VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY UNIVERSITY OF SCIENCE HUYNH THI HONG DIEM APPROXIMATIONS, STABILITY AND OPTIMALITY CONDITIONS IN NONSMOOTH OPTIMIZATION PhD THESIS IN MATHEMATICS Hochiminh City - 2015 VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY UNIVERSITY OF SCIENCE HUYNH THI HONG DIEM APPROXIMATIONS, STABILITY AND OPTIMALITY CONDITIONS IN NONSMOOTH OPTIMIZATION Major: Optimization Theory Code: 62 46 20 01 First examiner: Associate Prof Dr NGUYEN DINH HUY Second examiner: Associate Prof Dr NGUYEN NGOC HAI Third examiner: Dr NGUYEN DINH TUAN First independent examiner: Prof DSc NGUYEN DONG YEN Second independent examiner: Dr DINH NGOC QUY SCIENTIFIC SUPERVISOR: Prof DSc PHAN QUOC KHANH Hochiminh City - 2015 Declarization of originality I hereby declare that this thesis, done under the supervision of Professor Phan Quoc Khanh, is entirely the result of my own work To the best of my knowledge, this study and its findings have never been published by any other researchers There is no part of this thesis which overlaps parts of other works submitted for the award of any degree or diploma I also obtained the consent of Dr Le Thanh Tung, co-author of the joint paper [D4] referred to in Chapter 5, to let me include in my thesis some of the results of this joint paper, which were not included in his thesis defended three years before their publication Hochiminh City, November, 2015 The author Huynh Thi Hong Diem i Acknowledgements The completion of this doctoral dissertation would not have been possible without the support of many people I would like to express my sincere gratitude to all of them First, I want to express my deepest gratitude to my supervisor, Professor Phan Quoc Khanh, for his valuable guidance, scholarly inputs, and consistent encouragement I received throughout my research work From deciding on the research topic in the beginning to the process of actual writing of the thesis, he offered his unreserved help and guided me through every step of my work He provided me with inspiring and insightful guidance, without which this study would never have been brought to a successful completion Second, I am very pleased to extend my thanks to the reviewers of this thesis Their comments, remarks and questions have truly improved the quality of this manuscript I would like to thank the professors who agreed to be on the jury judging my thesis defense I would like to express my thankfulness to the members of the Group of Optimization in Southern Vietnam Their discussions during the seminar activities have provided me with valuable encouragements and suggestions In particular, I would like to appreciate Dr Le Thanh Tung who extended his support in a very special way during the work on our joint paper, and I benefited a lot from his scholarly interactions and suggestions My thanks also go to the University of Science, Vietnam National University Hochiminh City, and the College of Can tho for their support during my work to complete the PhD program Last but not least, I owe a lot to my mother and my aunts, who support, encourage and help me at every stage of my personal and academic life, and long to see this achievement come true They always make sure I am provided with a carefree ii environment where I can devote myself entirely to my study (by, for example, taking good care of my two children from their infant days so that I can focus on my PhD study) I really appreciate so much my children sacrified by agreeing to stay with their grandmother missing their busy mother’s warmth and care Hochiminh City, November, 2015 The author Huynh Thi Hong Diem iii Contents Certificate of originality i Acknowledgements ii List of symbols and notations vi Preface ix Basic notations and preliminary facts Variational convergence of finite-valued bifunctions 2.1 Introduction 2.2 Epi-convergence 2.2.1 Epi-convergence 2.2.2 Legendre-Fenchel transform and its continuity 10 2.3 Epi/hypo convergence and lopsided convergence, geometric characterizations 12 2.4 A characterization by e/h-convergence of proper bifunctions 23 2.5 A characterization by continuity of partial Legendre-Fenchel transform 25 2.6 Conclusions 32 Variational properties of epi/hypo convergence and approximations of optimization problems 33 3.1 