Research themes
- Optimisation on Manifolds
- Convex Optimisation
- Equilibrium Systems
- Ordered Vector Spaces
- Multicriteria Decision Problems
- Artificial Intelligence
Research activity
In the past Sándor has initiated the concepts of monotone vector fields, scalar derivatives, lattice-like operations and isotone retractions and together with his co-authors has connected them to optimisation on manifolds, convex optimisation, projection operators, equilibrium problems and isotonic regression. Sándor has also contributed to the mathematics part of important papers about fitness sharing and limitation of code growths in genetic programming. His most recent interests are in the following topics:
Optimisation on Riemannian Manifolds and Applications:
Sándor is interested in constrained optimization problems on Riemannian manifolds and related methods such as the projected gradient algorithm, the difference-of-convex (DC) algorithm, and the Frank–Wolfe algorithm. He is working on variants of the latter two methods on Hadamard manifolds (e.g., the positive-definite manifold and hyperbolic space) that use the Busemann function.
Sándor is interested in the geodesic convexity and the optimisation of quadratic functions on the sphere. This topic is motivated by a wide range of applications such as solid mechanics, signal processing, computational anatomy and quantum mechanics. Besides the practical interest, many optimisation problems are naturally posed on the sphere, which has an underlying algebraic structure that can be greatly exploited to find efficient solutions.
Sándor is interested in convexity/optimisation on the Hyperbolic space. The hyperbolic space is of central interest in relativity, but recently became important in specific problems of machine learning, financial mathematics and geology too. He is working on the convexity of quadratic functions, duality, existence of solutions and their approximation.
Sándor is interested in the study of the convergence of the first order gradient based algorithms for manifolds of unbounded sectional curvature, which is an important open question.
Sándor is initiating the study of the Stiefel manifold of orthonormal 2-frames, and its connections with complementarity problems and duality in optimisation.
New Cones, Ordered Vector Spaces, Complementarity Problems, Variational Inequalities and Equilibrium, and Applications
Sándor is interested in extensions of the second order cone. He has introduced two new cones the extended second order cone and the monotone extended second order cone. Both of them can be used to model specific portfolio optimisation problems. With his coauthors he studied the structure of these cones and designed efficient methods to project onto them.
The cones mentioned in the previous item have good ordering properties which can be used to solve general complementarity problems and variational inequalities. Solving such problems is important because many equilibrium problems in economics, finance, physics, mechanics and traffic can be described as complementarity problems or variational inequalities. The Karush-Kuhn-Tucker conditions for the portfolio optimisation problems related to the aforementioned cones are complementarity problems on them. The study/solution of these portfolio optimisation problems can be done via the corresponding complementarity problems.