Aharon Ben-Tal (Faculty of Data and Decision Sciences, Technion Israel Institute of Technology)
Title: An Algorithm for Maximizing a Convex
Function Based on its Minimum, and Beyond
Abstract: In this talk the COMAX algorithm for maximizing a convex
function over a convex feasible set (an NP-hard problem) is
proposed. The algorithm consists of two phases: in phase 1 a
feasible starting point is obtained that is used in a Gradient
Descent algorithm in phase 2. The main contribution of the paper
is connected to phase 1; five different methods are used to
approximate the original NP-hard problem of maximizing a convex
function (MCF) by a tractable convex optimization problem. All
the methods use the minimizer of the convex objective
function in their construction. The performance of the overall
algorithm is tested on a wide variety of MCF problems,
demonstrating its efficiency.
As a next step COMAX is used in the design of new algorithms of
optimizing a difference of convex functions.
Coralia Cartis (Mathematical Institute, University of Oxford)
Title: Tensor methods for nonconvex optimization
Abstract: We consider the advantages of having and incorporating higher- (than second-) order derivative information inside regularization frameworks, generating higher-order regularization algorithms that have better complexity, universal properties and can certify higher-order criticality of candidate solutions. We discuss practical implementations of some of these methods, which involve the challenging solution of polynomial subproblems. Theoretical and numerical results for the latter will be presented.
Russell Luke (Institute for Numerical and Applied Mathematics, University of Goettingen)
Title: Proximal Splitting Algorithms in Nonlinear Spaces
Abstract: Motivated by the problem of computing Fréchet means of
points in spaces of phylogenetic trees, we examine proximal splitting
algorithms in general uniformly convex spaces, possibly with positive curvature.
Several key notions from linear space theory no longer apply in this setting and
force a reconfiguration of regularity concepts that explicitly accounts for the
regularity of the space in which the algorithms run. We present a few
convergence results for proximal splitting algorithms that benefit from the
analysis of these algorithms when applied to nonconvex problems in linear spaces.
Renata Sotirov (Department of Econometrics & Operations Research, Tilburg School of Economics and Management, Tilburg University)
Title: Mixed-Integer Semidefinite Programming - a New Perspective
Abstract: Mixed-integer semidefinite programming can be viewed as a generalization of mixed-integer programming where the vector of variables is replaced by mixed-integer positive semidefinite matrix variables. The combination of positive semidefiniteness and integrality allows to formulate various nonlinear optimization problems as (linear) mixed-integer semidefinite programs (MISDPs). Although MISDPs appeared already in the last century, they received little attention until recently. In this talk we present new results on MISDPs and resulting continuous relaxations. Finally, we present novel approaches for solving MISDPs.
