Abstract
Hypersurfaces of the type z=F(x1,...,xn), where F are single-valued functions of n real variables, cannot be visualized directly due to our inability to perceive dimensions higher than three. However, by projecting them down to two or three dimensions many of their properties can be revealed. In this paper a method to generate such projections is proposed, requiring successive global minimizations and maximizations of the function with respect to n-1 or n-2 variables. A number of examples are given to show the usefulness of the method, particularly for optimization problems where there is a direct interest in the minimum or maximum domains of objective functions.
Similar content being viewed by others
References
Ackley DH (1987) A connectionist machine for genetic hill climbing. Kluwer, Boston
Arabeyre J, Steiger J, Teather W (1969) The airline crew scheduling problem: a survey. Transp Sci 3(2):140-163
Asimov D (1985) The grand tour: a tool for viewing multidimensional data. SIAM J Sci Stat Comput 6(1):128–143
Back T (1997) Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, New York
Bajaj CL (1990) Rational hypersurface display. ACM Comput Graph 24(2):117–127
Banchoff TT (1986) Visualizing two-dimensional phenomena in four-dimensional space. In: Wegeman EJ, DePriest DJ (eds) Statistical image processing and graphics. Dekker, New York, pp 187–202
Bhaniramka P, Wenger R, Crawfis R (2000) Isosurfacing in higher dimensions. In: Proceedings of the IEEE Visualization Conference, Salt Lake City, UT, pp 267–273, 566, 8–13 October 2000
Brent RP (1973) Algorithms for minimization without derivatives. Prentice-Hall, Upper Saddle River, NJ
Carr DB, Nicholson WL, Littlefeld RJ, Hall DL (1986) Interactive color display methods for multivariate data. In: Wegeman EJ, DePriest DJ (eds) Statistical image processing and graphics. Dekker, New York, pp 215–250
Chen JX, Wang S, (2001) Data visualization: parallel coordinates and dimension reduction. Comput Sci Eng 3(5):110–113
Cornell JA (2002) Experiments with mixtures. Wiley, New York
Craven MW, Shavlik J (1992) Visualization learning and computation in artificial neural networks. Int J Artif Intell Tools 1(3):399–425
Encarnação JL, Lindner R, Schlechtendahl EG (1990) Computer Aided Design. Fundamentals and System Architecture. Springer, Berlin Heidelberg New York
Feiner S, Beshers C (1990) Visualizing n-dimensional virtual worlds with n-Vsion. Comput Graph 24(2):37–38
Feiner S, Beshers C (1990) Worlds within worlds: metaphors for exploring n-dimensional virtual worlds. In: Proceedings of the 3rd ACM SIGGRAPH Symposium on User Interface Software and Technology, Snowbird, UT, pp 76–83, 3–5 October 1990
Grüne L, Metscher M, Ohlberger M (1999) On numerical algorithm and interactive visualization for optimal control problems. Comput Vis Sci 1(4):221–229
Haimes YY, Lasdon LS, Wismer DA (1971) On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst, Man Cybern 1(3):296–297
Hansen P, Jaumard B, Lu SH (1989) Some further results on monotonicity in globally optimal design. ASME J Mech Trans Automat Des 111(3):345–352
Hinton GE, Sejnowski TJ, Ackley DH (1984) Boltzman machines: constraint satisfaction networks that learn. Technical Report CMU-CS-84-119, Carnegie-Mellon University, Pittsburgh, PA
Inselberg A (1985) The plane with parallel coordinates. Vis Comput 1(2):69–91
Inselberg A, Dimsdale B (1990) Parallel coordinates: a tool for visualizing multi-dimensional geometry. In: Proceedings of the 1st IEEE Conference on Visualization, San Francisco, CA, pp 361–378, 23–26 October 1990
Jones CV (1996) Visualization and optimization. Kluwer, Boston
Jones CV (1998) Interactive transactions of OR/MS. http://catt.bus.okstate.edu/jones98/
