Mathematical Programming

, Volume 58, Issue 1–3, pp 415–428

D.c. sets, d.c. functions and nonlinear equations

• Phan Thien Thach
Article

Abstract

Ad.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in ℝ n into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.

Key words

D.c. sets d.c. functions nonlinear equations—inequations

References

1. 
B.C. Eaves and W.I. Zangwill, “Generalized cutting plane algorithms,”SIAM Journal on Control 9 (1971) 529–542.Google Scholar
2. 
J.E. Falk and K.R. Hoffman, “A successive underestimation method for concave minimization problems,”Mathematics of Operations Research 1 (1976) 251–259.Google Scholar
3. 
R.M. Freund and J.B. Orlin, “On the complexity of four polyhedral set containment problems,”Mathematical Programming 33 (1985) 139–145.Google Scholar
4. 
J.B. Hiriart-Urruty, “Generalized differentiability, duality and optimization for problems dealing with differences of convex functions,”Lecture Notes in Economics and Mathematical Systems No. 256 (Springer, Berlin, 1984) pp. 37–70.Google Scholar
5. 
K.L. Hoffman, “A method for globally minimizing concave functions over convex sets,”Mathematical Programming 20 (1981) 22–32.Google Scholar
6. 
R. Horst, J. de Vries and N.V. Thoai, “On finding new vertices and redundant constraints in cutting plane algorithms for global optimization,”Operations Research Letters 7 (1988) 85–90.Google Scholar
7. 
L.A. Istomin, “A modification of Hoang Tuy's method for minimizing a concave function over a polytope,” Žurnal Vyčislitel'noi Matematiki i Matematičesko428-2 17 (1977) 1592–1597. [In Russian.]Google Scholar
8. 
B.M. Mukhamediev, “Approximate method for solving the concave programming problem,” Žurnal Vyčislitel'noi Matematiki i Matematičesko428-4 22 (1982) 727–731 [In Russian.]Google Scholar
9. 
R.T. Rockafellar, “Favorable classes of Lipschitz continuous functions in subgradient optimization,” Working Paper, IIASA (Laxenburg, 1983).Google Scholar
10. 
R.T. Rockafellar,,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
11. 
J.B. Rosen and P.M. Pardalos, “Global minimization of large scale constrained concave quadratic problems by separable programming,”Mathematical Programming 34 (1986) 163–174.Google Scholar
12. 
P.T. Thach and H. Tuy, ‘Global optimization under Lipschitzian constraints,”Japan Journal of Applied Mathematics 4 (1987) 205–217.Google Scholar
13. 
P.T. Thach, “Concave minimization under nonconvex constraints with special structure,”Essays on Nonlinear Analysis and Optimization Problems (Hanoi Institute of Mathematics Press, Hanoi, 1987) pp. 121–139.Google Scholar
14. 
T.V. Thieu, B.T. Tam and V.T. Ban, “An outer approximation method for globally minimizing a concave function over a compact convex set,”Acta Mathematica Vietnamica 8 (1983) 21–40.Google Scholar
15. 
N.V. Thoai and H. Tuy, “Convergent algorithm for minimizing a concave function,”Mathematics of Operations Research 5 (1980) 556–566.Google Scholar
16. 
H. Tuy, “Concave programming under linear constraints,”Doklady Akademii Nauk SSR 159 (1964) 32–35. [English translation in:Soviet Mathematics (1964) 1437–1440.]Google Scholar
17. 
H. Tuy, “On outer approximation methods for solving concave minimization problems,”Acta Mathematica Vietnamica 8 (1983) 3–34.Google Scholar
18. 
H. Tuy, T.V. Thieu and N.Q. Thai, “A conical algorithm for globally minimizing a concave function over a closed convex set,”Mathematics of Operations Research 10 (1985) 489–514.Google Scholar
19. 
H. Tuy, “Concave Minimization under linear constraints with special structure,”Optimization 26 (1985) 335–352.Google Scholar
20. 
H. Tuy, “A general deterministic approach to global optimization via d.c. programming,” in: J.B. Hiriart-Urruty, ed.,Mathematics Studies No. 129 (North-Holland, Amsterdam, 1987) pp. 273–303.Google Scholar
21. 
N.S. Vassiliev, “Active computing method for finding the global minimum of a concave function,” Žurnal Vyčislitel'noi Matematiki i Matematičesko428-6 23 (1983) 152–156. [In Russian.]Google Scholar