Farkas type results for nonconvex systems and applications to optimization = một số kết quả dạng farkas cho các hệ không lồi và áp dụng vào lý thuyết tối ưu

VIETNAM NATIONAL UNIVERSITY-HO CHI MINH CITY UNIVERSITY OF SCIENCE TRAN HONG MO FARKAS-TYPE RESULTS FOR NONCONVEX SYSTEMS AND APPLICATIONS TO OPTIMIZATION PhD Thesis: Optimization Theory Code: 62 46 20 01 Reviewer 1: Assoc.Prof Dr Nguyen Dinh Huy Reviewer 2: Assoc.Prof Dr Dinh Ngoc Thanh Reviewer 3: Assoc.Prof Dr Pham Hoang Quan Independent Reviewer 1: Dr Bui Trong Kien Independent Reviewer 2: Dr Nguyen Xuan Hai Supervisor: Assoc.Prof Dr.Sc NGUYEN DINH Ho Chi Minh City - 2015 Originality statement I hereby declare that this submission is my own work, done at the University of Sciences - VNU HCMC under the supervision of Assoc.Prof Dr.Sc Nguyen Dinh, International University - VNU HCMC, and, to the best of my knowledge, it contains no materials previously published and written by another person Ho Chi Minh City, 2015 The author Tran Hong Mo i Acknowledgements First and foremost, I am deeply grateful to my supervisor, Assoc.Prof.Dr.Sc Nguyen Dinh, International University - VNU HCMC, for his patient guidance throughout the writing of this thesis I also wish to express my thanks to Prof Dr.Sc Phan Quoc Khanh, University of Sciences - VNU HCMC, for his valuable and insightful lectures in Optimization in my doctorate programme Next, a very special thanks goes to the Management of the University of Sciences VNU HCMC, and the Office of Graduate Admission, for their support and assistance all the time of my PhD programme and thesis project I will never forget how the Management of the University of Tien Giang, its Office for Personnel and Administration, and Faculty of Basic science have been constantly providing me with favourable conditions for the completion of this thesis My gratitude also goes to Assoc.Prof Dr Le Hoan Hoa, HCMC University of Education, whose supervision of my Master’s thesis inspired and motivated me to further and advance my research work In addition, I owe a lot to my friend Pham Duy Khanh, the University of Education, who played a big role in persuading me to choose Optimization for my research project Finally, I am fully indebted to my family, especially my father and my wife for nurturing my learning and supporting my dream I dedicate this thesis to the memory of my late mother Without her, I would never have pursued mathematical studies and become a PhD candidate Ho Chi Minh City, 2015 Tran Hong Mo ii Contents Glossary of Notations v Introduction vii Notations and Preliminaries Farkas-type results for systems involving composite functions 2.1 Dual qualification conditions and their relations 2.1.1 Dual qualification conditions in purely algebraic setting 2.1.2 Dual qualification conditions in convex setting 2.2 Characterizations of dual conditions–Generalized Moreau-Rockafellar results 2.2.1 Dual conditions characterizing generalized Moreau-Rockafellar results 2.2.2 Special cases 2.3 Nonconvex Farkas-type results 2.3.1 Nonconvex Farkas-type results 2.3.2 Special cases 2.4 Applications 2.4.1 Alternative-type theorems 2.4.2 Set containments 2.4.3 Fenchel-Rockafellar duality formula 7 11 13 13 16 19 19 24 27 27 28 29 New versions of Farkas lemma and Hahn-Banach theorem under Slatertype conditions 32 3.1 New versions of the Farkas lemma under Slater-type conditions 33 3.1.1 Farkas lemma for cone-convex systems 33 3.1.2 Farkas lemma for sublinear-convex systems 34 3.2 New versions of the Hahn-Banach theorem under Slater-type conditions 38 3.2.1 Extended Hahn-Banach-Lagrange theorem 38 iii 3.2.2 3.3 Extension of the Hahn-Banach theorem, the sandwich theorem, and the Mazur-Orlicz theorem 3.2.3 The equivalence between extended versions of the Farkas lemma and the Hahn-Banach-Lagrange theorem Applications to optimization and convex analysis 3.3.1 Generalized optimization problems involving sublinear-convex mappings 3.3.2 A Special case - Penalty problem in convex programming 3.3.3 Generalized Fenchel duality theorem and a separation theorem 3.3.4 A conjugate formula for the supremum of a family of convex functions From Farkas lemma to Hahn-Banach theorem 4.1 Characterizing extended Farkas lemmas for cone-convex systems 4.2 Charactering extended Farkas lemmas for sublinear-convex systems 4.3 