Electronic Issue "Scientific Research"

In this paper, we consider the problem of minimizing a strictly convex separable function over a feasible region defined by a convex inequality constraint and two-sided bounds on the variables (box constraints). Also, the convex separable program with a strictly convex objective function subject to linear equality constraints and bounded variables is considered. These problems are interesting from both theoretical and practical point of view because they arise in some mathematical programming problems and in various practical problems. Characterization theorems (necessary and sufficient conditions) for the optimal solution to the considered problems are proved. Some illustrative examples are also presented.

[1] G.R. Bitran and A.C. Hax, Disaggregation and resource allocation using convex knapsack problems with bounded variables, Management Science 27 (1981), No. 4, 431-441. [2] K. Dahiya, S.K. Suneja, and V. Verma, Convex programming with single separable constraint and bounded variables, Computational Optimization and Applications. An International Journal 36 (2007), No. 1, 67-82. [3] J.-P. Dussault, J. Ferland, and B. Lemaire, Convex quadratic programming with one constraint and bounded variables, Mathematical Programming 36 (1986), No. 1, 90-104. [4] R. Helgason, J. Kennington, and H. Lall, A polynomially bounded algorithm for a singly constrained quadratic program, Mathematical Programming 18 (1980), No. 3, 338-343. 9 [5] N. Katoh, T. Ibaraki, and H. Mine, A polynomial time algorithm for the resource allocation problem with a convex objective function, Journal of the Operational Research Society 30 (1979), No. 5, 449-455. [6] H. Luss and S.K. Gupta, Allocation of effort resources among competing activities, Operations Research 23 (1975), No. 2, 360-366. [7] S.M. Stefanov, On the implementation of stochastic quasigradient methods to some facility location problems, Yugoslav Journal of Operations Research 10 (2000), No. 2, 235-256. [8] S.M. Stefanov, Convex separable minimization subject to bounded variables, Computational Optimization and Applications. An International Journal 18 (2001), No. 1, 27-48. [9] S.M. Stefanov, Separable Programming: Theory and Methods, Applied Optimization, vol. 53, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001. [10] S.M. Stefanov, Method for solving a convex integer programming problem, International Journal of Mathematics and Mathematical Sciences, vol. 2003, No. 44, 2829-2834. [11] S.M. Stefanov, Convex quadratic minimization subject to a linear constraint and box constraints, Applied Mathematics Research Express, vol. 2004, No. 1, 17-42. [12] S.M. Stefanov, Polynomial algorithms for projecting a point onto a region defined by a linear constraint and box constraints in IRn, Journal of Applied Mathematics, vol. 2004, No. 5, 409-431. [13] S.M. Stefanov, Convex separable minimization problems with a linear constraint and bounded variables, International Journal of Mathematics and Mathematical Sciences, vol. 2005, No. 9, 1339-1363. [14] S.M. Stefanov, An efficient method for minimizing a convex separable logarithmic function subject to a convex inequality constraint or linear equality constraint, Journal of Applied Mathematics and Decision Sciences, vol. 2006, 1-19, article ID 89307. [15] S.M. Stefanov, Minimizing a convex separable exponential function subject to linear inequality or equality constraint and bounds on the variables, Journal of Interdisciplinary Mathematics 9 (2006), No. 1, 207-226. [16] S.M. Stefanov, Minimization of a convex linear-fractional separable function subject to a convex inequality constraint or linear inequality constraint and bounds on the variables, Applied Mathematics Research Express, vol. 2006, 1-24, article ID 36581. [17] S.M. Stefanov, An efficient method for minimizing a convex separable function subject to convex separable inequality constraint and bounds on the variables, Journal of Complexity, to appear. [18] S.M. Stefanov, Solution of some separable production planning and resource allocation problems with bounds on the variables, Applied Mathematics Research Express, to appear. [19] T.V. Tchemisova, On the constructive solution of convex programming problems in separable form, Journal of Mathematical Sciences 120 (2004), No. 1, 1016-1031. [20] P.H. Zipkin, Simple ranking methods for allocation of one resource, Management Science 26 (1980), No. 1, 34-43. 10