Study On Fenchel Dualities And Lagrange Dualities For Optimization Problems

In this paper, we consider the Fenchel duality of the following optimization problem and the Lagrange duality of the infinite optimization problem Minimize h(x), where f,h,ht,t∈T and g are proper functions defined on locally convex Hausdorff topo-logical vector spaces X and Y, respectively, and A is a linear operator from X to Y, T is an arbitrary (possibly infinite) index set, C is a nonempty convex subset of X. Under some new constraint qualifications, we give the sufficient and necessary conditions for the weak dualities, the strong dualities and total dualities to hold. The main works done in this dissertation are organized as follows.In the first part, we study the Fenchel duality for the optimization problem (PA).First, we study the case when f and g are convex functions (not necessarily lower semicontinuous). By using the properties of the epigraph of the conjugated functions, some sufficient and necessary conditions for the strong Fenchel duality and the strong converse Fenchel duality of (PA) are provided. Sufficient and necessary conditions for the stable Fenchel duality and for the total Fenchel duality are also derived. Further, we study the optimization problem (PA) but with f and g being two DC (difference of two convex functions) functions. Adopting different tactics, two types of the Fenchel dual problems of (PA) are given. By using the epigraph technique, we provide some new regularity conditions, which characterize the weak dualities, the strong dualities, the stable strong dualities as well as the total dualities between the prime problem and the two dual problems. Most of results seem new and are proper extensions of the results in the convex case. In the second part, we study the Lagrange duality for the infinite optimization problem (Ph).For an inequality system defined by a possibly infinite family of proper convex func-tions (not necessarily lower semicontinuous); we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we obtain characterizations of those reverse-convex inequal-ities which are consequence of the constrained system, and we provide necessary and/or sufficient conditions for a stable Farkas lemma to hold. Similarly, we provide characteriza-tions for constrained minimization problems to have the strong/stable strong/total/stable total Lagrangian dualities. Moreover, we provide necessary and/or sufficient conditions for KKT rule to hold. Several known results in the conic programming problem are extended and improved. Further, we consider the optimization problem with DC objection function and infinite many DC inequality constraints. Adapting different tactics, two types of La-grange dual problems are defined. By using the properties of the epigraph of the conjugate functions, some characterizations for DC programming to have the weak dualities, the strong dualities and the total dualities are provided and some necessary and/or sufficient conditions for a stable Farkas lemma to hold are also obtained.