New theoretical and algorithmic challenges in vector optimization
Final Report Abstract
We have shown that under some weak additional hypotheses the generalized weakly minimal elements of a set with respect to a convex cone with possibly empty interior and with nonempty quasi or quasi-relative interior can be characterized by means of linear continuous functionals. Moreover, we have extended the vector duality approach usually considered in the case when the ordering cone of the image space has a nonempty interior to the situation when only the quasi or quasi-relative interior of the cone is nonempty. We have also delivered characterizations via the linear scalarization of minimal and different types of properly minimal elements of a set with respect to a convex cone. The properties of the gauge functions associated to convex cones (that have an important role in constructing iterative methods for solving some classes of vector optimization problems) with possibly empty interiors were investigated as well, providing characterizations for them when the cones have nonempty quasi or quasi-relative interiors. These results may have potential applications in mathematical finance, where cones with empty interiors occur quite often in various situations. We have also delivered some new insights on vector duality for linear vector optimization problems, semidefinite multiobjective optimization problems and general constrained convex vector optimization problems where the image space is partially ordered by an arbitrary convex cone. Moreover, we have compared different proper minimality concepts with respect to a convex cone that was not necessarily pointed. Optimality conditions for general nonsmooth vector optimization problems by means of general scalarization functions formulated via a generic subdifferential were also delivered. The role of the closedness type regularity conditions in convex optimization was investigated and we have provided different other insights into duality theory for different classes of optimization problems, for instance for the ones having multi-composed objective functions, and new formulae for biconjugates of infimal and multi-composed functions. An inertial proximal point method for solving unconstrained vector optimization problems with simple objective functions whose image space is partially ordered by a convex cone with possibly empty interior and nonempty quasi interior and an inertial forward-backward iterative method for solving unconstrained vector optimization problems with objective functions consisting of the sum of a smooth and a nonsmooth cone-convex vector function were also delivered. They should find applications in fields like mathematical finance, image processing and location theory. In order to prepare the future applications of our results in these fields, we have provided new subdifferential formulae for convex deviation measures and risk functions as well as duality statements for linear single minimax location problems with geometric constraints, nonlinear single minimax location problems with geometric constraints and set-up costs, multifacility location problems with and without set-up costs and constrained multifacility minimax location problems with gauges of closed convex sets, respectively. These theoretical results on location problems are accompanied by some splitting algorithms for solving single as well as multifacility minimax location problems defined by the Euclidean norm or by polyhedral gauges. As pleasant surprises we can mention the fact that we have succeeded in proving that several results in vector optimization known so far only when the ordering cone had nonempty interior can be successfully extended to situations where the ordering cone has only some nonempty generalized interior. In this matter we have successfully completed all the planned investigations, a fact which was not so foreseeable because of the diffculties encountered when working with the generalized interior notions, which often lack reasonable calculus properties. Regarding the investigations in scalar optimization, perhaps the biggest surprise was the alternative formula for the biconjugate of a composed function we obtained while we were convinced that in that special case of the main statement on biconjugates of infimal functions one could only rediscover an older formula from the literature. The surprises regarding our work in the framework of Objective 5 concern some beautiful apparently simple but hard to get ideas that enabled us to surpass the diffculties encountered in adapting the iterative methods considered so far for solving scalar optimization problems in order to deliver efficient solutions to vector optimization problems and in providing closed formulae for projections on epigraphs of complex combinations of functions involving norms and gauges.
Publications
- Vector duality for convex vector optimization problems by means of the quasi interior of the ordering cone, Optimization 63:21–37 (2014)
S.-M. Grad, E.L. Pop
(See online at https://doi.org/10.1080/02331934.2013.775283) - On biconjugates of infimal functions, Optimization 64:1759–1775 (2015)
S.-M. Grad, G. Wanka
(See online at https://doi.org/10.1080/02331934.2015.1046873) - Vector Optimization and Monotone Operators via Convex Duality, Springer-Verlag, Cham (2015)
S.-M. Grad
(See online at https://doi.org/10.1007/978-3-319-08900-3) - On gauge functions for convex cones with possibly empty interiors, Journal of Convex Analysis 24:519–524 (2017)
S.-M. Grad
