摘要: In the current chapter, we study problem of minimizing a real-valued function \(f(\boldsymbol{x})\) subject to constraints $$\displaystyle\begin{array}{rcl} g_{i}(\boldsymbol{x})& =& 0,\quad \quad 1 \leq i p \\ h_{j}(\boldsymbol{x})& & j q.\end{array}$$ All these functions share some open set U ⊂R n as their domain. Maximizing is equivalent \(-f(\boldsymbol{x})\), so there no loss generality in considering minimization. The called objective function, \(g_{i}(\boldsymbol{x})\) are equality constraints, and \(h_{j}(\boldsymbol{x})\) inequality constraints. Any point \(\boldsymbol{x} \in U\) satisfying all said be feasible.
springer.com
sci-hub.se 参考文章(14)
William Karush, Minima of Functions of Several Variables with Inequalities as Side Conditions Traces and Emergence of Nonlinear Programming. pp. 217- 245 ,(2014) , 10.1007/978-3-0348-0439-4_10
Imre Pólik, Tamás Terlaky, Interior Point Methods for Nonlinear Optimization Lecture Notes in Mathematics. pp. 215- 276 ,(2010) , 10.1007/978-3-642-11339-0_4
H. L. Le Roy, L. Lecam, J. Neyman, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Vol. IV Revue de l'Institut International de Statistique / Review of the International Statistical Institute. ,vol. 37, pp. 230- ,(1969) , 10.2307/1402306
Osman Güler, Foundations of Optimization Graduate Texts in Mathematics. ,(2010) , 10.1007/978-0-387-68407-9
J. Michael Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities American Mathematical Monthly. ,vol. 112, pp. 575- 579 ,(2004) , 10.1017/CBO9780511817106
Jan Brinkhuis, Vladimir M. Tikhomirov, Optimization Insights and Applications ,(2010)
E. J. McShane, The Lagrange Multiplier Rule American Mathematical Monthly. ,vol. 80, pp. 922- 925 ,(1973) , 10.1080/00029890.1973.11993409
I. Ekeland, On the variational principle Journal of Mathematical Analysis and Applications. ,vol. 47, pp. 324- 353 ,(1974) , 10.1016/0022-247X(74)90025-0
Anders Forsgren, Philip E. Gill, Margaret H. Wright, Interior Methods for Nonlinear Optimization SIAM Review. ,vol. 44, pp. 525- 597 ,(2002) , 10.1137/S0036144502414942
Olga A. Brezhneva, Alexey A. Tret’yakov, Stephen E. Wright, A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization Optimization Letters. ,vol. 3, pp. 7- 10 ,(2009) , 10.1007/S11590-008-0096-3