A nonlinear Knapsack problem

The nonlinear Knapsack problem is to minimize a separable concave objective function, subject to a single ''packing'' constraint, in nonnegative variables. We consider this problem in integer and continuous variables, and also when the packing constraint is convex. Although the nonlinear Knapsack problem appears difficult in comparison with the linear Knapsack problem, we prove that its complexity is similar. We demonstrate for the nonlinear Knapsack problem in n integer variables and knapsack volume limit B, a fully polynomial approximation scheme with running time [email protected]?((1/@e^2) (n + 1/@e^2)) (omitting polylog terms); and for the continuous case an algorithm delivering an @e-accurate solution in O(n log(B/@e)) operations.

[1]  Donald B. Johnson,et al.  The Complexity of Selection and Ranking in X+Y and Matrices with Sorted Columns , 1982, J. Comput. Syst. Sci..

[2]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[3]  Hans Ziegler,et al.  Solving certain singly constrained convex optimization problems in production planning , 1982, Oper. Res. Lett..

[4]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[5]  Toshihide Ibaraki,et al.  Resource allocation problems - algorithmic approaches , 1988, MIT Press series in the foundations of computing.

[6]  P. Brucker.  Review of recent development: An O( n) algorithm for quadratic knapsack problems , 1984 .

[7]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[8]  Dorit S. Hochbaum,et al.  Strongly Polynomial Algorithms for the Quadratic Transportation Problem with a Fixed Number of Sources , 1994, Math. Oper. Res..

[9]  Eugene L. Lawler,et al.  Parameterized Approximation Scheme for the Multiple Knapsack Problem , 2009, SIAM J. Comput..

[10]  Harvey M. Salkin,et al.  A branch and search algorithm for a class of nonlinear knapsack problems , 1983 .

[11]  R. E. Marsten,et al.  An Algorithm for Nonlinear Knapsack Problems , 1976 .

[12]  Siddhartha S. Syam,et al.  A Branch and Bound Algorithm for Integer Quadratic Knapsack Problems , 1995, INFORMS J. Comput..

[13]  Bala Shetty,et al.  The Nonlinear Resource Allocation Problem , 1995, Oper. Res..

[14]  Hanan Luss,et al.  Technical Note - Allocation of Effort Resources among Competing Activities , 1975, Oper. Res..

[15]  Thomas L. Morin,et al.  A hybrid approach to discrete mathematical programming , 2015, Math. Program..

[16]  Dorit S. Hochbaum,et al.  Lower and Upper Bounds for the Allocation Problem and Other Nonlinear Optimization Problems , 1994, Math. Oper. Res..

[17]  J. George Shanthikumar,et al.  Convex separable optimization is not much harder than linear optimization , 1990, JACM.

[18]  L. Khachiyan,et al.  The polynomial solvability of convex quadratic programming , 1980 .