CEOC - Center for Research on Optimization and Control

Optimization, Graph Theory and Combinatorics

Project's objectives:

The primary goals of this project include carrying out research and providing advanced training in optimization, graph theory and combinatorics. Aiming to capture the interest of Portuguese students and junior researchers to discrete mathematics, a Portuguese book on this topic it is in preparation. The main research objectives cover graph spectra, continuous optimization and graphs, combinatorics and related topics, stochastic optimization and stochastic approximation and nonstandard optimization techniques. We search for practical implementation of results of our work and for further development of internship programs in cooperation with industry.

Objectives on Graph Spectra
Results about adjacency and Laplacian eigenvectors and eigenvalues of graphs with regularity constraints and about their relations with the existence of subsets of vertices and subsets of edges with regularity properties.

Objectives on continuous optimization and graphs
Recognition of graphs with convex-QP stability number (that is, for which the stability number may be determined by convex quadratic programming), improvements on the determination of lower and upper bounds on the stability number of graphs obtained by convex quadratic programming and extensions to combinatorial optimization problems. Furthermore we intend to apply Jordan algebras on self-concordant barriers over symmetric cones and to study necessary and suficient conditions for abnormal extrema without constraint qualification.

Objectives on combinatorics and related topics
Results about the existence of the fourth graph of Moore. Conditions under which a ranking solution is a linear extension of a quasi-order extension of a preference weighted sum relation defined by a multiattribute ranking problem. Concordance graphs and concordance pairs of linear orders. Application of specialization orders and digital topologies in shape reorganization in digitalized image.

Objectives on stochastic optimization and stochastic approximation
Algorithms for the stochastic optimal path problem. Algorithms of accelerated convergence, aiming to achieve better performance by speeding up the transient stage in stochastic approximation.

Objectives on nonstandard optimization techniques
To apply hyperfinite discretization and non-standard hull methods in critical point theory as well as to control in differentiable manifolds.

Members

Research Team and Plan 2003-2005

Scientific Reports 2003-2005