex function on [u, v], and the proof follows a similarargument as above.

Example 1.4.1 Consider f (x, y) = ln(x2 + 2y2) on R2 and u = (1, 1),v = (2, 2) (see Figure 1.6) Set

g(t) = f

u + t v− u∥v − u∥

, t∈ [0, ∥v − u∥] = [0,√2]= f

1 + √t

2, 1 +t√2

= ln"

1 + √t2

2#.

Similarly to Example 1.3.2, we have

Λ = {p ∈ R : g(x) + px is quasiconvex on D}= (−∞, −√2]∪

√22 , +∞

Therefore, g + ξ is quasiconvex on D for ξ ∈ R∗n, ∥ξ∥ ≤√

2 By Definition1.4.1 and Proposition 2.1.4, the function f is robustly quasiconvex on [u, v].

1.0 1.2

1.4 1.6

1.8 2.0 1.01.21.41.6

f(x, y) = ln(x2+ 2y2)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.5

g(t) = f

(u + v u||v u||t

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.48

(g + )(t)

1.0 1.21.4 1.6

1.8 2.0 1.01.21.41.6

f(x, y) = ln(x2+ 2y2)

g(t) = f

(u + v u||v u||t

(g + )(t)

u + t v− u∥v − u∥

Figure 1.6: The function f (x, y) = ln(x2 + 2y2) and f

u + t v− u∥v − u∥

,where u = (1, 1), v = (2, 2)

Chapter 2

Computing the Robustness

Index of Quasiconvex Functions

Definition 2.1.1 ([3, p 194], [6, p 5]) Suppose f : X ⊂ Rn → R is aquasiconvex function on D ⊂ X The robustness index of f on D, denotedsf(D) or just sf when the domain is fixed, is defined to be

sf(D) = sup {ϵ > 0 : f + ξ is quasiconvex on D, ξ ∈ ϵB}

By Definition 2.1.1, it is clear that a quasiconvex function f is robustlyquasiconvex onD if and only if sf(D) > 0, and f isn’t robustly quasiconvexif and only if sf(D) = inf ∅ := −∞ Proposition 2.1 in [8, p 197] impliesthat f is convex if and only if sf(D) = ∞.

In Example 1.3.2, we have

• If 0 < ϵ ≤ 1, then fp is quasiconvex on D for |p| < ϵ.

• If ϵ > 1, there exists 1 < |p| < ϵ such that fp is not quasiconvex onD.Therefore, f is robustly quasiconvex on D and sf(D) = 1 Moreover, ifξ ∈ R∗n and ∥ξ∥ = sf(D) = 1, f + ξ is also quasiconvex on D In general,we have the following proposition.

Proposition 2.1.1 ([6, p 5]) Suppose f : D ⊂ Rn → R is a robustlyquasiconvex function on D Then f + ξ is quasiconvex on D for ξ ∈ R∗n,∥ξ∥ ≤ sf.

Proof For ξ∈ R∗n,∥ξ∥ ≤ sf, we have f + ξ is quasiconvex If∥ξ∥ = sf, wecan choose ξm ∈ R∗nsuch that∥ξm∥ < sf and ξmconverges to ξ Proposition1.3.3 implies that the set Λ = {ξ ∈ R∗n : f + ξ is quasiconvex onD} isclosed Since ξm ∈ Λ and ξm converges to ξ, it follows that ξ∈ Λ Hence, fis robustly quasiconvex if and only if f + ξ is quasiconvex onD for ξ ∈ R∗n,∥ξ∥ ≤ sf.

Proposition 2.1.2 ([6, p 6]) Suppose f : D ⊂ Rn → R is a quasiconvexfunction Then

sf(D) = sup {σ > 0 : σ in Theorem 1.4.1}

Proof Set ¯sf(D) := sup {σ > 0 : σ in Theorem 1.4.1} If f isn’t robustlyquasiconvex, Definition 2.1.1 and Theorem 1.4.1 show that

sf(D) = sf(D) = −∞.

