Breakpoint searching algorithms for the continuous quadratic knapsack problem

We give several linear time algorithms for the continuous quadratic knapsack problem. In addition, we report cycling and wrong-convergence examples in a number of existing algorithms, and give encouraging computational results for large-scale problems.

[1]  R. Cottle,et al.  A Lagrangean relaxation algorithm for the constrained matrix problem , 1986 .

[2]  Krzysztof C. Kiwiel,et al.  On Floyd and Rivest's SELECT algorithm , 2005, Theor. Comput. Sci..

[3]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[4]  K. Kiwiel.  Variable Fixing Algorithms for the Continuous Quadratic Knapsack Problem , 2008 .

[5]  P. Pardalos.  Complexity in numerical optimization , 1993 .

[6]  Jose A. Ventura,et al.  Computational development of a lagrangian dual approach for quadratic networks , 1991, Networks.

[7]  Dorit S. Hochbaum,et al.  About strongly polynomial time algorithms for quadratic optimization over submodular constraints , 1995, Math. Program..

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

[9]  Stavros A. Zenios,et al.  Massively Parallel Algorithms for Singly Constrained Convex Programs , 1992, INFORMS J. Comput..

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

[11]  C. Michelot.  A finite algorithm for finding the projection of a point onto the canonical simplex of ∝n , 1986 .

[12]  A. G. Robinson,et al.  On the continuous quadratic knapsack problem , 1992, Math. Program..

[13]  Bala Shetty,et al.  A Parallel Projection for the Multicommodity Network Model , 1990 .

[14]  Panos M. Pardalos,et al.  An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds , 1990, Math. Program..

[15]  Arnoldo C. Hax,et al.  Disaggregation and Resource Allocation Using Convex Knapsack Problems with Bounded Variables , 1981 .

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

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

[18]  Siddhartha S. Syam,et al.  A Projection Method for the Integer Quadratic Knapsack Problem , 1996 .

[19]  Geraldo Galdino de Paula,et al.  A linear-time median-finding algorithm for projecting a vector on the simplex of Rn , 1989 .

[20]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[21]  P. Berman,et al.  Algorithms for the Least Distance Problem , 1993 .

[22]  Michel Minoux,et al.  A $O(n)$ algorithm for projecting a vector on the intersection of a hyperplane and $R^n_+$ , 1997 .

[23]  Bala Shetty,et al.  Quadratic resource allocation with generalized upper bounds , 1997, Oper. Res. Lett..

[24]  J. J. Moré,et al.  Quasi-Newton updates with bounds , 1987 .

[25]  Jeffery L. Kennington,et al.  A polynomially bounded algorithm for a singly constrained quadratic program , 1980, Math. Program..

[26]  Paul H. Zipkin,et al.  Simple Ranking Methods for Allocation of One Resource , 1980 .

[27]  Nimrod Megiddo,et al.  Linear time algorithms for some separable quadratic programming problems , 1993, Oper. Res. Lett..