Abstract
A minimization problem with convex and separable objective function subject to a separable convex inequality constraint “≤” and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality “≥” constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.
Similar content being viewed by others
References
P. Berman, N. Kovoor, and P.M. Pardalos, “Algorithms for the least distance problem, ” in Complexity in Numerical Optimization, P.M. Pardalos (Ed.), World Scientific, Singapore, 1993, pp. 33–56.
D.P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints, ” SIAM Journal on Control and Optimization, vol. 20, pp. 221–246, 1982.
G.R. Bitran and A.C. Hax, “Disaggregation and resource allocation using convex knapsack problems with bounded variables, ” Management Science, vol. 27, pp. 431–441, 1981.
J.R. Brown, “Bounded knapsack sharing, ” Mathematical Programming, vol. 67, pp. 343–382, 1994.
P. Brucker, “An O(n) algorithm for quadratic knapsack problems, ” Operations Research Letters, vol. 3, pp. 163–166, 1984.
P.H. Calamai and J.J. Moré, “Quasi-Newton updates with bounds, ” SIAM Journal on Numerical Analysis, vol. 24, pp. 1434–1441, 1987.
A. Charnes and W.W. Cooper, “The Theory of Search: Optimum Distribution of Search Effort, ” Management Science, vol. 5, pp. 44–49, 1958.
R.W. Cottle, S.G. Duval, and K. Zikan, “A Lagrangean relaxation algorithm for the constrained matrix problem, ” Naval Research Logistic Quarterly, vol. 33, pp. 55–76, 1986.
R.S. Dembo and U. Tulowitzki, “On the minimization of quadratic functions subject to box constraints, ” Working Paper Series B # 71, School of Organization and Management, Yale University, New Haven, 1983.
J. Dussault, J. Ferland, and B. Lemaire, “Convex quadratic programming with one constraint and bounded variables, ” Mathematical Programming, vol. 36, pp. 90–104, 1986.
J.A. Ferland, B. Lemaire, and P. Robert, “Analytic solutions for nonlinear programs with one or two equality constraints, ” Publication # 285, Departement d'informatique et de recherche operationnelle, Université de Montréal, Montréal, Canada, 1978.
S.M. Grzegorski, “Orthogonal projections on convex sets for Newton-like methods, ” SIAM Journal on Numerical Analysis, vol. 22, pp. 1208–1219, 1985.
M. Held, P. Wolfe, and H.P. Crowder, “Validation of subgradient optimization, ” Mathematical Programming, vol. 6, pp. 62–88, 1974.
R. Helgason, J. Kennington, and H. Lall, “Apolynomially bounded algorithm for a singly constrained quadratic program, ” Mathematical Programming, vol. 18, pp. 338–343, 1980.
G.T. Herman and A. Lent, “A family of iterative quadratic optimization algorithms for pairs of inequalities, with application in diagnostic radiology, ” Mathematical Programming Study, vol. 9, pp. 15–29, 1978.
N. Katoh, T. Ibaraki, and H. Mine, “A polynomial time algorithm for the resource allocation problem with a convex objective function, ” Journal of the Operations Research Society, vol. 30, pp. 449–455, 1979.
H. Luss and S.K. Gupta, “Allocation of effort resources among competing activities, ” Operations Research, vol. 23, pp. 360–366, 1975.
R.K. McCord, “Minimization with one linear equality constraint and bounds on the variables, ” Technical Report SOL 79-20, System Optimization Laboratory, Dept. of Operations Research, Stanford University, Stanford, 1979.
C. Michelot, “A finite algorithm for finding the projection of a point onto the canonical simplex of Rn, ” Journal of Optimization Theory and Applications, vol. 50, pp. 195–200, 1986.
J.J. Moré and G. Toraldo, “Algorithms for bound constrained quadratic programming problems, ” Numerische Mathematik, vol. 55, pp. 377–400, 1989.
J.J. Moré and S.A. Vavasis, “On the solution of concave knapsack problems, ” Mathematical Programming, vol. 49, pp. 397–411, 1991.
P.M. Pardalos and N. Kovoor, “An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds, ” Mathematical Programming, vol. 46, pp. 321–328, 1990.
P.M. Pardalos, Y. Ye, and C.-G. Han, “Algorithms for the solution of quadratic knapsack problems, ” Linear Algebra and its Applications, vol. 152, pp. 69–91, July 1991.
A.G. Robinson, N. Jiang, and C.S. Lerme, “On the continuous quadratic knapsack problem, ” Mathematical Programming, vol. 55, pp. 99–108, 1992.
R.T. Rockafellar and R.J.-B. Wets, “Anote about projections in the implementation of stochastic quasigradient methods, ” in Numerical Techniques for Stochastic Optimization, Yu. Ermoliev and R.J.-B. Wets (Eds.), Springer Verlag, Berlin, 1988, pp. 385–392.
S.M. Stefanov, “Convex knapsack problem with bounded variables, ” in Applications of Mathematics in Engineering, vol. 19, D. Ivanchev (Ed.), Technical University of Sophia, Sofia, 1994, pp. 189–194.
S.M. Stefanov, “On the implementation of stochastic quasigradient methods to some facility location problems, ” YUJOR, vol. 10, no. 2, pp. 235–256, 2000.
S.A. Vavasis, “Local minima for indefinite quadratic knapsack problems, ” Mathematical Programming, vol. 54, pp. 127–153, 1992.
P. Wolfe, “Algorithm for a least-distance programming problem, ” Mathematical Programming Study, vol. 1, pp. 190–205, 1974.
P.H. Zipkin, “Simple ranking methods for allocation of one resource, ” Management Science, vol. 26, pp. 34–43, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stefanov, S.M. Convex Separable Minimization Subject to Bounded Variables. Computational Optimization and Applications 18, 27–48 (2001). https://doi.org/10.1023/A:1008739510750
Issue Date:
DOI: https://doi.org/10.1023/A:1008739510750