# zbMATH — the first resource for mathematics

Dual method for solving a special problem of quadratic programming as a subproblem at nonlinear minimax approximation. (English) Zbl 0621.65061
This paper concerns the nonlinear minimax approximation problem, where a point $$x^*\in R^ n$$ is sought such that $$F(x^*)=\min_{x\in R^ n}(\max_{i\in M}f_ i(x))$$ where $$f_ i(x)$$, $$i\in M$$ are real-valued functions defined in the n-dimensional vector space $$R^ n$$, with continuous second-order derivatives, and $$M=\{1,...,m\}$$. To solve this problem the method of recursive quadratic programming is applied where the most important step is the solution of a special quadratic programming subproblem. In this paper a dual method is presented in detail for solving this quadratic programming problem and its finite step convergence is proved.
Reviewer: T.Rapcsák

##### MSC:
 65K05 Numerical mathematical programming methods 90C20 Quadratic programming
Full Text:
##### References:
 [1] M. S. Bazaraa C. M. Shetty: Nonlinear programming. Theory and algorithms. New York: Wiley 1979. · Zbl 0476.90035 [2] C. Charalambous J. W. Bandler: Nonlinear minimax optimization as a sequence of least p-th optimization with finite values of p. Faculty Engn., McMaster University, Hamilton, Ontario, Canada, Kept. SOC-3, 1973. · Zbl 0349.90108 [3] C. Charalambous: Acceleration of the least p-th algorithm for minimax optimization with engineering applications. Math. Programming 17, 270-297) · Zbl 0418.90072 [4] V. F. Demyanov V. N. Malozemov: Introduction to minimax. Chap. 3, § 5. New York: Wiley 1974. [5] D. Goldfarb: Extension of Davidon’s variable metric method to maximization under linear inequality and equality constraints. SIAM J. Appl. Math. 17, 739-764) · Zbl 0185.42602 [6] D. Goldfarb A. U. Idnani: A numerically stable dual method for solving strictly convex quadratic programs. The City College of New York, Dept. of Computer Sci., Rept. 81- 102) · Zbl 0537.90081 [7] J. Hald K. Madsen: Combined LP and Quasi-Newton methods for minimax optimization. Math. Programming 20, 49-62) · Zbl 0441.90097 [8] S. P. Han: Variable metric methods for minimizing a class of nondifferentiable functions. Math. Programming 20, 1 - 13) · Zbl 0441.90095 [9] L. Lukšan: Variable metric methods for linearly constrained nonlinear minimax approximation. Computing 30, 315-334) · Zbl 0514.49016 [10] K. Madsen: An algorithm for minimax solution of overdetermined systems of nonlinear equations. J. Inst. Math. Appl. 16, 321-328) · Zbl 0355.65038 [11] M. J. D. Powell: A fast algorithm for nonlinearly constrained optimization calculations. In ”Numerical analysis, Dundes 1977”, (G. A. Watson, Lecture Notes in Mathematics 630, Berlin: Springer-Verlag 1978. · Zbl 0374.65032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.