238 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA B. M. M ukhamediev REFERENCES 1. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA STOER. J.. On the relation between quadratic termination and convergence properties zyxwvutsrqponmlkjihgfedcbaZ of minimization algorithms. Part 1. Theory,Numer. Marh., 28, No. 3,343-365, 1977. 2. BAPTIST, P., and STOER. J.. On the relation between quadratic termination and convergence properties of minimization algorithms. Part II. Applications, Numer. Math., 28, No. 4, 367-392, 1977. 3. FLETCHER, H., and REEVES. C. M., Function minimization by conjugate gradients, Compur. J.. 7, 149-154.1964. 4. POLYAK. B. T., The method of conjugate gradients in extrernum problems, Zh. vychisl. Mar. mar. Fiz., 9, No. 4.807-821,1969. 5. HUANG, H. J.. Unified approach to quadratically convergent algorithms for function minimization, J. Optimizat. Theory Appl., 5, No. 6,405-423, 1970. 6. PSHENICHNYI, B. N., and DANILIN, YU. M.. Numerical methods in extremd problems (Chislennye metody v ekstremal’nykh zadachakh), Nauka, Moscow, 1975. 7. ZOUTENDIJK, G.,Merhods ofFeasible Directions, Amer. Elsevier, New York, 1960. 8. BULANYI. A. P.. and DANILIN. YU. M., Quasi-Newtonian algorithms of minimization, based on the construction of systems of conjugate vectors, Zh. v.?chisl. Mat. mat. Fir., 18, No. 4,877- 885, 1978. 9. POWELL. J. M. D.. Convergent properties of a class of minimization algorithms, in: Nonlinear Programming 2. pp. l-28, Acad. Press. New York, 1975. 10. POWELL, hl. J. D.. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J., 7, 155-162, 1964. 11. POLYAK, B. T., and TRET’YAKOV. N. V.. The method of penalty estimates for constrained extremum problems, Zh. v);chisl. Mat. mar. Fiz., 13, No. 1) 34- 36, 1973. U.S.S.R. Compuf. Maths. Marh. Ph,,s. Vol. 22, No. 3: pp. 238-245,1982. Printed in Great Britain 0041-5553,‘82%07.50+.00 01983. Pergamon Press Ltd. APPROXIMATE METHOD OF SOLVING CONCAVE PROGRAMMING PROBLEMS* B. M. MUKHAMEDIEV Alma-Ata (Received 5 June 1980; revised 28 Junuq~ 198 I ) A FINITE-STEP method is described for seeking with given accuracy the global minimum of a concave function in a convex set. Introduction In economic studies, the concave properties of the target function to be minimized reflect the economic effects of amalgamation. e.g. in the problem of the optimal disposition of production [ I]. Linear min-max problems with connected variables [2] , the problem of bilinear programming l Zh. vychisl. Mat. mar. Fiz., 22. 3: 727-732. 1982.