Characterizing extended Hahn-Banach theorems 4.4 The equivalence between new versions of the Farkas lemma and HahnBanach-Lagrange theorem 41 44 45 45 53 56 62 66 66 68 72 77 Sequential Farkas lemmas and approximate Hahn-Banach theorems 80 5.1 Sequential Farkas lemma for cone-convex systems 81 5.2 Sequential Farkas lemma for sublinear-convex systems 84 5.3 Approximate Hahn-Banach theorems 87 5.4 Sublinear-convex optimization problems without any constraint qualification conditions 91 5.5 An application: limiting conjugate formula for the supremum of a family of convex functions 96 Conclusion and suggested further research 99 Author’s publications related to the thesis 101 Author’s conferences 102 Bibliography 103 iv Glossary of Notations Spaces and Sets X, Y X ∗, Y ∗ M⊥ cl A int A K+ lin D R |.| ∈ ⊂ ∅ ∩, ∪, × lim inf the locally convex topological spaces the topological dual spaces of X, Y , respectively the orthogonal subspace to M the closure of the subset A the interior of the subset A the dual cone of K the linear hull of the subset D of Y the norm of a vector the set of real numbers the absolute value of a real number the membership of an element in a set the set inclusion the empty set intersection, union, Cartesian product the limit inferior of a net (ai )i∈I lim sup the limit superior of a net (ai )i∈I i∈I i∈I Functions dom f epi f epis f Γ(X) f∗ f ∗∗ iA ∂ f (¯ x) ∂f (¯ x) the the the the the the the the the domain of the function f : X → R ∪ {+∞} epigraph of the function f strict epigraph of the function f set of proper lower semi-continuous convex functions conjugate function of f biconjugate function of f indicator function of the subset A -subdifferential of f at x¯ in dom f subdifferential of f at x¯ in dom f v NC (¯ x) φ ψ y∗, y L(X, Y ) IdR PrY the the the the the the normal cone at x¯ infimal convolution of two function φ, ψ image of y ∗ at y set of all continuous linear operators from X to Y identity function on R function from X × Y into Y defined by PrY (x, y) = y Partial orders ≤K ∞K Y• dom h epiK h h−1 (−K) g◦h ≤S K-convex S-convex the partial order on Y by a cone K in Y the greatest element with respect to ≤K , i.e., y ≤K ∞K for all y ∈ Y the space which we add ∞K to Y the domain of the mapping h : X → Y • , i.e., the set {x ∈ X : h(x) ∈ Y } the K-epigraph of the mapping h : X → Y • , i.e., the set epiK h := {(x, y) ∈ X × Y : y ∈ h(x) + K} the set {x ∈ X : h(x) ∈ −K} the composite function of g : Y → R ∪ {+∞} and h : X → Y • the partial order on Y by the sublinear function S : Y → R ∪ {+∞} convex with respect to a convex cone K convex with respect to a sublinear function S vi Introduction The statement of the original Farkas lemma was proposed by the Hungarian mathematician and physicist Gyula Farkas in 1894, inspired from his work concerning certain equilibrium problems in mechanics However, the first correct proof was published eight years later, in 1902 [35] The lemma states that given any vectors a1 , a2 , · · · , am and c in Rn , the following statements are equivalent: (i) x ∈ Rn , aTi x ≥ 0, i = 1, 2, , m =⇒ cT x ≥ 0; (ii) ∃λi ≥ 0, i = 1, 2, , m, c = m i=1 λ i In the 1950s of the last century, after the time when Gale, Kuhn, and Tucker successfully applied this lemma to establish duality results and optimality conditions for linear programming and nonlinear programming, the Farkas lemma had become one of the well-known tools in optimization and in applied mathematics as well Since then, many efforts have been made to generalize this lemma, and as a result, many new extended versions of the lemma have been discovered and many of their applications have been found, not only in applied mathematics but also in other fields such as finance and economics [33], [36] In the theoretical aspect, it was proved in [38, Corollary 2, p.92] that the convex extended version of the Farkas lemma, namely the so-called Farkas-Minkowski lemma, is equivalent to the Hahn-Banach theorem Moreover, the Farkas lemma is none other than a “mathematical version” of the First Fundamental Principle of financial markets [33] Because of its importance, it was continuously extended in the last decades from linear systems to convex systems, and also nonconvex systems; from finite to infinite dimensional spaces and also, from systems involving single-valued functions