Abstract
In this two-part article, nonlinear coordinate transformations are discussed in order to simplify global unconstrained optimization problems and to test their unimodality on the basis of the analytical structure of the objective functions. If the transformed problems can be quadratic in some or all the variables, then the optimum can be calculated directly, without an iterative procedure, or the number of variables to be optimized can be reduced. Otherwise, the analysis of the structure can serve as a first phase for solving global unconstrained optimization problems.
The first part treats real-life problems where the presented technique is applied and the transformation steps are constructed. The second part of the article deals with the differential geometrical background and the conditions of the existence of such transformations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avriel, M. (1976),Nonlinear Programming, Analysis and Methods, Prentice-Hall, New Jersey.
Avriel, M. and Zang, I. (1980), Generalized Arcwise Connected Functions and Characterizations of Local-Global Properties,Journal of Optimization Theory and Applications 32, 407–425.
Ben-Tal, A. (1977), On Generalized Means and Generalized Convex Functions,Journal of Optimization Theory and Applications 21, 1–13.
Castagnoli, E. and Mazzoleni, P. (1987), Generalized Connectedness for Families of Arcs,Optimization 18, 3–16.
Gabay, D. (1982), Minimizing a Differentiable Function over a Differentiable Manifold,Journal of Optimization Theory and Applications 37, 117–219.
Hansen, P., Jaumard, B., and Lu, S.-H. (1989), An Analytical Approach to Global Optimization, Rutcor Research Report 4-89, Rutgers University, New Brunswick.
Hansen, P., Jaumard, B., and Lu, S.-H. (1991) An Analytical Approach to Global Optimization, Mathematical Programming52, 227–254.
Hicks, N. J. (1965)Notes on Differential Geometry, Van Nostrand Publishing Company, Princeton, New Jersey.
Horst, R. (1982), A Note on Functions Whose Local Minima Are Global,Journal of Optimization Theory and Applications 36, 457–463.
Horst, R. (1984) Global Optimization in Arcwise Connected Metric Spaces,Journal of Optimization Theory and Applications 104, 481–483.
Horst, R. and Thach, P. T. (1988) A Topological Property of Limes-Arcwise Strictly Quasiconvex Functions,Journal of Mathematical Analysis and Applications 134, 426–430.
Horst, R. and Tuy, H. (1990),Global Optimization, Springer-Verlag, Berlin, Heidelberg, New York.
Iri, M. (1991), Integrability of Vector and Multivector Fields Associated with Interior Point Methods for Linear Programming,Mathematical Programming 52, 511–525.
Luenberger, D. G. (1972), The Gradient Projection Methods Along Geodesics,Management Science 18, 620–631.
Luenberger, D. G. (1973),Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading.
Martin, D. H. (1982), Connected Level Sets, Minimizing Sets and Uniqueness in Optimization,Journal of Optimization Theory and Applications 36, 71–93.
Matsushima, Y. (1972),Differentiable Manifolds, Marcel Dekker, Inc., New York.
Milnor, J. W. (1969),Morse Theory, Princeton University Press, Princeton, New Jersey, 1969.
Mishchenko, A. and Fomenko, A. (1988),A Course of Differential Geometry and Topology, Mir Publishers Moscow, Moscow.
Ortega, J. M. and Rheinboldt, W. C. (1970),Iterative Solution of Nonlinear Equations, Academic Press, New York.
Perekatov, A. E. and Redkovskii, N. N. (1989), Method for Minimization of Unimodal Non-Convex Functions,Dokladi AN USSR Ser. A Physical-Mathematics and Technical Sciences 10, 36–38 (in Russian).
Prenowitz, W. and Jantosciak, J. (1979),Join Geometries, Springer, New York.
Rapcsák, T. (1986), Convex Programming on Riemannian Manifold, System Modelling and Optimization,Proceedings of 12-th IFIP Conference, Edited by A. Prékopa, J. Szelezsán and B. Strazicky, Springer Verlag, Berlin, Heidelberg, 733–741.
Rapcsák, T. (1987), Arcwise-Convex Functions on Surfaces,Publicationes Mathematicae 34, 35–41.
Rapcsák, T. (1989), Minimum Problems on Differentiable Manifolds,Optimization 20, 3–13.
Rapcsák, T. (1990), Tensor Optimization, MTA SZTAKI Report, 34/1990.
Rapcsák, T. (1991), Geodesic Convexity in Nonlinear Optimization,Journal of Optimization Theory and Applications 69, 169–183.
Ratschek, H. and Rokne, J. (1993), Solving a Global Optimization Problem,J. of Global Optimization (to appear).
Redkovskii, N. N. (1989), Nonlinear Transformations of Coordinates in Unconstrained Optimization Problems,Issledovanii Operatsii UAS 34, 60–65 (in Russian).
Schnabel, R. B. and Frank, P. (1984), Tensor Methods for Nonlinear Equations,SIAM Journal on Numerical Analysis 21, 815–843.
Schnabel, R. B. and Chowe, T.-T. (1991) Tensor Methods for Unconstrained Optimization Using Second Derivatives,SIAM Journal on Optimization 1, 293–315.
Singh, C. (1983), Elementary Properties of Arcwise Connected Sets and Functions,Journal of Optimization Theory and Applications 41, 377–387.
Stoutemyer, D. R. (1978), Automatic Categorization of Optimization Problems: An Application of Computer Symbolic Mathematics,Operations Research 26, 773–788.
Udriste, C. (1976) Convex Functions on Riemannian Manifolds,St. Cerc. Mat. 6, 735–745 (in Roumanian).
Udriste, C. (1977), Continuity of Convex Functions on Riemannian Manifolds,Bulletine Mathematique de Roumanie 21, 215–218.
Udriste, C. (1979), Directional Derivatives of Convex Functions on Riemannian Manifolds,Revue Roumaine de Mathématiques Pures et Appliqées 24, 1385–1388.
Udriste, C. (1984), Kuhn-Tucker Theorem on Riemannian Manifolds, inColloquia Mathematica Societatis János Bolyai, 46. Topics in Differential Geometry, Debrecen, Hungary, 1247–1259.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rapcsák, T., Csendes, T. Nonlinear coordinate transformations for unconstrained optimization II. Theoretical background. J Glob Optim 3, 359–375 (1993). https://doi.org/10.1007/BF01096776
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01096776
Key words
- Global optimization
- smooth unconstrained optimization
- geodesic convexity
- unimodal function
- nonlinear coordinate transformation