Research team
Project’s summary
We also intend to study Jordan algebras and their applications to cone optimization (namely self-concordant barriers and their parameters) and the anomalous extrema phenomenon in nonlinear programming. In addition, we intend to offer continuing education in optimization and discrete mathematics and to develop internships and research collaboration with industry.
Project’s description objectives
Stochastic optimization and stochastic approximation algorithms will be developed, as well as applications of non-standard analysis to optimization and control. We expect to obtain results on adjacency and Laplacian eigenvectors and eigenvalues of graphs with regularity constraints and on their relations with the existence of subsets of vertices and subsets of edges with regularity properties.
Relationship with the state of the art
In addition, only a few computational experiments were performed on lower and upper bounds on the stability number of graphs determined by convex quadratic programming [19, 20]. On the other hand, the relationship between convex quadratic limits of the stability number and the Lovasz theta number [18] still lacks clarification.
Tasks
The recognition of these graphs is related to the existence of (k, τ)-regular sets, where k= 0 and τ is equal to the symmetric value of the minimal eigenvalue of the adjacency. Special attention will be paid to a short family of Moore graphs of diameter 2, namely the possible existence of a fourth Moore graph.
Problems of minimal resistance and problems of mass transfer
- Research team
- Project’s summary
- Project’s description objectives
- Relationship with the state of the art
- Tasks
This problem seems more realistic; it turns out to be closely related to billiards theory and the mass transfer problem. Mass transfer theory has been on the rise in recent years; yet the number of exactly solvable problems, even in one dimension, is relatively small. Also, the relationship between finite-dimensional transport problems and the mass transfer problems is not fully understood.
Our goal is to discover new exactly solvable problems, as well as to represent the mass transfer problem as the limiting case of finite-dimensional transport problem. Solution of Newton's problem in the classes of non-convex axisymmetric and non-axisymmetric bodies, in two and three dimensions. The body with minimal resistance in a medium of positive temperature; limiting cases of zero and of infinite temperature.
It is planned to obtain exact solutions about mass transfer in two dimensions, using methods of finite-dimensional transport problem.
Computability and algorithms
Research team
Project’s Summary
Project’s Description Objectives
In the field of quantum computing, our work focuses on the design of quantum computing circuits, especially on the use of generalized redundant number systems, as well as on the problem of quantum computer simulation.
Relationship with the state of the art
Tasks
We want to explore a new perspective on Lipschitz regularity results of minimizing trajectories in the calculus of variations using the appropriate conservation laws (Torres'02). We propose a new look at the results regarding the correctness of the minimizing trajectories. We intend to obtain a proper expansion in contrast to the recent development of Noether's symmetric theorems for the optimal control setting, which give rise to improper expansions when specified for calculus of variations problems.
One way to solve the problem is to obtain conservation laws to lower the order of the equations. Plakhov, Newton's problem of the body of minimal resistance with a finite number of collisions, Russ. Plakhov, “Exact solutions to the one-dimensional Monge-Kantorovich problem, Cadernos de Matemática, Universidade de Aveiro, CM03/I p.
In the quiet case, all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives an appropriate extension of Noether's classical theorem. As conclusions, results are obtained for first-order and higher-order discrete problems of the calculus of variations [3,9,14]. Torres, A proper extension of Noethers symmetry theorem for non-smooth extrema of the calculus of variations.
Torres, a correct extension of Noether's symmetry theorem for non-smooth extremals in the calculus of variations. Torres, On the Constancy of the Pontryagin Hamiltonian for Autonomous Problems, Cadernos de Matemática CM03/I-14, Dep.
Control theory group
Research team
Project’s Summary
Project’s Description Objectives
Relationship with the state of the art
Tasks
Optimization, graph theory and combinatorics
Activities during 2003
An MSc course was also prepared in collaboration with Adam Mickiewicz University (namely Michal Karonski and Jerzy Szymanski) and Universitat Politicnica de Catalunya (namely Marc Noy and Oriol Serra). Some results concerning the convex quadratic upper bound of the stability number with the Lovasz theta number were obtained and submitted for publication. Some relations between the optimal parameter of a self-concordant barrier over a symmetric cone, the carathédorian number of the cone and the rank of each underlying Euclidean Jordan algebra were also obtained and submitted for publication (this study was carried out in collaboration with Luis A. Vieira of the Universidade do Porto, who is completing his PhD thesis under the supervision of Domingos M. Cardoso).
A multi-attribute ranking solution procedure was developed and applied to confirm the results obtained by a multi-attribute ranking decision methodology on a tender for the supply of buses to the Porto Public Transport Operator (STCP) (this research was carried out in collaboration with Jorge F. Sousa of Universidade do Porto and submitted for publication). A dynamic step size adjustment algorithm was constructed that allows to increase the step size when the algorithm is relatively far from the solution, in the 'deterministic' phase, and . We have shown that there are sufficient second-order optimality conditions for nonlinear programming problems with mixed constraints that, in the case of anomalous extrema, ensure the isolation of the extreme point in the allowable set.