to the ones defined by multi-valued functions (see [9], [19], [23], [30], [31], [37], [41], [42], [50], [56], and the recent survey paper [20] for more details) It is worth noting that extended versions of the Farkas lemma for infinite dimensional spaces or for nonlinear systems often hold under some kinds of constraint qualification conditions such as the Slater condition or its generalizations (called interior-type conditions) (see [5], [37], [38], and references therein) In the recent years, the authors vii V Jeyakumar, N Dinh, R Burachik, R I Bot, G Wanka introduced some kinds of qualification conditions called “closedness conditions” which are much weaker than the interior-type ones ( see [7], [9], [10], [11], [14], [15], [29], [31], and references therein) Better still, in some of the works published by these authors, it was shown that the latter type conditions characterize extended Farkas lemma in some concrete settings, i.e., these conditions are necessary and sufficient conditions to ensure the validity of extended Farkas lemmas, not only the sufficient conditions as usual (see, e.g., [42]) All the extended versions of the Farkas lemma are known under the common name “Farkas-type results” and have had many applications to the theory of optimization Specifically, versions of Farkas lemma have been applied to get duality results and optimality conditions for cone-convex problems, DC (difference of convex functions) problems, bilevel problems, variational inequalities, equilibrium problems, best approximation problems, etc (see [27], [28], [29], [30], [31], [42], [46], and references therein) As mentioned above, many new extended versions of the Farkas lemma were proposed in the last decades with successful applications to optimization theory However, there still remain many problems and open questions not yet solved or answered, e.g., whether it is possible that some Farkas lemma version can be established for systems involving composite functions? for systems involving vector functions? These expected versions, if any, may help us a lot in studying classes of problems with composite functions or vector optimization problems On the other hand, there appeared a few works in recent years on some new versions of the Farkas lemma holding without any qualification conditions [21], [44] Any more extended versions of the Farkas lemma holds without any qualification conditions? Last but not least, are there any relation between extended versions of Farkas lemma existing in the literature of fundamental mathematics? Motivated by these observations, we plan to study the following problems in this thesis: Problem Farkas lemma for systems involving composite functions: Studying and establishing some generalized versions of the Farkas lemma for systems involving composite functions with/without convexity and lower semi-continuity from the data Applying the results obtained to convex/nonconvex composite optimization problems Problem Extended Farkas lemmas and fundamental mathematics: As mentioned above, some earlier version of the Farkas lemma is equivalent to the HahnBanach theorem (see [38]) So, a natural question arises: Do there exist extended versions of the Hahn-Banach theorem which are equivalent to recent/new extended versions of the Farkas lemma? viii Problem Sequential Farkas lemmas and approximate Hahn-Banach theorems: Establishing new versions of the Farkas lemma without any qualification condition (called “sequential Farkas lemma”) Combining these versions with the Problem 2, i.e., finding new versions of Hahn-Banach theorem (called “approximate Hahn-Banach theorem”) holding without qualification condition that are equivalent to the mentioned sequential Farkas lemmas During the years of studying to realize this thesis, we found part of the answers for the Problems mentioned above The results obtained will be presented in Chapters 2, 3, 4, and of this thesis Specifically, we get the following results concerning the Problems 1,2, and 3: Farkas-type results for systems involving composite functions (Chapter 2) We consider the functional inequality concerning composite functions of the form: f (x) + g(x) + (k ◦ H)(x) ≥ h(x) ∀x ∈ X, (1) where X, Y are locally convex Hausdorff topological spaces, f, g, h : X → R ∪ {+∞} are proper functions, H : dom H ⊂ X → Y is a mapping, and k : Y → R ∪ {+∞} is a proper function We get the following main results: • We introduce six dual conditions (of closedness-type) that will serve as qualification conditions in establishing versions of Farkas lemma concerning system (1) It is worth mentioning that for these conditions, all the functions, mappings considered are just proper but not necessarily convex nor lower semi-continuous • We establish characterizations of the inequality of the form (1) These characterizations are precisely Farkas-type results involving the inequality of the mentioned form • Generalized Moreau-Rockafellar results involving composite functions are established These results generalize the classical ones in three ways: the functions are not necessarily convex, some assumptions are weakened, and the conditions are not only sufficient but also necessary • Several alternative theorems, characterizations of set containments of a convex set in a DC set or in a reverse convex set, and generalized Fenchel-Rockafellar duality results are derived These results are presented in Chapter Concerning Problem 2, we get the following results, which will be divided into two parts due to the qualification conditions considered and also the level of their generalization These results will be presented, one after another, in Chapter and Chapter ix • For the strong duality, according to the assumption (5.16), we have inf (SP) < +∞ If inf (SP) = −∞ then by the weak duality, inf (SP) = sup (DFSP) = −∞, in this case any (x∗1i , x∗2i , x∗3i , yi∗ , γ i )i∈I ⊂ (X ∗ )3 × Y ∗ × R+ satisfying yi∗ ≤ γ i S on Y for all i ∈ I and (x∗1i + x∗2i + x∗3i , γ i ) −→ (0X ∗ , 1) is a solution of (DFSP), for instance, (0X ∗ , 0X ∗ , 0X ∗ , yi∗ , 1)i∈I such that yi∗ ≤ S on Y for all i ∈ I is a solution (note that such an (yi∗ )i∈I ⊂ E ∗ exists by the approximate Hahn-Banach theorem, Corollary 5.3.1, where X := Y , F := {0Y }, φ := on F , and F ∩ domS = ∅ as S(0Y ) = 0) So, we can assume that inf (SP) ∈ R, which is (i) in Theorem 5.3.1 The mentioned theorem ensures the existence of (x∗1i , x∗2i , x∗3i , yi∗ , γ i )i∈I ⊂ (X ∗ )3 × Y ∗ × R+ satisfying yi∗ ≤ γ i S on Y for all i ∈ I and (x∗1i + x∗2i + x∗3i , γ i ) −→ (0X ∗ , 1) such that lim inf i∈I − f ∗ (x∗1i ) − (yi∗ ◦ g)∗ (x∗2i ) − i∗C (x∗3i ) = inf (SP) ≥ sup (DFSP), which means that max (DFSP) = inf (SP) The proof is complete We now apply Theorem 5.3.1 again, with lim inf i∈I − f ∗ (x∗1i ) − (yi∗ ◦ g)∗ (x∗2i ) − i∗C (x∗3i ) = inf (SP) replaced by inf x∈C f (x)+lim inf i∈I (yi∗ ◦g)(x) = inf (SP), to obtain Theorem 5.4.2 [Approximate strong Lagrange duality for (SP)] Assume that the set (5.5) is closed in the product space X × Y × R and (5.16) holds Then the approximate strong Lagrange duality holds for (SP), i.e., inf {f (x) + (S ◦ g)(x)} = x∈C max inf (yi∗ ,γ i )i∈I ⊂Y ∗ ×R+ x∈C yi∗ ≤γ i S for all i∈I, γ i −→1 f (x) + lim inf (yi∗ ◦ g)(x) i∈I As a consequence of Theorem 5.2.1, we get optimality condition for the problem (SP) Theorem 5.4.3 (Approximate optimality condition for (SP)) Let x¯ ∈ C Assume that the set (5.5) is closed in the product space X × Y × R and (5.16) holds Then the following statements are equivalent: (i) x¯ is a solution of (SP), (ii) there exists a net (x∗1i , x∗2i , x∗3i , yi∗ , δ i , γ i )i∈I ⊂ X ∗ × X ∗ × X ∗ × Y ∗ × R+ × R+ , with yi∗ ≤ γ i S on Y for all i ∈ I such that  ∗  x), x∗2i ∈ ∂δi (yi∗ ◦ g)(¯ x), x∗3i ∈ Nδi (C, x¯),  x1i ∈ ∂δi f (¯ (5.24) γ i (S ◦ g)(¯ x) ≥ (yi∗ ◦ g)(¯ x) ≥ γ i (S ◦ g)(¯ x) − δ i ∀i ∈ I,   ∗ ∗ ∗ (x1i + x2i + x3i , γ i , δ i ) −→ (0X ∗ , 1, 0+ ), 93 (iii) there exists a net (x∗i , yi∗ , ξ i , γ i )i∈I ⊂ X ∗ × Y ∗ × R+ × R+ with yi∗ ≤ γ i S on Y for all i ∈ I such that  ∗ ∗  x),  xi ∈ ∂ξi (f + yi ◦ g + iC )(¯ ∗ (5.25) x) ≥ γ i (S ◦ g)(¯ x) − ξ i ∀i ∈ I, γ i (S ◦ g)(¯ x) ≥ (yi ◦ g)(¯   ∗ (xi , γ i , ξ i ) −→ (0X ∗ , 1, 0+ ) Proof We shall apply Theorem 5.2.1 where ψ(λ) = λ for all λ ∈ R It is clear that ψ is proper convex continuous function and ψ ∗ (γ) = if γ = 1, +∞ else (5.26) It is easy to see that the assumption (5.6) follows from the assumption (5.16) in this circumstance •[(i) ⇒ (ii)] Assume that (i) holds, i.e., x¯ is a solution of (SP) Then one has f (x) + (S ◦ g)(x) ≥ f (¯ x) + (S ◦ g)(¯ x) ∀x ∈ C, or equivalently, f (x) − β + (S ◦ g)(x) ≥ ∀x ∈ C, where β := f (¯ x) + (S ◦ g)(¯ x) Consequently, for any x ∈ C, α ∈ R with (S ◦ g)(x) ≤ α, one has f (x) − β + ψ(α) = f (x) − β + α ≥ f (x) − β + (S ◦ g)(x) ≥ 0, which is (a) in Theorem 5.2.1 (with f −β playing the role of f ) The mentioned theorem now yields the existence of nets (yi∗ , γ i )i∈I ⊂ Y ∗ × R+ and (x∗1i , x∗2i , x∗3i , η i , i )i∈I ⊂ X ∗ × X ∗ × X ∗ × R × R with yi∗ ≤ γ i S on Y for all i ∈ I such that i ≥ (f − β)∗ (x∗1i ) + (yi∗ ◦ g)∗ (x∗2i ) + i∗C (x∗3i ) + ψ ∗ (η i + γ i ) ∀i ∈ I (5.27) and (x∗1i + x∗2i + x∗3i , η i , i ) −→ (0X ∗ , 0, 0) (5.28) Note that (η i + γ i )i∈I ⊂ domψ ∗ follows from (5.27) Hence, by (5.26), one gets ψ ∗ (η i + γ i ) = and η i + γ i = for all i ∈ I As η i −→ we have γ i −→ (5.29) We next apply the Young-Fenchel For this purpose, we rewrite (5.27) as follows (note that (f − β)∗ (x∗1i ) = f ∗ (x∗1i ) + β = f ∗ (x∗1i ) + f (¯ x) + (S ◦ g)(¯ x)) i − x∗1i + x∗2i + x∗3i , x¯ + (γ i − 1)(S ◦ g)(¯ x) ≥ f ∗ (x∗1i ) + f (¯ x) − x∗1i , x¯ + (yi∗ ◦ g)∗ (x∗2i ) + (yi∗ ◦ g)(¯ x) − x∗2i , x¯ + γ i (S ◦ g)(¯ x) − (yi∗ ◦ g)(¯ x) ∀i ∈ I 94 + i∗C (x∗3i ) + iC (¯ x) − x∗3i , x¯ Set δ i := i − x∗1i + x∗2i + x∗3i , x¯ + (γ i − 1)(S ◦ g)(¯ x) for all i ∈ I By (5.28) and the fact that γ i −→ 1, one has δ i −→ Since the four brackets above are nonnegative, each of them is less or equal than δ i for all i ∈ I, i.e.,  ∗ ∗ x) − x∗1i , x¯ ≥ 0,   δ i ≥ f (x1i ) + f (¯   δ ≥ (y ∗ ◦ g)∗ (x∗ ) + (y ∗ ◦ g)(¯ x) − x∗2i , x¯ ≥ 0, i i 2i i (5.30) ∗ ∗ ∗  δ x ) − x , x ¯ ≥ i ≥ iC (x3i ) + iC (¯  3i   x) δ i ≥ γ i (S ◦ g)(¯ x) − (yi∗ ◦ g)(¯ According to (1.4), it follows from (5.30) that x), x∗3i ∈ Nδi (C, x¯) x), x∗2i ∈ ∂δi (yi∗ ◦ g)(¯ x∗1i ∈ ∂δi f (¯ (5.31) Moreover, combining the last inequality in (5.30) with the fact that yi∗ ≤ γ i S on Y for all i ∈ I, we get γ i (S ◦ g)(¯ x) ≥ (yi∗ ◦ g)(¯ x) ≥ γ i (S ◦ g)(¯ x) − δ i ∀i ∈ I (5.32) From (5.28), (5.29), (5.31) and (5.32), we obtain (5.35) •[(ii) ⇒ (iii)] Assume that there exists a net (x∗1i , x∗2i , x∗3i , yi∗ , δ i , γ i )i∈I ⊂ (X ∗ )3 × Y ∗ × R+ × R+ , with yi∗ ≤ γ i S on Y for all i ∈ I such that (5.35) holds By (1.5) and (1.6), (5.35) gives rise to  ∗ ∗ ∗  x) + ∂δi (yi∗ ◦ g)(¯ x) + Nδi (C, x¯) ⊂ ∂3δi (f + yi∗ ◦ g + iC )(¯ x),  x1i + x2i + x3i ∈ ∂δi f (¯ γ i (S ◦ g)(¯ x) ≥ (yi∗ ◦ g)(¯ x) ≥ γ i (S ◦ g)(¯ x) − δ i ≥ γ i (S ◦ g)(¯ x) − 3δ i ∀i ∈ I,   ∗ ∗ ∗ (x1i + x2i + x3i , γ i , δ i ) −→ (0X ∗ , 1, 0+ ) Set x∗i := x∗1i + x∗2i + x∗3i and ξ i := 3δ i for all i ∈ I Then we get (iii) •[(iii) ⇒ (i)] Assume that (iii), i.e., there exists a net (x∗i , yi∗ , ξ i , γ i )i∈I ⊂ X ∗ × Y ∗ × R+ × R+ with yi∗ ≤ γ i S on Y for all i ∈ I such that (5.25) holds By the definition of -subdifferential, it follows from (5.25) that for any i ∈ I,  ∗ ∗ ∗  x) − ξ i ∀x ∈ X,  (f + yi ◦ g + iC )(x) ≥ xi , x − x¯ + (f + yi ◦ g + iC )(¯ ∗ (5.33) γ i (S ◦ g)(¯ x) ≥ (yi ◦ g)(¯ x) ≥ γ i (S ◦ g)(¯ x) − ξ i ,   ∗ (xi , γ i , ξ i ) −→ (0X ∗ , 1, 0+ ) The first inequality and the second inequality of (5.33) imply that for any i ∈ I, f (x) + (yi∗ ◦ g)(x) ≥ x∗i , x − x¯ + f (¯ x) + γ i (S ◦ g)(¯ x) − 2ξ i ∀x ∈ C Combining this inequality and the fact that yi∗ ≤ γ i S on Y for all i ∈ I, one gets f (x) + γ i (S ◦ g)(x) ≥ x∗i , x − x¯ + f (¯ x) + γ i (S ◦ g)(¯ x) − 2ξ i ∀x ∈ C for all i ∈ I Passing to the limit on i for each fixed x ∈ C, we obtain f (x) + (S ◦ g)(x) ≥ f (¯ x) + (S ◦ g)(¯ x) ∀x ∈ C, which is (i) The proof is complete 95 (5.34) Special case We now consider the case where X, Y are locally convex Hausdorff topological vector spaces, C is a nonempty closed convex subset of X, K is a closed convex cone in Y , f ∈ Γ(X), and g : X → Y • is a K-convex and K-epi closed mapping Set S := i−K It is easy to see that S is an lsc sublinear function (as K is closed convex cone) and g is S-convex Then the following cone-convex problem, namely (CP), is a special case of (SP) with S, f , g, K and C defined as above: (CP) inf f (x) x∈C, g(x)∈−K Moreover, since g is K-epi closed, it follows that the set (5.5) is closed in the product space X × Y × R Then Corollary 5.4.1 is a consequence of Theorem 5.4.2, and Corollary 5.4.2 is a consequence of Theorem 5.4.3 Corollary 5.4.1 [Approximate strong Lagrange duality for (CP)] Assume that (dom f ) ∩ C ∩ g −1 (−K) = ∅ holds Then the approximate strong Lagrange duality holds for (CP), i.e., inf x∈C∩g −1 (−K) f (x) = max inf (yi∗ )i∈I ⊂K + x∈C f (x) + lim inf (yi∗ ◦ g)(x) i∈I Corollary 5.4.2 [Approximate optimality for (CP)] Assume that (dom f ) ∩ C ∩ g −1 (−K) = ∅ holds Then the following statements are equivalent: (i) x¯ is a solution of (CP), (ii) there exists a net (x∗1i , x∗2i , x∗3i , yi∗ , δ i )i∈I ⊂ X ∗ × X ∗ × X ∗ × K + × R+ , such that  ∗  x), x∗2i ∈ ∂δi (yi∗ ◦ g)(¯ x), x∗3i ∈ Nδi (C, x¯),  x1i ∈ ∂δi f (¯ (5.35) ≤ (yi∗ ◦ g)(¯ x) ≤ δ i ∀i ∈ I,   ∗ ∗ ∗ (x1i + x2i + x3i , δ i ) −→ (0X ∗ , 0+ ), It is worth noting that Corollary 5.4.1 was given in [23, Corollary 4] and Corollary 5.4.2 was given in [19, Proposition 5] under the assumption that y ∗ ◦ g ∈ Γ(X) for all y ∗ ∈ K + while g is K-epi closed and K-convex here Note that the assumption that y ∗ ◦ g is lsc for all y ∗ ∈ K + is stronger than the one that g is K-epi closed (see [5, p 24]) 5.5 An application: limiting conjugate formula for the supremum of a family of convex functions This section provides a limit conjugate formula for the supremum of a (possibly infinite) family of convex functions 96 Let X be a locally convex Hausdorff topological vector space, T be an arbitrary (possibly infinite) index set, and g t : X → R ∪ {+∞} be proper lsc convex function for all t ∈ T such that dom supt∈T g t = ∅ (this implies that t∈T dom g t = ∅) We consider the product space RT endowed with the product topology and use R(T ) to denote the space of real tuples λ = (λt )t∈T with only finitely many λt = Note that (T ) R(T ) is the topological dual of RT ([18], [34], [55]) We represent by R+ the positive cone in R(T ) , that is (T ) R+ = {(λt )t∈T ∈ R(T ) : λt ≥ for all t ∈ T } (T ) Note that R+ is also the dual cone of the cone RT+ in the product space RT The supporting set of λ ∈ R(T ) is suppλ := {t ∈ T : λt = 0}, and λt ut := λ(u) = t∈T λt ut ∀u = (ut )t∈T ∈ RT , ∀λ = (λt )t∈T ∈ R(T ) t∈suppλ As a consequence of Theorem 5.4.1, we get the following result which is approximate version of Proposition 3.3.1 Corollary 5.5.1 For any x∗ ∈ X ∗ , one has sup g t t∈T ∗ (x∗ ) = lim sup (T ) (x∗i ,(λti )t∈T )i∈I ⊂X ∗ ×R+ (x∗i , λti )−→(x∗ ,1) t∈T i∈I λti g t ∗ (x∗i ) t∈T Proof Let Y := RT , g : X → Y • := RT ∪ {∞RT+ } be the mapping defined by   (gt (x))t∈T if x ∈ domgt , t∈T g(x) =  ∞ T otherwise R+ Define S : RT → R ∪ {+∞} by S(y) := sup yt for all y = (yt )t∈T ∈ RT As shown in t∈T Proposition 3.3.1, S is an lsc extended sublinear function and g is S-convex Now we check that the set (5.5) in Theorem 5.4.1 is closed in the product space X × Y × R Indeed, one has {(x, y, α) ∈ X × Y × R : S(g(x) − y) ≤ α} = (x, (y t )t∈T , α) ∈ X × RT × R : sup{g t (x) − y t } ≤ α t∈T = (x, (y t )t∈T , α) ∈ X × RT × R : sup{g t (x, y)} ≤ α , t∈T where for each t ∈ T, the function g t : X × RT → R ∪ {+∞} is defined by g t (x, y) = g t (x) − y t ∀x ∈ X, ∀y = (y t )t∈T ∈ RT 97 (5.36) It is obvious that g t ∈ Γ(X × RT ) for all t ∈ T as g t ∈ Γ(X) for all t ∈ T, and hence, supt∈T g t ∈ Γ(X × RT ) Thus the set (5.36) is closed in the product space X × RT × R Moreover, since dom supt∈T g t = ∅, it follows that (5.16) in Theorem 5.4.1 holds We now apply Theorem 5.4.1 to the function S, the mapping g defined as above, C = X, and the function f : X → R with f (x) := − x∗ , x for all x ∈ X Then the mentioned theorem ensures the following equality holds: max (x∗1i ,x∗2i ,x∗3i ,λi ,γ i )i∈I ⊂(X ∗ )3 ×R(T ) ×R+ λi ≤γ i S on RT for all i∈I, (x∗1i +x∗2i +x∗3i ,γ i )−→(0X ∗ ,1) = inf x∈X lim inf i∈I − f ∗ (x∗1i ) − (λi ◦ g)∗ (x∗2i ) − i∗X (x∗3i ) − x∗ , x + sup g t (x) f (x) + (S ◦ g)(x) = inf x∈X (5.37) t∈T Observe that for any x∗1i , x∗2i , x∗3i ∈ X ∗ and λi = (λti )t∈T ∈ R(T ) , one has if x∗3i = 0X ∗ , +∞ otherwise, i∗X (x∗3i ) = sup { x∗3i , x } = x∈X f ∗ (x∗1i ) = sup { x∈X x∗1i , x ∗ + x ,x } = and (λi ◦ g)∗ (x∗2i ) = λti g t if x∗1i = −x∗ , +∞ otherwise, ∗ (x∗2i ) t∈T Fix i ∈ I and λi = t (λti )t∈T (T ) ∈ R such that λi (y) ≤ γ i S(y) = γ i sup y t for all T t t∈T t k y = (y )t∈T ∈ R Then for any k ∈ T , u¯ = (u )t∈T with u = −1 and u = for all t = k, one gets   −λki = λi (¯ u) ≤ γ i S(¯ u) =    t λi = λi (1)t∈T ≤ γ i S (1)t∈T = γ i t∈T    λti = λ (−1)t∈T ≤ γ i S (−1)t∈T = −γ i  − t∈T Hence, ≤ λki and λti = γ i −→ t∈T On the other hand, by the definition of the conjugate function, it is easy to see that − sup g t ∗ (x∗ ) = inf − x∗ , x + sup g t (x) x∈X t∈T t∈T By what were