If f is robustly quasiconvex, Theorem 1.4.1 implies that ¯sf(D) > 0 For0 < σ < ¯sf(D), we have σ satisfying Theorem 1.4.1 For ξ ∈ σB, x0, x1 ∈D, x0 ̸= x1, if (f + ξ)(x0)≤ (f + ξ)(x1), we set

δ = ξ

x1− x0∥x0− x1∥

.We have|δ| =

x1− x0∥x0− x1∥

≤ ∥ξ∥ < σ andf (x0)− f(x1)

x0− x1∥x0− x1∥

xλ− x1∥xλ− x1∥

, for all λ∈ (0, 1).Theorem 1.4.1 shows that

f (xλ)− f(x1)

xλ− x1∥xλ− x1∥

(f + ξ)(xλ)≤ (f + ξ)(x1).

Hence, f + ξ is quasiconvex on D for ξ ∈ σB, and ¯sf(D) ≤ sf(D) Forevery 0 < ϵ < sf(D), we can similarly prove that ϵ satisfies Theorem 1.4.1.Therefore, ¯sf(D) ≥ sf(D) Thus, ¯sf(D) = sf(D).

Proposition 2.1.3 ([6, p 6]) Suppose f : X ⊂ Rn → R is quasiconvex onD ⊂ X If A ⊂ D and A is a nonempty and convex set, then sf(A)≥ sf(D).Proof Suppose g : X ⊂ Rn → R is quasiconvex on D ⊂ X For everyx0, x1 ∈ A ⊂ D, since g is quasiconvex on D,

g(xλ)≤ max{f(x0), f (x1)}, xλ = (1− λ)x0 + λx1, λ∈ [0, 1].Therefore, the restriction of g on A is quasiconvex on A.

Suppose that sf(A) < sf(D), we choose ϵ > 0 such that sf(A) < ϵ <sf(D) For every ξ ∈ ϵB, f + ξ is quasiconvex on D Therefore, f + ξ isquasiconvex on A Hence,

f + ξ is a quasiconvex on A, for all ξ ∈ ϵB.Definition 2.1.1 shows that

sf(A)≥ ϵ,

which contradicts to sf(A) < ϵ Hence, sf(A)≥ sf(D).

Proposition 2.1.4 ([6, p 7]) Let f be a function on [u, v] ⊂ Rn Definefunction

g(t) := f

u + t v− u∥v − u∥

, t ∈ [0, ∥v − u∥] Then sf([u, v]) = sg([0,∥v − u∥]).

Proof Proposition 1.4.3 shows that sf(I) = sg([0,∥v − u∥]).

Proposition 2.1.5 ([6, p 8]) Suppose f : X ⊂ Rn → R is a quasiconvexfunction on D ⊂ X, then

i) sf(D) = inf{sf([u, v]) : (u, v)∈ D × D, u ̸= v}.

ii) sf(D) = inf{sf([u, v]) : (u, v)∈ ∂D × ∂D, u ̸= v}, if D is compact.Proof i) By Proposition 2.1.3, we have

sf(D) ≤ sf([u, v]) for all (u, v)∈ D × D, u ̸= v.Therefore,

sf(D) ≤ ¯sf(D) := inf{sf([u, v]) : (u, v)∈ D × D, u ̸= v}.

If ¯sf(D) = −∞, then sf(D) = −∞ If ¯sf(D) > 0, for 0 < ϵ < ¯sf(D), forevery (u, v)∈ D × D,

sf([u, v]) > ϵ.

So, there exits ϵu,v > ϵ such that f + ξ is quasiconvex on [u, v], ξ ∈ ϵu,vB.Therefore, if ξ∈ ϵB, then ξ ∈ ϵu,vB, for every (u, v)∈ D × D, u ̸= v Hence,f + ξ is quasiconvex on line segments [u, v] ⊂ D, ξ ∈ ϵB By Proposition1.4.2, we have f + ξ is quasiconvex on D, ξ ∈ ϵB By Definition 2.1.1, wehave

sf(D) = sup{ϵ > 0 : f + ξ is quasiconvex on D, ξ ∈ ϵB}.Therefore, sf(D) ≥ ϵ for all 0 < ϵ < ¯sf(D) It implies that

sf(D) ≥ ¯sf(D).Hence, sf(D) = ¯sf(D).

ii) If D is compact, we have ∂D ⊂ D and

{sf([u, v]) : (u, v)∈ ∂D × ∂D, u ̸= v} ⊂ {sf([u, v]) : (u, v)∈ D × D, u ̸= v}.Therefore,

sf(D) ≤ ¯¯sf(D) := inf{sf([u, v]) : (u, v)∈ ∂D × ∂D, u ̸= v}.