Essentially fundamental studies have been made by attempting approximate minimization with embeddings in hyperfinite dimensional normed spaces as well as from hyperfinite families of curves, in collaboration with Maria João Borges from IST, Lisbon.
List of publications
Preparatory studies on generalized manifolds were made by beginning PhD student Ricardo Almeida and were presented at the weekly assignment seminar.
List of talks
Cardoso, "Graphs with convex-QP stability number", 18th International Symposium on Mathematical Programming, August Copenhagen, Denmark. Luis Gouveia, Thomas Magnanti and Cristina Requejo, "An intersecting tree model for odd-diameter-constrained minimum spanning and Steiner trees", Proceedings of the International network optimization conference - INOC 2003, October 27-29, Evry/Paris, France. Luís Gouveia, Thomas Magnanti and Cristina Requejo, "MA 2-paths approach for odd-diameter-constrained minimum spanning and Steiner trees", 18th International Symposium on Mathematical Programming, August Copenhagen, Denmark.
Natália Martins and Vítor Neves, "Discrete Nonstandard Derivatives and Existence Theorems for ODES", Second European Junior Meeting on Control Theory and Stabilization, December 2003, University of Turin, Italy. Vítor Neves, "Nonstandard Calculus of Variations", Variational Analysis and Applications, (June/July 2003), Erice, Sicily. Vítor Neves, "Nonstandard Calculus of Variations", 12th St. Petersburg Summer Meeting on Mathematical Analysis, August 2003.
Cardoso, "An Algebraic Approach to Strongly Regular Graphs", Sessão em Homenagem ao Professor Mário da Silva Rosa, 15 de Outubro de 2003, Coimbra, Portugal.
List of reports (including proceedings)
List of organized seminars and conferences
Miguel Constantino, Planeamento Integrado da Produção e Sequenciação de Lotes na Indústria de Tintas: Programação Inteira Mista, Decomposição e Programação por Restrições", Seminários CEOC, 20 de Junho de 2003, Universidade de Aveiro. Leal Duarte, "Múltiplos autovalores de matrizes cujo grafo é um degrau ", Seminários CEOC, 4 de Abril de 2003, Universidade de Aveiro.Carvalho, "Contributos da optimização combinatória na protecção de dados estatísticos", Seminários CEOC, 28 de Março de 2003, Universidade de Aveiro.
Luz, "Relacionar o número Lovász vartheta com alguns limites quadráticos convexos no número de estabilidade de um grafo", Seminários do CEOC, 7 de Março de 2003, Universidade de Aveiro.
List of PhD and MSc dissertations
Activities during 2003
List of publications
List of talks
List of reports (including proceedings)
List of organized seminars and conferences
Computability and Algorithms
Activities during 2003
List of publications
List of talks
List of organized seminars and conferences
Control theory group
Activities during 2003
We prove that Emmy Noether's theorem on the calculus of variations is still valid in the broader class of Lipschitz functions, so long as one restricts the Euler-Lagrange extremals to those satisfying the DuBois-Reymond necessary condition. Noether's theorem in Optimal Control, which states that integrals of motion associated with the discrete-time Pontryagin maximum principle can be computed from the quasi-invariance properties of the discrete-time Lagrangian and discrete-time control system. We extend Noether's second theorem to optimal control problems that are invariant under symmetries depending on k random functions of the independent variable and their derivatives up to a certain order m.
As far as we consider a semi-invariance concept, and the transformation group can also depend on the control variables, the result is new even in the classical context of the calculus of variations [6]. By rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle for the later problem and the Lipschitzian regularity conditions for the original problem. Under existence theory's standard hypotheses of constraint theory, the conditions imply that the optimal controls are essentially bounded, ensuring the applicability of the classical necessary optimality conditions such as the Pontryagin maximum principle.
We provide [16] a new, simpler and more direct proof of the well-known fact that for autonomous optimal control problems the Pontryagin extremal evolves on a smooth surface of the respective Pontryagin Hamiltonian.
List of publications
Talks at International Conferences
List of reports (including proceedings)
List of organized seminars and conferences
Margarida Camarinha, "Polinómios cúbicos em espaços de curvatura constante", Seminários CEOC, 6 de Junho de 2003, Universidade de Aveiro. Manuel Guerra, "Generalized Discontinuous and Extremal Hamiltonian Flows for Nonlinear Control Systems", Seminários CEOC, 23 de Maio de 2003, Universidade de Aveiro. Davydov, "Sobre singularidades genéricas de otimização média paramétrica de sistemas de controle", Seminários do CEOC, 21 de fevereiro de 2003, Universidade de Aveiro.
Torres was member of the organizing committee of the Second Junior European Meeting on "Control Theory and Stabilization", Dipartimento di Matematica del Politecnico di Torino, Torino, Italy, December 3-5, 2003. 15] Jakubczyk, B.: Critical Hamiltonians and feedback invariants. eds.) et al., Geometry of feedback and optimal control. A quadratic programming approach to determining an upper bound on the weighted stability number.
