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The structure of many real-world optimization problems includes minimizationof a nonlinear (or quadratic) functional subject to bound and singly linearconstraints (in the form of either equality or bilateral inequality) which arecommonly called as continuous knapsack problems. Since there are efficientmethods to solve large-scale bound constrained nonlinear programs, it isdesirable to adapt these methods to solve knapsack problems, while preservingtheir efficiency and convergence theories. The goal of this paper is tointroduce a general framework to extend a box-constrained optimization solverto solve knapsack problems. This framework includes two main ingredients whichare O(n) methods; in terms of the computational cost and required memory; forthe projection onto the knapsack constrains and the null-space manipulation ofthe related linear constraint. The main focus of this work is on the extensionof Hager-Zhang active set algorithm (SIAM J. Optim. 2006, pp. 526--557). Themain reasons for this choice was its promising efficiency in practice as wellas its excellent convergence theories (e.g., superlinear local convergence ratewithout strict complementarity assumption). Moreover, this method does not useany explicit form of second order information and/or solution of linear systemsin the course of optimization which makes it an ideal choice for large-scaleproblems. Moreover, application of Birgin and Mart{\'\i}nez active setalgorithm (Comp. Opt. Appl. 2002, pp. 101--125) for knapsack problems is alsobriefly commented. The feasibility of the presented algorithm is supported bynumerical results for topology optimization problems.

Active Set Algorithm for Large-Scale Continuous Knapsack Problems with Application to Topology Optimization Problems