For every (u, v) ∈ D × D, u ̸= v, there exits (u∗, v∗)∈ ∂D × ∂D such that[u, v]⊂ [u∗, v∗] By Proposition 2.1.3, we have

sf([u, v]) ≥ sf([u∗, v∗])≥ ¯¯sf(D).Therefore,

sf(D) ≥ ¯¯sf(D).Hence,

sf(D) = ¯¯sf(D).

Proposition 2.1.6 ([6, p 10]) Suppose that a continuous function f :[a, b]⊂ R → R is quasiconvex Let x∗

∈ [a, b] such that f(x∗) = min{f(x) :x ∈ [a, b]} If there exist x0, x1 ∈ (a, b), x0 ̸= x1 satisfying f (x) = f (x0) =f (x1) > f (x∗) for x∈ [x0, x1], then sf =−∞.

Proof By Proposition 1.3.2, without loss of generality, let’s assume thatx∗ < x0 < x1,

i) sf([a, b]) = inf{|f′(t)| : t ∈ (a, b) satisfying ft is non-quasiconvex on [a, b]}.ii) sf([a, b]) = inf{|f′(t)| : t ∈ (a, b) satisfying t is a local maximum point of ft}.where f′(t) is the derivative of the function f at the point t.

Proof By Definition 2.1.1, we have

sf([a, b]) = sup{ϵ > 0 : f(x) + px is quasiconvex on [a, b], |p| < ϵ}

= inf{|p| : p ∈ R satisfying f(x) + px is non-quasiconvex on [a, b]}

≤ inf {|f′(t)| : t ∈ (a, b) satisfying f(x) − f′(t)x is non-quasiconvex on [a, b]}≤ inf {|f′

(t)| : t ∈ (a, b) satisfying t is a local maximum point of ft}

For p ∈ R such that Fp(x) := f (x) + px is non-quasiconvex on [a, b].Therefore, there exist x0, x1 ∈ [a, b] and λ ∈ (0, 1) such that Fp(xλ) >max{Fp(x0), Fp(x1)} Moreover, since Fp is continuous on [a, b], there existst∈ (x0, x1) such that t is a local maximum point of Fp Then

Fp′(t) = 0.Therefore,

p =−f′(t).Hence,

p∈ {−f′(t) : t∈ (a, b) satisfying t is a local maximum point of ft},so

(t)| : t ∈ (a, b) satisfying t is a local maximum point of ft}

Corollary 2.1.1 ([6, p 11]) Suppose that twice continuously differentiablefunction f : [a, b] ⊂ R → R is quasiconvex on [a, b] Then, α ≤ sf ≤ β,where

α := inf{|f′(t)| : t ∈ (a, b), f′(t)̸= 0, f′′(t)≤ 0}β := inf{|f′(t)| : t ∈ (a, b), f′(t)̸= 0, f′′(t) < 0} ,and f′′(t) is the twice derivative of the function f at the point t.Proof For t∈ (a, b), set ft(x) = f (x)− xf′(t) for x∈ [a, b] We have

ft′(x) = f′(x)− f′

(t) and ft′′(x) = f′′(x).Therefore, f′

t(t) = 0 Morever, we have• If f′′

t(t) < 0, t is a local maximum point of ft.• If t is a local maximum point of ft, f′′

t(t)≤ 0.By Proposition 2.1.7,

sf([a, b]) = inf{|f′

(t)| : t ∈ (a, b) satisfying t is a local maximum point of ft} Hence, α≤ sf ≤ β.

2.2Approximating Robustness Index of QuasiconvexFunctions

Consider a convex and compact set D ⊂ Rn (n > 1) such that there existsa continuous mapping C : I → Rn satisfying

∂D = {C(t1, t2, , tn−1) : ti ∈ [ai, bi], i = 1, 2, , n− 1},

where I = [a1, b1]×[a2, b2]×· · ·×[an−1, bn−1]⊂ Rn−1 Let m = (m1, m2, , mn−1),mi ∈ N, mi > 0 and

∂D(m) :=

C (t1, t2, , tn−1) : ti = ai+ bi− ai

2mi ki for ki = 0, 1, , 2mi

,s(m)f (D) := min{sf([u, v]) : u, v∈ ∂D(m), u̸= v}.

