Đề tài Some qualitative problems in optimization

1Introduction The problem of determining a point in a given set to be an extremal point of an op- timization problem is a topic which has attracted many mathematicians for a long time. It might say that the Fermat theorem (1638) was the first necessary condition for obtaining extremal points of a class of unconstrained optimization problems. When searching a method for solving a mechanics problem in the form of constrained opti- mization problems, Lagrange proposed his method (1788) which is known as Lagrange Theory nowadays. This theory relates to many questions of optimality conditions, duality, and saddle-points of optimization problems. Since 1960s, it has been realized that convex optimization have many applications in areas such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, and finance. The results on convex opti- mization have also found wide application in combinatorial optimization and global optimization, where they are used to find bounds on the optimal value as well as approximate solutions (see [9]). The Lagrange multiplier condition of Kuhn-Tucker type is an essential optimality condition for optimization problems. The condition would be established for the prob- lem provided that some constraint qualification (CQ) such as Slater’s condition, Robin- son’s condition, or Mangasarian-Fromovitz’s condition is satisfied. However, there exist convex optimization problems which achieve their optimal value but the (CQ) is not satisfied. In the last decades, there were several results concerning Lagrange multi- plier condition of convex programming problems in which the constraint qualification conditions were weaken or absent (see [3], [4], [40], [49]). 2The search for Lagrange multiplier conditions of Kuhn-Tucker type of a class of convex problems in which the (CQ) is not satisfied has attracted many mathemati- cians. In [85], Thibault presented a concept of sequential Lagrange multipliers for establishing the optimality conditions of convex problems without constraint qualifi- cation condition. Afterwards, Jeyakumar, Lee, and Dinh developed this concept to establish new sequential Lagrange multiplier conditions for characterizing optimality condition without constraint qualification of the convex problem [43] in the form of Minimize{f(x) | g(x) ∈ −S} where X is a reflexive Banach space, Z is a Banach space, S is a closed convex cone in Z, f : X → R is a continuous convex function, and g : X → Z is a continuous and S-convex mapping. In that paper, they also proved that the Lagrange multiplier condition for the problem above is still valid under a general constraint qualification condition known as closed cone constraint qualification (CCCQ) stating that the cone⋃ λ∈S+ epi(λg)∗ is weak∗-closed. Recently, a development of (CCCQ) for the class of problems above, but with an additional closed convex set constraint, was presented:⋃ λ∈S+ epi(λg)∗ + epiδ∗C is weak ∗-closed. In these cases, (CCCQ) is known as the constraint qualification which is weaker than some generalized Slater’s constraint qualifications (for details see [42], [43]). Dinh, Goberna and Lopez have considered a class of optimization problems [20] of the model (P) Minimize f(x) subject to ft(x) ≤ 0, t ∈ T, x ∈ C, where X is a locally convex Hausdorff topological vector space, T is an arbitrary (possibly infinite) index set, C is a closed convex subset ofX, f, ft : X → R∪{+∞}, t ∈ 3T are proper, convex and lower semi-continuous (l.s.c) functions. For Problem (P), the generalized constraint qualification is the following cone{ ⋃ t∈T epif∗t }+ epiδ∗C is weak∗-closed (0.1) and it is called the Farkas-Minkowski (FM) condition. Such problems are called convex infinite problems. We note that an optimization problem is called infinite problem if it has an infinite number of constraints and the working space is an infinite dimensional space. If exactly one of these assumptions is finite then it is called semi-infinite prob- lem. In [20], the authors have emphasized that the (FM) condition is strictly weaker than several known interior-type constraint qualifications. Very recently, it is proved in [25] that (FM) condition is strict weaker than the Slater constraint qualification condition. It has been developed effectively to several classes of optimization problems such as: convex semi-infinite vector optimization problems, DC programs with convex constraints, DC and bilevel infinite and semi-infinite programs (see [22], [23], [24], [25], and [54]). In this thesis we study the Lagrange Theory and some related questions for a class of convex infinite problems in locally convex Hausdorff topological vector spaces. The first part of the thesis is devoted to Lagrange multiplier