Introduction 33 3.2 Variational properties of epi-convergence 35 iv 3.3 Variational properties of epi/hypo convergence 37 3.4 Approximations of equilibrium problems 43 3.5 Approximations of multi-objective optimization 47 3.6 Approximations of Nash equilibria 50 3.7 Conclusions 53 Variational convergence of bifunctions on nonrectangular domains and approximations of quasiequilibrium problems 55 4.1 Introduction 55 4.2 Variational convergence of bifunctions on nonrectangular domains 56 4.3 Variational properties of epi/hypo convergence 62 4.4 Approximations of quasiequilibrium problems 69 4.5 Conclusions 72 Higher-order sensitivity analysis in nonsmooth vector optimization 74 5.1 Introduction 74 5.2 Higher-order radial-contingent derivatives 76 5.3 Properties of higher-order contingent-type derivatives 87 5.4 Higher-order contingent-type derivatives of perturbation maps 92 5.5 Conclusions 100 Optimality conditions for a class of relaxed quasiconvex minimax problems 101 6.1 Introduction 101 6.2 Optimality conditions for minimax problem (P) 103 6.3 Conclusions 111 General conclusions 112 List of the author’s papers related to the thesis 114 List of the author’s conference reports related to the thesis 115 References 116 v List of symbols and notations N the set of the natural numbers Q the set of the rational numbers R the set of the real numbers ¯ = R ∪ {−∞, ∞} R the set of the extended straight line Rn an n-dimensional normed space X∗ the dual/conjugate space of a normed space X f∗ the Legendre-Fenchel conjugate of a numerical function f clA the closure of a set A intA the interior of a set A bdA the boundary of a set A coA the convex hull of a set A dist(x, A) the distance from x to a subset A ∀ for all ∃ there exists ✷ the end of a proof U(¯ x) the collection of the neighbourhoods of x¯ fcn(Rn ) the collection of the extended real-valued unifunctions pfcn(Rn ) the collection of the proper functions of fcn(Rn ) biv(Rn × Rm ) the collection of the extended real-valued bifunctions e/h epi/hypo fv-fcn(Rn ) the collection of the finite-valued unifunctions fv-biv(Rn × Rm ) the collection of the finite-valued bifunctions on rectangles e f ν → f or f = e-limν f ν f ν epi converge to f vi e-lsν f ν the upper epi-limit of functions f ν e-liν f ν the lower epi-limit of functions f ν usc upper semicontinuous lsc lower semicontinuous coneA the conical hull of a set A limsupx →x f (x ) infδ>0 supx ∈B(x,δ) f (x ) for f : X → R liminfx →x f (x ) supδ>0 infx ∈B(x,δ) f (x ) for f : X → R argminf the set of the minimizers of a function f argmaxf the set of the maximizers of a function f infA the infimum value of a numerical set A supA the supremum value of a numerical set A domf the domain of a numerical function f epif the epigraph of a numerical function f hypof the hypograph of a numerical function f domF the domain of a set-valued map F+ the profile map of a set-valued map gphF the graph of F epiF :=epiF+ Limsupν K ν the upper/outer limit of a sequence of set-valued maps K ν Liminfν K ν the lower/inner limit of a sequence of set-valued maps K ν MinC A the collection of the local (Pareto) minimal points of A WMinC A the collection of the local weak minimal points of A HeC A the collection of the Henig-proper minimal points of A QMinC A the collection of the local Q-minimal poinst of A m D F (x0 , y0 , u1 , v1 , , um−1 , vm−1 ) the mth-order contingent derivative of F at (x0 , y0 ) wrt (u1 , v1 ), , (um−1 , vm−1 ) ∈ X × Y Dm F (x0 , y0 ) the mth-order contingent derivative of F at (x0 , y0 ) m DR F (x0 , y0 ) the mth-order radial derivative of F at (x0 , y0 ) DSm F (x0 , y0 ) the mth-order radial-contingent derivative of F at (x0 , y0 ) Dbm F (x0 , y0 ) the mth-order adjacent-type derivative of F at (x0 , y0 ) vii Dlm F (x0 , y0 ) the mth-order strong adjacent-type derivative of F at (x0 , y0 ) L(X, Y ) the space of the bounded linear maps from X to Y Lf (¯ x) the sublevel set of f at x¯ L< x) f (¯ the strict sublevel set of f at x¯ x) Laf (¯ the adjusted sublevel set of f at x¯ ∂ < f (¯ x) the lower subdifferential or Plastria subdifferential ∂ ≤ f (¯ x) the infradifferential or Gutierrez subdifferential ∂ ∗ f (¯ x) the Greenberg-Pierskalla subdifferential ∂ ν f (¯ x), ∂ f (¯ x) the normal-cone subdifferentials ∂ a f (¯ x) the adjusted