## Abstract

We present a global algorithm for indefinite knapsack separable quadratic programs with bound constraints. The upper bounds on variables with nonconvex terms are assumed to be infinite in the algorithmic development. By characterizing optimal solutions of the problem, we enumerate a subset of KKT points to determine a global optimum. The enumeration is made efficient by developing a theory for shrinking and partitioning the search domain of KKT multipliers. The global algorithmic procedure is developed based on interval and point testing techniques that have roots in solving strictly convex problems. Our algorithm has quadratic (worst-case) complexity and its performance is demonstrated on a broad range of randomly generated problem instances and comparisons with existing global and local solvers. It turns out that our method can solve these nonconvex problems of extremely large size in a fraction of the time (relative to commercial software such as CPLEX 12.6) on a desktop computer. Furthermore, a feasible upper bounding scheme is developed for the case when the variables with nonconvex terms are allowed to have finite upper limits, using an iterative application of the above global algorithm, and the computational experience is presented.

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## Notes

- 1.
If some \(a_k<0\), then perform the transformation \(x_k\leftarrow -x_k,\,a_k\leftarrow -a_k,\,c_k\leftarrow -c_k,\,l_k\leftarrow -u_k\), and \(u_k\leftarrow -l_k\). If \(a_k=0\), then solve (QP) without \(x_k\) and the optimal value of \(x_k\) is determined by the univariate minimization: \(\min \{0.5d_kx^2_k-c_kx_k\;:\; l_k\le x_k \le u_k \}\).

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## Acknowledgments

Without implicating them, we sincerely thank Stephen Vavasis and Samuel Burer for sharing their computer codes, and Yu-Hong Dai and Roger Fletcher for clarifying certain aspects of their papers. We are also indebted to the anonymous referees for their careful and insightful reviews. All remaining errors in this paper are ours.

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Edirisinghe, C., