CCOR Three-Part Mini Course and Seminar Talk

Dates of the three-part mini course: 

In this three-part minicourse we discuss some recent developments in Vector Optimization. 

In short, during the first course, we talk about a vector/multiobjective optimization problem when several (usually conflicting) objective functions need to be simultaneously optimized (in a certain sense). Various applications from fields like Economics, Finance, Medicine or Logistics can be modeled as vector optimization problems. The classical example of a vector optimization problem relies on Markowitz’s portfolio optimization model where the risk needs to be minimized while at the same time the expected return should be maximized. 

The first meeting will be dedicated to introducing/recalling some basic facts in Multiobjective/Vector Optimization, such as solution notions, useful tools, fundamental results, scalarization approaches, examples of problems. 

During the second session the focus will be switched towards duality and optimality. Various duality approaches in the linear and convex cases (that extend in various ways their counterparts from scalar optimization or formulated by involving set-valued mappings) and optimality conditions formulated by means of generalized subdifferentials in the nonsmooth nonconvex case will be discussed. 

In the third part, emphasis will be placed on recent contributions on algorithmic methods for solving vector optimization problems. Extensions of Newton’s algorithm and of gradient descent methods for solving multiobjective optimization problems will be discussed. In particular we will talk about the existing generalizations of proximal point methods towards Vector Optimization and about approaches towards solving vector optimization problems by means of dynamical systems. (The contributions of the speaker to the algorithmic side of Vector Optimization will be presented with more details in the subsequent CCOR research seminar talk.) 

Various open questions will be highlighted during the presentation. 

Date of the seminar talk: 

The title and abstract of the seminar talk: 

Proximal point type algorithms for solving vector optimization problems 

We present iterative proximal point type algorithms for determining weakly efficient solutions to unconstrained convex vector optimization problems. After discussing the vector-minimization of a single cone-convex vector function by means of an inertial proximal point algorithm, we turn to the case when the objective function is the sum of a differentiable vector function with a nonsmooth cone-convex one. An inertial forward-backward method with memory effects based on recent advances in solving scalar convex optimization problems and monotone inclusions is proposed, by making use of some adaptive linear scalarization techniques. During the talk, the difficulties encountered while formulating the algorithm and proving its convergence will be stressed, while the related discussion of extending the celebrated FISTA method from scalar to vector optimization problems will be mentioned, too. Several (still unsolved) challenges in extending proximal point type methods from scalar to vector optimization problems will be addressed as well. 

This talk is based on joint work with Radu Ioan Bot. 

You can inquire about the possibility of connecting online at the email address marianna.eisenberg-nagy@uni-corvinus.hu.