subdifferential viii Lg (¯ x) = [0, ∞), and f2 is usc in L< x) Furthermore, (6.1) holds since ∂ f2 (¯ x) = f2 (¯ {x∗ |∀x ∈ (−∞, 0), x∗ , x ≤ 0} = [0, ∞), and ∂ ν g(¯ x) = {x∗ |∀x ∈ [0, ∞), x∗ , x ≤ 0} = (−∞, 0] Theorem 6.2.2 asserts that x¯ = is a solution, and this can also be seen directly Now we move on to considering unique solutions Theorem 6.2.3 (necessary optimality condition, unique solution) Let the assumptions of Theorem 6.2.1 be fulfilled If, additionally, x¯ is a unique solution and C \ {¯ x} = ∅, then ∂ fi , i ∈ I(¯ x), used in Theorem 6.2.1, can be replaced by the smaller subdifferential ∂ ν fi Proof The fact that x¯ is a unique solution of (P) means that C ∩ i∈I q ) = {¯ x} Lfi (¯ Hence, x¯ cannot be in the interior of ∩i∈I Lfi (¯ q ), which is nonempty by (iv) Therefore, the separation theorem yields x∗ ∈ X ∗ \{0} such that, for x ∈ C and w ∈ ∩i∈I(¯x) Lfi (¯ q ), x∗ , x − x¯ ≥ ≥ x∗ , w − x¯ (6.9) The rest of the proof is similar to the corresponding part for Theorem 6.2.1 But, here we are working with Lfi instead of L< fi as in Theorem 6.2.1, and hence we obtain that ∂ ν fi replaces ∂ fi in the conclusion The following example says that we cannot replace ∂ fi , i ∈ I(¯ x), by ∂ ν fi in Theorem 6.2.1 (as we in Theorem 6.2.3) Example 6.2.5 Let all the data be as in Example 6.2.3, except that now x¯ = It is not hard to verify that the assumptions of Theorem 6.2.1 are satisfied Since Lf2 (¯ x) = (−∞, 0]∪[2, ∞) and hence ∂ ν f2 (¯ x) = {0}, we have ∂ ν f2 (¯ x)∩(−∂ ν g(¯ x)) = {0} It is seen in this case that the sublevel set Lf2 (¯ x) at the considered point is not convex, but Theorem 6.2.1 still works Theorem 6.2.4 (sufficient optimality condition, unique solution) Let the assumptions of Theorem 6.2.2 be satisfied Suppose furthermore that fi−1 (fi (¯ x)) = {¯ x} for all i ∈ I(¯ x) Then, the necessary condition stated in Theorem 6.2.3 is a sufficient one 109 Proof The mentioned necessary condition implies the existence of x∗ ∈ X ∗ \ {0} satisfying (6.9) Suppose x¯ is not a unique solution, i.e., one finds v ∈ C ∩ different from x¯ Then, v ∈ C ∩ i∈I(¯ x) i∈I Lfi (¯ q) Lfi (¯ x) By (6.9), x∗ , v − x¯ = On the other hand, as fi−1 (fi (¯ x)) = {¯ x}, Lfi (¯ x) \ {¯ x} = L< x) for i ∈ I(¯ x) By the upper fi (¯ semicontinuity of fi , L< x) is open and so is (∩i∈I(¯x) Lfi (¯ x)) \ {¯ x} Consequently, for fi (¯ any d ∈ X and small positive t, v + td ∈ ∩i∈I Lfi (¯ x) Therefore, t < x∗ , d >=< x∗ , v − x¯ + td > − < x∗ , v − x¯ >≤ Thus, x∗ = 0, a contradiction Theorem 6.2.4 confirms that a candidate is a solution in the next example Example 6.2.6 Let f1 and g be as in Example 6.2.1 Let x¯ = and    1/2(x + 1) if x ≤ 0,   f2 (x) = x − x + if < x ≤ 2,     if x > Clearly the assumptions of Theorem 6.2.2 are fulfilled We have I(¯ x) = {2}, q¯ = 1/2, and f2−1 (f2 (¯ x)) = {¯ x} Furthermore, as ∂ ν f2 (¯ x) ∩ (−∂ ν g(¯ x)) = [0, ∞) = {0}, Theorem 6.2.4 asserts that x¯ = is the unique solution, which is also seen directly In Theorem 6.2.4, the condition that fi−1 (fi (¯ x)) = {¯ x} for all i ∈ I(¯ x) is essential as shown by the following Example 6.2.7 Let f1 , g be as in Example 6.2.1, x¯ = 0, and   x+1 if x ≤ 0, f2 (x) =  x2 − x + if x > We have I(¯ x) = {1, 2}, q¯ = 1, L< x) = (−∞, 2), L< x) = (−∞, 0), Lg (¯ x) = [2, ∞), f1 (¯ f2 (¯ and fi is u.s.c in L< x) for i = 1, Furthermore, Lf1 (¯ x) = (−∞; 2], ∂ ν f1 (¯ x) = IR+ , fi (¯ Lf2 (¯ x) = (−∞; 0] ∪ {2}, ∂ ν f2 (¯ x) = IR+ , ∂ ν g(¯ x) = IR− , and hence ∂ ν fi (¯ x) ∩ i=1,2 (−∂ ν g(¯ x)) = [0, ∞) as Theorem 6.2.4 requires But, x¯ = is not a unique solution (0 is another one) The cause is that f2−1 (f2 (¯ x)) = {0; 2} = {¯ x} 110 6.3 Conclusions This chapter considers optimality conditions for a particualar model of minimax problems with a finite index set for the involved maximization The main assumption is that the data of the problem are relaxed quasiconvex in the sense that the involved functions have convex sublevel sets at the considered point, not necessary at a whole region Thanks to this assumption, the naturally chosen derivertive-like tools are generalized subdifferentials of the mentioned functions based on their sublevel sets Both necessary optimality conditions and sufficient ones are established When applied even to the special case of minimization problem, i.e., the