Variable Fixing Algorithms for the Continuous Quadratic Knapsack Problem

Abstract We study several variations of the Bitran–Hax variable fixing method for the continuous quadratic knapsack problem. We close the gaps in the convergence analysis of several existing methods and provide more efficient versions. We report encouraging computational results for large-scale problems.

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

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

[3]  Arnoldo C. Hax,et al.  On the Solution of Convex Knapsack Problems with Bounded Variables. , 1977 .

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

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

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

[7]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[23]  N. Maculan,et al.  An O(n) Algorithm for Projecting a Vector on the Intersection of a Hyperplane and a Box in Rn , 2003 .

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

[25]  K. Kiwiel.  On Linear-Time Algorithms for the Continuous Quadratic Knapsack Problem , 2007 .