Abstract
The nonlinear multidimensional knapsack problem is defined as the minimization of a convex function with multiple linear constraints. The methods developed for nonlinear multidimensional programming problems are often applied to solve the nonlinear multidimensional knapsack problems, but they are inefficient or limited since most of them do not exploit the characteristics of the knapsack problems. In this paper, by establishing structural properties of the continuous separable nonlinear multidimensional knapsack problem, we develop a multi-tier binary solution method for solving the continuous nonlinear multidimensional knapsack problems with general structure. The computational complexity is polynomial in the number of variables. We presented two examples to illustrate the general application of our method and we used statistical results to show the effectiveness of our method.
Highlights
The nonlinear multidimensional knapsack problem is defined as minimizing a convex function with multiple linear constraints
The methods developed for nonlinear multidimensional programming problems are often applied to solve the nonlinear multidimensional knapsack problems, but they are inefficient or limited since most of them do not exploit the characteristics of the knapsack problems
The nonlinear knapsack problem is a class of nonlinear programming, and some methods designed for nonlinear programming can be applied for solving the nonlinear multidimensional knapsack problems
Summary
The nonlinear multidimensional knapsack problem is defined as minimizing a convex function with multiple linear constraints. The nonlinear knapsack problem is a class of nonlinear programming, and some methods designed for nonlinear programming can be applied for solving the nonlinear multidimensional knapsack problems. [9] and the filter method [10] [11] These methods are designed for nonlinear programming problems, and some of them are inefficient or limited for solving the nonlinear knapsack problems since they do not consider the characteristics of the knapsack problems. Two basic specialized methods are mainly applied for solving the single-constraint nonlinear knapsack problem. Zhang and Hua developed a united method for solving a class of continuous separable nonlinear knapsack problems [16]. Sharkey et al studied a general class of nonlinear non-separable continuous knapsack problem [18]
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