case with the index set for the maximization being a singleton, the obtained results are advatageous over several recent existing optimality conditions 111 General conclusions This thesis contains a study of stability in the broad sense of approximations and sensitivity analysis and of optimality conditions for nonsmooth optimizations The approximations of the considered problems, instead of its pertubations in usual stability studies, are defined in terms of variational convergence of finite-valued bifunctions Hence, we first develop full characterizations of the kinds of such convergence Then, we continue with variational properties of epi/hypo convergence, the main variational convergence investigated in the first two chapters before moving on to applications of these properties to approximations of selected typical variational problems In the subsequent Chapter 4, we extend the definitions of epi/hypo and lopsided convergence to the case of bifunctions defined on nonrectangular domains and investigate variational properties in order to apply them in considering approximations of quasivariational problems After three chapters devoted to derivative-free studies, in Chapter we apply generalized derivatives to sensititity analysis, including a new kind of derivative we introduce in order to compute approximately or find bounds for the derivatives of the pertubation and weak pertubation maps of a general multivalued vector optimization problem The last Chapter is devoted to establishing optimality conditions for a relaxed quasiconvex minimax problem Here, as derivative-like objects, we use several cone subdifferentials defined suitably for quasiconvex functions, which allow us to apply convex analysis tools In all the chapters, we obtain new results or extend or improve recent existing ones in the literature We discuss in detail the results and explain relations with known ones to clarify our findings 112 As follow-ups, we think that the following issues can be the topics of our near future study (i) Variational convergence of bifunctions on nonrectangular domains and applications in approximations of quasivariational models are important The result obtained in Chapter are only the first step of development (ii) In Chapters 2-4, we restrict ourselves to finite dimensional settings We can expect important and interesting extensions to general infinite dimensional cases Stability and approximations for problems on infinite dimensional are clearly crucial because such problems are more frequent in practice (iii) For sensitivity analysis, applications of other generalized derivatives and derivativelike objects are possible, since many of them have been successfully applied in studies of optimality conditions, but have not been used in sensitivity analysis so far Furthermore, our results in Chapter can be directly developed for particular optimization problems such as mathematical programming, which is more often met in practice (iv) The results in Chapter can be developed for more general minimax problems with infinite index sets 113 List of the author’s papers related to the thesis [D1] Huynh Thi Hong Diem, Phan Quoc Khanh, Optimality conditions for a class of relaxed quasiconvex minimax problems, Control Cyber 43 (2014), 249-260 [D2] Huynh Thi Hong Diem, Phan Quoc Khanh, Criteria for epi/hypo convergence of finite-valued bifunctions, Vietnam J Math 43 (2015), 439-458 [D3] Huynh Thi Hong Diem, Phan Quoc Khanh, Approximations of optimizationrelated problems in terms of variational convergence, Vietnam J Math onlinefirst, DOI: 10.1007/s10013-015-0148-9 (2015) [D4] Huynh Thi Hong Diem, Phan Quoc Khanh, Le Thanh Tung, On higherorder sensitivity analysis in nonsmooth vector optimization, J Optim Theory Appl 162 (2014), 463-488 [D5] Huynh Thi Hong Diem, Phan Quoc Khanh, Variational convergence of bifunctions on nonrectangular domains and approximations of quasivariational problems, J Global Optim., submitted for publication 114 List of the author’s conference reports related to the thesis [1] 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[5, 12, 72] The latter was proposed in [8] and rigorously treated in [11] In [9] a modified and stronger form than epi/hypo convergence, called lopsided convergence (lop-convergence) was given