Abstract
One classic problem in air traffic management (ATM) has been the problem of detection and resolution of conflicts between aircraft. Traditionally, a conflict between two aircraft is detected whenever the two protective cylinders surrounding the aircraft intersect. In Trajectory-based Air Traffic Management, a baseline for the next generation of air traffic management system, we suggest that these protective cylinders be deformable volumes induced by variations in weather information such as wind speed and directions subjected to uncertainties of future states of trajectory controls. Using contact constraints on deforming parametric surfaces of these protective volumes, a constrained minimization algorithm is proposed to compute collision between two deformable bodies, and a differential optimization scheme is proposed to resolve detected conflicts. Given the covariance matrix representing the state of aircraft trajectory and its control and objective functions, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables where the coefficients in this combination are required to be non-negative, and the number of non-zero coefficients is constrained (e.g. state of trajectory and estimated time of arrival over one change point). Using convex relaxation and re-weighted l 1 technique, we reduce the problem to solving some semi-definite programming ones, and reinforce the non-negative principal components that satisfy the sparsity constraints. Numerical results show that the method presented in this paper is efficient and reliable in practice. Since the proposed method can be applied to a wide range of dynamic modeling problems such as collision avoidance in dynamic autonomous robots environments, dynamic interactions with 4D computer animation scenes, financial asset trading, or autonomous intelligent vehicles, we also attempt to keep all descriptions as general as possible.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alizadeh, F.: Interior point methods in semi-definite programming with applications to combinatorial optimization. SIAM J. Optim. 5, 13–51 (1995)
Badea, L., Tilivea, D.: Sparse factorizations of gene expression guided by binding data. In: Pacific Symposium on Biocomputing (2005)
Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)
Cadima, J., Jolliffe, I.T.: Loadings and correlations in the interpretation of principal components. J. Appl. Statist. 22, 203–214 (1995)
Candes, E.J., Wakin, M.B., Boyd, S.: Enhancing sparsity by re-weighted l 1 minimization (preprint)
D’Aspremont, A., El Ghaoui, L., Jordan, M.I., Lanckriet, G.R.G.: A direct formulation for sparse PCA using semi-definite programming. SIAM Rev. 49, 434–448 (2007)
Duong, V.: Dynamic models for airborne air traffic management capability: State-of-the-art analysis (Internal report). Eurocontrol Experimental Centre, Bretigny (1996)
Fazel, M., Hindi, H., Boyd, S.: A rank minimization heuristic with application to minimum order system approximation. In: Proceedings of the American Control Conference, Arlington, VA., vol. 6, pp. 4734–4739 (2001)
Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24, 417–441 (1933)
Jagannathan, R., Ma, T.: Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance 58, 1651–1684 (2003)
Jeffers, J.: Two case studies in the application of principal components. Appl. Statist. 16, 225–236 (1967)
Jolliffe, I.T.: Rotation of principal components: Choice of normalization constraints. J. Appl. Statist. 22, 29–35 (1995)
Jolliffe, I.T., Trendafilov, N.T., Uddin, M.: A modified principal component technique based on the LASSO. J.Comput. Graphical Statist. 12, 531–547 (2003)
Jolliffe, I.T.: Principal component analysis. Springer, New York (2002)
Lemarechal, C., Oustry, F.: Semi-definite relaxations and lagrangian duality with application to combinatorial optimization. Rapport de recherche 3710, INRIA, France (1999)
Lovasz, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190 (1991)
Moghaddam, B., Weiss, Y., Avidan, S.: Spectral Bounds for Sparse PCA: Exact & Greedy Algorithms. In: Advances in Neural Information Processing Systems, vol. 18, pp. 915–922. MIT Press, Cambridge (2006)
Nesterov, Y.: Smoothing technique and its application in semi-definite optimization. Math. Program. 110, 245–259 (2007)
Pearson, K.: On lines and planes of closest fit to systems of points in space. Phil. Mag. 2, 559–572 (1901)
Sturm, J.: Using SEDUMI 1.0x, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11, 625–653 (1999)
Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3 - a MA TLAB software package for semi-definite programming. Optim. Methods Softw. 11, 545–581 (1999)
Vines, S.: Simple principal components. Appl. Statist. 49, 441–451 (2000)
Zass, R., Shashua, A.: Non-negative Sparse PCA. In: Advances In Neural Information Processing Systems, vol. 19, pp. 1561–1568 (2007)
Zhang, Z., Zha, H., Simon, H.: Low-rank approximations with sparse factors I: Basic algorithms and error analysis. SIAM J. Matrix Anal. Appl. 23, 706–727 (2002)
Zhang, Z., Zha, H., Simon, H.: Low-rank approximations with sparse factors II: Penalized methods with discrete Newton-like iterations. SIAM J. Matrix Anal. Appl. 25, 901–920 (2004)
Zou, H., Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graphical Statist. 15, 265–286 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duong, T.D.X., Duong, V.N. (2008). Non-negative Sparse Principal Component Analysis for Multidimensional Constrained Optimization. In: Ho, TB., Zhou, ZH. (eds) PRICAI 2008: Trends in Artificial Intelligence. PRICAI 2008. Lecture Notes in Computer Science(), vol 5351. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89197-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-89197-0_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89196-3
Online ISBN: 978-3-540-89197-0
eBook Packages: Computer ScienceComputer Science (R0)