Abstract
The fact that two disjoint convex sets can be separated by a plane has a tremendous impact on optimization theory and its applications. We begin the paper by illustrating this fact in convex and partly convex programming. Then we look beyond convexity and study general nonlinear programs with twice continuously differentiable functions. Using a parametric extension of the Liu-Floudas transformation, we show that every such program can be identified as a relatively simple structurally stable convex model. This means that one can study general nonlinear programs with twice continuously differentiable functions using only linear programming, convex programming, and the inter-relationship between the two. In particular, it follows that globally optimal solutions of such general programs are the limit points of optimal solutions of convex programs.
Similar content being viewed by others
References
Asgharian, M. and Zlobec, S. (2002), Abstract parametric programming, Optimization, 51, 841–861.
Ben-Israel, A., Ben-Tal, A. and Zlobec, S. (1981), Optimality in Nonlinear Programming: A Feasible Directions Approach, Wiley-Interscience, New York.
Canon, M.D., Cullum Jr., C.D. and Polak, E. (1970), Theory of Optimal Control and Mathematical Programming, McGraw-Hill Series in Systems Science, New York.
Floudas, C. (2000), Deterministic Global Optimization, Kluwer Academic.
Floudas, C.A. and Zlobec, S. (1998), Optimality and duality in parametric convex lexicographic programming. in Migdalas et al. (eds.), Multilevel Optimization: Algorithms and Applications, Kluwer Academic, pp. 359–379.
Liu, W.B. and Floudas, C.A. (1993), A remark on the GOP algorithm for global optimization, J. Global Optimization, 3, 519–521.
Sethi, S.P. A survey of management science applications of the deterministic maximum principle, TIMS Studies in the Management Sciences, North-Holland v. 9 (1978), 33–67.
Trujillo-Cortez, R. and Zlobec, S. (2001), Stability and optimality in convex parametric programming, Mathematical Communications, 6, 107–121.
Vincent, T.L. and Grantham, W.J. (1981), Optimality in Parametric Systems, Wiley Interscience, New York.
Zangwill, W.I. (1969), Nonlinear Programming, Prentice-Hall Inc., Englewood Cliffs, N.J.
Zlobec, S. (1983) Characterizing an optimal input in perturbed convex programming, Mathematical Programming, 25, 109–121; Corrigendum: Ibid 35 (1986) 368–371.
Zlobec, S. (1985) Input optimization: I. Optimal realizations of mathematical models,Mathematical Programming, 31, 245–268.
Zlobec, S. (1995), Partly convex programming and Zermel's navigation problems, J. Global Optimization, 7, 229–259.
Zlobec, S. (2001), Stable Parametric Programming, Kluwer Academic; series: Applied Optimization, 57.
Zlobec, S. (2001), Stability in linear programming: An index set approach, Annals of Operations Research, 101, 363–382.
Zlobec, S. (2003), Estimating Convexifiers in Continuous Optimization, Mathematical Communications (forthcoming).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zlobec, S. Saddle-Point Optimality: A Look Beyond Convexity. Journal of Global Optimization 29, 97–112 (2004). https://doi.org/10.1023/B:JOGO.0000035004.66019.3b
Issue Date:
DOI: https://doi.org/10.1023/B:JOGO.0000035004.66019.3b