The book provides a deep insight into duality theory for scalar and vector optimization in the framework of convex analysis. It is based, essentially, on the habilitation thesis of the author [Recent advances in vector optimization and set-value analysis via convex duality. Faculty of Mathematics, Chemnitz University of Technology (2014)], where several of his recent contributions to the fields of vector optimization and monotone operators are collected.
The book is structured into seven chapters.
The first chapter is devoted to some preliminary notions and results needed in the sequel.
Chapter 2 is dedicated to scalar optimization problems. The duality properties are investigated by embedding the scalar optimization problem into a family of general perturbed scalar optimization problems to which scalar corresponding conjugate dual problems are assigned. The notion of \(\epsilon\)-duality gap between the primal and the dual problem, i.e., when the difference between the optimal objective values of the two problems is less than \(\epsilon\), is considered, and \(\epsilon\)-duality gap statements are investigated and proved in terms of two different approaches: via suitable epigraph inclusions, and via subdifferential inclusions. Then, in both cases, the general scalar optimization problem is particularized to be constrained, and unconstrained.
In order to deal with optimization problems with a vector-valued objective function, in Chapter 3 various minimality concepts for sets in a Hausdorff locally convex space partially ordered by a (possibly not pointed) convex cone \(K\) are introduced, and the different minimality sets are compared. The case of a pointed cone is then considered, and weak conditions entailing characterizations of different notions of proper minimality via linear scalarization are provided. The notions of weakly minimal and relatively minimal elements are subsequently introduced in the case where \(K\) has a nonempty quasi interior. It is shown that some properties fulfilled by weakly minimal points with respect to a cone with nonempty interior are inherited. A characterization of relatively minimal elements via linear scalarization is proved.
In Chapter 4, vector duality via a general scalarization for general vector optimization problems is considered. The general scheme introduced in the previous chapter for defining several minimality notions for sets is used in order to introduce different types of properly efficient solutions with respect to which vector dual problems are assigned to the original vector optimization problems. Some of the classical scalarization functions are studied in details, and the corresponding vector duals and vector duality results are stated. Specific duality schemes are then obtained by specializing the initial problem to be constrained, or unconstrained.
Chapter 5 is devoted to a detailed analysis of the Wolfe and Mond-Weir type duality schemes for scalar and vector convex nonsmooth optimization problems. In the literature these two duality concepts were considered, in general, for constrained optimization problems involving differentiable convex functions. In the scalar case, a duality approach via perturbations is presented, and it leads, in the case of a primal constrained differentiable problem, to the classical Wolfe and Mond-Weir duals. Several Wolfe and Mond-Weir type duals can be built for both the constrained and unconstrained cases, by specializing the perturbation functions; generalized convex functions are allowed, too. Subsequently, this approach is extended to vector optimization problems. Two different ways of assigning vector dual problems of Wolfe and Mond-Weir type to vector optimization problems are considered. The first one follows a classical approach, bringing, under suitable regularity assumptions, to some of the Wolfe and Mond-Weir type vector dual problems, while the second one follows an alternative approach by making use of the idea below the construction of vector dual problems arising in convex vector optimization. It turns out that, in the latter case, the image sets of these vector duals are larger than those in the classical ones, and this proves to be an advantage when solving them numerically. The image sets of several different vector dual problems assigned to the same primal problem are compared.
Chapter 6 focuses on two particular and important vector optimization problems: the linear vector optimization problem and the semidefinite vector optimization problem. First, the vector duality for the linear vector optimization problem is revised in the framework of finite-dimensional spaces; weak, strong and converse duality results are proved for a new vector dual with respect to efficient solutions, with the image space of the primal problem partially ordered by an arbitrary pointed closed convex cone. This vector dual is subsequently extended to the framework of infinite-dimensional vector spaces, where most of the properties are maintained. Vector duality statements in finite- and infinite-dimensional settings are delivered also with respect to weakly efficient solutions, and weak, strong and converse duality results are proved as well. Finally, vector duality for the vector minimization of a matrix function with respect to the cone of the symmetric positive semidefinite matrices subject to both geometric and semidefinite inequality constraints is considered.
The last chapter is devoted to some recent results on monotone operators in Banach spaces. Some Brézis-Haraux type results are presented for the range of the sum of a monotone operator with another one composed with a linear mapping, in both a general, and a reflexive Banach space; some earlier results are corrected, and some generalizations are stated. The special case of the subdifferential of some proper, convex and lower semicontinuos functions is then considered. Two concrete applications to optimization problems and complementarity problems are presented. In the setting of a reflexive Banach space, the surjectivity of the sum of a maximally monotone operator with the translation of another one is investigated and characterized by means of closedness type regularity conditions involving representative functions. Regularity conditions for different results dealing with ranges of sums of both general and particular maximally monotone operators are derived, by weakening results known in the literature. Finally, the last section is devoted to maximal monotonicity of bifunctions investigated via representative functions. This approach leads to the extension of some statements from the literature, and to the proof of some recent conjecture.