Keiper J, Wickam-Jones T (1994) Design tools for visualization and optimization. ORSA J Comput 6(3):273–277
Messac A, Chen X (2000) Visualizing the optimization process in real-time using physical programming. Eng Optim 32(6):721–747
Mihalisin T, Timlin J, Schwengler J (1991) Visualizing multivariate functions, data and distributions. IEEE Comput Graph Appl 11(3):28–35
Napel S, Marks MP, Rubin GD, Jeffrey RB, Dake MD, Enzmann DR, McDonnell CH, Song SM (1992) CT angiography using spiral CT and maximum intensity projections. Radiology 185(2):607-610
Nielson GM, Foley TA, Hamann B, Lane D (1991) Visualizing and modeling scattered multivariate data. IEEE Comput Graph Appl 11(3):47–55
Noll AM (1967) A computer technique for displaying n-dimensional hyperobjects. Commun ACM 10(8):469–473
Papalambros PY, Wilde DJ (2000) Principles of optimal design: modeling and computation. Cambridge University Press, Cambridge, UK
Parkinson AR, Balling RJ (2002) The OptdesX design optimization software. Struct Multidisc Optim 23(2):127–139
Press WH, Flannery BP, Teukolsky S, Vetterling W T (1989) Numerical recipes. Cambridge University Press, Cambridge, UK
Pudmenzky A (1998) A visualization tool for N-dimensional error surfaces. Australian Conference on Neural Networks ACNN’98, University of Queensland, Brisbane, Australia, 11–13 February 1998
Rosenbrock HH (1960) An automatic method for finding the greatest or least value of a function. Comput J 3(3):175–184
Rossnick S, Laub G, Braeckle R, Bachus R, Kennedy D, Nelson A, Dzik S, Starewicz A (1986) Three dimensional display of blood vessels in MRI. In: Proceedings of the IEEE Computers in Cardiology Conference, New York, pp 193–196
Shaffer CA, Knill DL, Watson LT (1998) Visualization for multiparameter aircraft designs. In: Proceedings of the IEEE Visualization Conference, Triangle Research Park, NC, pp 491–494,575, 18–23 October 1998
Streeter M, Ward M, Alvarez SA (2001) N2VIS: an interactive visualization tool for neural networks. In: Proceedings of the International Society for Optical Engineering (SPIE) Visual Data Exploration and Analysis VIII, San Jose, CA, pp 234–241, 22–23 January 2001
Van Wijk JJ, Van Liere R (1993) HyperSlice: visualization of scalar functions of many variables. In: Proceedings of the IEEE Visualization Conference, San Jose, CA, pp 119–125, 25–29 October 1993
Wegenkittl R, Löffelmann H, Gröller E (1997) Visualizing the behavior of higher dimensional dynamical systems. In: Proceedings of the IEEE Visualization Conference, Phoenix, AZ, pp 291–296, 569, 19–24 October 1997
Wejchert J, Tesauro G (1990) Neural network visualization. In: Touretzky DS (ed) Advances in neural information processing systems. Morgan Kaufmann, San Mateo, CA, pp 465–472
Winer EH, Bloebaum CL (2001) Visual design steering for optimization solution improvement. Struct Multidisc Optim 22(3):219–229
Winer EH, Bloebaum CL (2002) Development of visual design steering as an aid in large-scale multidisciplinary design optimization part I & II. Struct Multidisc Optim 23(6):412–435
Wright H, Brodlie K, David T (2000) Navigating high-dimensional spaces to support design steering. In: Proceedings of the IEEE Visualization Conference, Salt Lake City, UT, pp 291–296, 569, 8–13 October 2000
Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102
Wolfram Research (2004) Math-World, http://mathworld.wolfram.com/topics/MultidimensionalGeometry.html
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Simionescu, P., Beale, D. Visualization of hypersurfaces and multivariable (objective) functions by partial global optimization. Vis Comput 20, 665–681 (2004). https://doi.org/10.1007/s00371-004-0260-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-004-0260-4