Constrained Extremum Problems, Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case
- 61 Downloads
Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.
KeywordsImage space analysis Regularity condition Constraint qualification Separation function
Mathematics Subject Classification49N15 90C30 90C46
The author is grateful to Professor Franco Giannessi and Professor Letizia Pellegrini for their constructive comments on this paper and their great help when this work was carried out during a stay of the author in the Department of Mathematics, University of Pisa. The author expresses his gratitude to the anonymous referees for their valuable comments and suggestions, which help to improve the paper. This research was supported by China Scholarship Council, the National Natural Science Foundation of China (Grants 11601437, 11526165) the Basic and Advanced Research of CQ CSTC (Grant cstc2015jcyja00038), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant KJ1501503) and the Chongqing Research Program of Basic Research and Frontier Technology (cstc2016jcyjA0270).
- 2.Karush, W.: Minima of functions of several variables with inequalities as side conditions. Master’s thesis, University of Chicago (1939)Google Scholar
- 3.John, F.: Extremum problems with inequalities as subsidiary conditions. In: Friedrichs, K.O., Neugebauer, O.E., Stoker, J.J. (eds.) Studies and Essays: Courant Anniversary Volume, pp. 187–204. Interscience, New York (1948)Google Scholar
- 4.Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
- 6.Eustaquio, R.G., Karas, E.W., Ribeiro, A.A.: Constraint qualification for nonlinear programming. Technical report, Federal University of Parana (2010)Google Scholar
- 19.Giannessi, F.: Some perspectives on vector optimization via image space analysis. J. Optim. Theory Appl. 177(3) (2018)Google Scholar
- 22.Gould, F.J., Howe, S.: A new result on interpreting Lagrange multipliers as dual variables. Technical report no. 738, The Institute of Statistics, University of North Carolina (1971)Google Scholar
- 23.Flegel, M. L.: Constraint qualifications and stationarity concepts for mathematical programs with equilibrium constraints. Ph.D. thesis, Institute of Applied Mathematics and Statistics, University of Würzburg (2005)Google Scholar