shown above, (5.37) can be rewritten as follows max (T ) (x∗2i ,(λti )t∈T )i∈I ⊂X ∗ ×R+ t ∗ ∗ (−x +x2i , t∈T λi )−→(0X ∗ ,1) lim inf i∈I λti g t − t∈T ∗ (x∗2i ) = − sup g t t∈T The conclusion of the corollary follows from this The proof is complete 98 ∗ (x∗ ) Conclusion and suggested further research In this thesis we obtain the following results concerning Problems 1,2, and 3: Farkas-type results for systems involving composite functions • We introduce six dual conditions (of closedness-type) that will serve as qualification conditions in establishing versions of Farkas lemma concerning system (1) but not necessarily convex nor lower semi-continuous • We establish characterizations of the inequality of the form (1) These characterizations are precisely Farkas-type results involving the inequality of the mentioned form • Generalized Moreau-Rockafellar results involving composite functions are established These results generalize the classical ones in three ways: the functions are not necessarily convex, some assumptions are weakened, and the conditions are not only sufficient but also necessary • Several alternative theorems and characterizations of set containments, and of generalized Fenchel-Rockafellar duality results are derived New versions of Farkas lemma and Hahn-Banach theorem under Slatertype conditions • New versions of the Farkas lemma for cone-convex systems and for sublinearconvex systems are established under Slater-type conditions and in the absence of the lower semi-continuity and the closedness of functions and constrained sets involved • An extended version of Hahn-Banach-Lagrange theorem (that generalizes the one in [62]) is established and it is shown to be equivalent to the extended Farkas lemma for cone-convex systems and for sublinear-convex systems just obtained • Our results lead to extensions of other fundamental theorems such as the sandwich theorem, the Mazur-Orlicz theorem, and also the Hahn-Banach theorem itself for the case involving extended sublinear functions (the situation where the celebrated Hahn-Banach theorem failed) • The results obtained are then applied to get duality results and optimality 99 conditions for a class of composite problems involving sublinear-convex mappings • As illustrative examples, we consider the penalty problems associated with convex programming problems, a formula for the conjugate of the supremum of a family (possibly infinite, not lsc) of convex functions, and a special class of problems inspired in the Fenchel duality theorem and a separation theorem for convex sets in normed spaces From Farkas lemma to Hahn-Banach theorem • We establish that some new closedness conditions characterize new versions of the Farkas lemma for systems defined by cone-convex mapping and the ones defined by sublinear-convex mappings These results inspired us to many versions of stable Farkas lemmas which were given in [22] • We show that the extended Farkas lemmas just obtained give rise to the characterization of an analytic version of the Hahn-Banach theorem, the analytic HahnBanach-Lagrange theorem, the analytic Mazur-Orlicz theorem for sublinear functions which may take the value +∞ (the case where the classical Hahn-Banach theorem fails) • We show that the mentioned versions of the Farkas lemma and the analytic Hahn-Banach-Lagrange theorem established above are actually equivalent to each other Sequential Farkas lemmas and approximate Hahn-Banach theorems • New versions of sequential Farkas lemma for cone-convex systems and sublinearconvex systems are established (without any constraint qualification condition) • Several versions of approximate Hahn-Banach-Lagrange theorem are established • It is shown that sequential Farkas lemmas and approximate Hahn-BanachLagrange theorems are equivalent to each other Versions of approximate Hahn-Banach theorems and approximate sandwich theorems for extended sublinear functions are derived from the results just obtained • The results are then applied to obtain approximate duality results and optimality conditions for optimization problem involving sublinear-convex mappings We provide an approximate conjugate formula for the supremum of a family of convex functions There still remain many open questions and problems to solve In the future, if provided favourable conditions, we