conditions and related results for Problem (P). We note that one type of Karush-Kuhn Tucker optimality condition of the problem was established recently in [20]. In this thesis we give optimality conditions for (P), which improve the results given in [20]. Concretely, we combine the (FM) condition with a closedness condition (CC) to deduce the necessary and sufficient optimality conditions of the problem under a representation of the epigraph of conjugate functions of convex functions [40]. Moreover, a new version of Farkas’ lemma is established, and then, Lagrange duality and stability for Problem (P) are also studied. In the second part we establish characterizations of solution sets for the model problem above. We note that the first results concerning characterizations of solution sets of convex programs were given by Mangasarian in [65] (1988), and later by Burke and Ferris in [12] (1991). Recently, there were many results concerning solution sets 4of optimization problems (see [19], [45], [46]). Here, we deal with convex infinite prob- lems. Several Lagrange multiplier characterizations of the solution set of Problem (P) are given. Characterizations of solution sets of cone-constrained convex problems are derived as well. Moreover, the procedure is adapted to a class of semi-convex problems with an infinite number of convex constraints and then we obtain characterizations of solution sets of these problems. At the end of this part we also introduce some results concerning characterization of solution sets of a different class of problems. We study characterization of solution sets of linear fractional problems via a duality scheme. The third and last part of the thesis is devoted to the approximate optimality results developed from the first part. These are introduced in two chapters. One is for convex infinite problems and another is for nonconvex infinite problems. For convex infinite problems of the model (P), we first establish approximate optimality conditions. Then, the approximate Lagrange duality and approximate saddle points of (P) are studied. These results were given, based on the assumptions that the Farkas-Minkowski (FM) condition is satisfied and that the closedness condition (CC) holds. From the problem (P), we consider a nonconvex problem by substituting the objective function of (P) with a locally Lipschitz functional, and the setting of the problem is now a Banach space. Several ε-optimality conditions are presented. We introduce the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker (KKT) conditions. These conditions are up to ε and obtained by weakening the classical complementarity conditions. Thanks to the Ekeland Variational Principle, we show that these KKT conditions are necessary for at least an almost regular ε-solution without any constraint qualification, and then that they are also sufficient for ε-optimality when the objective functional is ε-semiconvex. We also define quasi saddle-points associated with an ε- Lagrangian functional, and we investigate their relationship with generalized KKT conditions. This thesis is organized in the following manner. Chapter 1 is devoted to the preliminaries, a discussion on (FM) condition, and a short overview concerning convex infinite problems published recently. Chapter 2 5includes our main results on the Lagrange theory for convex infinite programming. In Chapter 3, we establish characterizations of solution sets of (P). Applications to cone-constrained convex programs are also presented. Moreover in this chapter, we deal in addition with characterization of solution sets of linear fractional problems. In Chapter 4, we present approximate optimality conditions and approximate duality as well as approximate saddle points condition for (P). In Chapter 5, we are concerned with ε-optimality conditions and ε-Lagrangian duality for a class of nonconvex infinite problems. Finally, we summarize the conclusions of the work and discuss briefly possible future research. Parts of this thesis were published in [21], [77], [26], [76], and submitted for publi- cation in [78]. They were reported at the following scientific meetings: – Workshops ”Optimization and Scientific Computing”, Hanoi 2005, Ba Vi (Hatay) 2006 and 2007. – Seminar Optimization I (Head: Prof. Phan Quoc Khanh), Department of Opti- mization and System Theory, Faculty of Mathematics and Informatics, University of Natural Sciences, VNU-HCM. – Seminar Optimization II (Head: Associate Prof. Nguyen Dinh), Department of Mathematics, International University, VNU-HCM. – Seminar of Department of Mathematics (Profs. V.H. Nguyen and J.J. Strodiot), University of Namur (FUNDP), Belgium. –The 6th Vietnam-Korea Workshop on Theory of Optimization and Applications, Nhatrang, 2008.