Proposition 2.2.1 ([6, p 12]) Suppose f :D ⊂ Rn → R is a quasiconvexfunction on D Then, s(m)f (D) converges to ∼sf(D) ≥ sf(D).

Proof By Proposition 2.1.5, we have

sf(D) = inf{sf([u, v]) : for all (u, v)∈ ∂D × ∂D, u ̸= v}.

It is clear that s(m)f (D) ≥ sf(D), for all m ∈ Nn−1, mi > 0 Furthermore, forevery mi ∈ N, mi > 0 we have

Suppose that the continuously twice differentiable function f : [a, b]⊂ R →R is quasiconvex In [3, p 198-199], an algorithm for computing the robust-ness index of a quasiconvex function f on [a, b] was presented (Algorithm 1).Then, sf ∈ [α1, α2], where f is convex on L(|f′

|, α1) :={x ∈ [a, b] : |f′(x)| ≤α1} and f is not convex on L(|f′

|, α2).

Example 2.3.1 Consider the quasiconvex function f (x) = ln(x4 + 1) onD = [−2, 3] For each x ∈ (−2, 3), we have

f′(x) = 4x3

x4+ 1 and f′′

(x) = 4x

2(−x4 + 3)(x4+ 1)2

Then, f′′(x) ≥ 0 if and only if x ∈ [−√3

3] Since L(|f′

|, 1) = {x ∈[−2, 3] : |f′(x)| ≤ 1} ⊂ [−√4

3], f is convex on L(|f′

|, 1) Since L(|f′|, 2) ={x ∈ [−2, 3] : |f′(x)| ≤ 2} ∋ 2 and f′′(2) < 0, f is not convex on L(|f′|, 2).By Algorithm 1, 1≤ sf ≤ 2 We have

β = inf{|f′(t)| : t ∈ (a, b), f′(t)̸= 0, f′′(t) < 0}= infn|f′(t)| : t ∈−2, −√4

3, 3o= |f′

(3)| = 5441.For

t∈ {t ∈ (a, b) : f′(t)̸= 0, f′′(t) = 0} =n−√4

3o,set ft(x) := f (x)− xf′

(t) for all x∈ D We have• t =√4

3, ft(x) = f (x)−√4

27x is quasiconvex on D.• t = −√4

3, ft(x) = f (x) +√4

27x is quasiconvex onD.By Proposition 2.1.7, sf = 54

41 ∈ [1, 2] In general, we present Algorithm 1-2to compute the robustness index of the function f on [a, b]⊂ R.

Algorithm 1: Finding the robustness index of quasiconvex functionson [a, b] (see [6, p 15])

Input : Quasiconvex function f (x) on [a, b];γ := step length;

zmin := real number allowed by the compiler;α := real number greater than zmin;

Output: sf, approximating the robustness index of f on [a, b].1 if f is convex on [a, b] then

Algorithm 2: Finding the robustness index of quasiconvex functionson [a, b]⊂ R (see [6, p 15])

Input : Quasiconvex function f (x) on [a, b].Output: sf, the robustness index of f on [a, b].1 sf :=−∞;

2 if f is convex on [a, b] then

ro-Algorithm 3: Finding the robustness index of quasiconvex functionsonD ⊂ Rn (n > 1) (see [6, p 16])

Input : Quasiconvex function f on D ⊂ Rn;

A continuous mapping C(t1, t2, , tn−1) such that∂D = {C(t1, t2, , tn−1) : ti ∈ [ai, bi]⊂ R, i = 1, 2, , n − 1};

m = (m1, m2, , mn−1), mi ∈ N, mi > 0,i = 1, 2, , n− 1.

Output: s(m)f (D), an approximation of the robustness index of f onD.

1 ∂D(m)=

C (t1, t2, , tn−1) : ti = ai+bi− ai

2mi k for k = 0, 1, , 2mi

;2 Sm =∅;

5 Sm := Sm∪ {sg([0,∥v − u∥])} ; //sg([0, ∥v − u∥]) is computed byAlgorithm 1 or 2

6 s(m)f (D) := min Sm;