will study the Farkas-type results for systems involving vector mappings and set-valued mappings, etc 100 Author’s publications related to the thesis N Dinh and T.H Mo, Qualification condition and Farkas-type results for systems involving composite functions, Vietnam J Math 40(4) (2012), 407–437 N Dinh, M.A Goberna, M.A L´opez, T.H Mo, From the Farkas lemma to the Hahn-Banach theorem, SIAM J Optim 24(2) (2014), 678–701 N.Dinh and T.H Mo, Farkas lemma for convex systems revisited and applications to sublinear-convex optimization problems, Vietnam J Math., DOI 10.1007/s10013014-0118-7 N Dinh and T.H Mo, Generalizations of the Hahn-Banach theorem revisited, Taiwanese J Math (to appear) 101 Author’s conferences The 9th Workshop on “Optimization and Scientific Computing”, April 2011, Ba Vi, Vietnam The 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epi(f +g+k◦H)∗ , A = D = E = epi(f +g+k◦H)∗ , and A = B = F = epi(f +g+k◦H)∗ This means that if (CA) holds then (CB), (CC), (CD), (CE), and (CF) do, too Also, by the same argument... B ⊂ F, and E ⊂ epi(f + g + k ◦ H)∗ , F ⊂ epi(f + g + k ◦ H)∗ In particular, if A = epi(f + g + k ◦ H)∗ then A = B = C = D = E = F Proof The proof is easy, using mainly (1.7) We first observe that epi g ∗ + epi g ∗ + epi(λH − k ∗ (λ))∗ epi(λH − k ∗ (λ))∗ = λ∈dom k∗ λ∈dom k∗ and, by (1.7) with φ = g and ψ = λH − k ∗ (λ), λ ∈ dom k ∗ , we have A ⊂ B Applying (1.7) one more time for φ = f and ψ = g + λH... conditions under which these Farkas- type results remain true One result of this type was found in the literature for the first time in [42] for a special case where g ≡ 0, k = i−K (K is a closed convex cone), H is a K-convex, continuous mapping, and f is a continuous and convex function 2.3.1 Nonconvex Farkas- type results As in the previous section, the functions f, g, k, and λH (for arbitrary λ ∈ Y ∗ )... of Farkas- type results for the inequality system in consideration Although this theorem gives only sufficient conditions for these types of results, it covers and extends several Farkas- type results in the literature (see e.g., [8], [21], [30], [31], [41], [37]) We now consider in more details each of Farkas- type results mentioned in Theorem 2.3.1 We will give in Theorem 2.3.2 variant necessary and. .. order to illustrate the significance of the general Farkas- type results established in the previous subsection, in this subsection, we will point out some special cases Here, we get new versions, extensions or find again Farkas- type results for convex, nonconvex systems, or for systems involving DC functions Necessary and sufficient conditions for strong stable Lagrange duality of a general convex optimization. .. the key results of this section: variant characterizations (necessary and sufficient conditions) of (1) in pure algebraic setting (i.e., without any convexity or topological assumptions) Results of this type are often known as Farkas- type results We will show that the dual conditions introduced in Section 2.1 are both necessary and sufficient for the validity of these Farkas- type results The results. .. theorem (that generalizes the one in [62]) is established and it is shown to be equivalent to the extended Farkas lemma for cone-convex systems and for sublinear-convex systems just obtained • Our results lead to extensions of other fundamental theorems such as the sandwich theorem, the Mazur-Orlicz theorem, and also the Hahn-Banach theorem itself for the case involving extended sublinear functions (the... necessary and sufficient conditions for) Farkas- type results We get the following results: • We establish that some new closedness conditions characterize new versions of the Farkas lemma for systems defined by convex cones and the ones defined by sublinear-convex mappings These results inspired us to obtain many versions of stable Farkas lemmas which were given in [22] • We show that the extended Farkas. .. y1 ∈ K We add to Y a greatest element with respect to ≤K , denoted by ∞K which does not belong to Y and let Y • = Y ∪ {∞K } Then one has y ≤K ∞K for every y ∈ Y • 3 Consider the following operations on Y • : y + ∞K = ∞K + y = ∞K for all y ∈ Y • , and α∞K = ∞K if α ≥ 0 A function h : X → Y • , we call domain of h the set dom h = {x ∈ X : h(x) ∈ Y }, and we say that h is proper if dom h = ∅ The K-epigraph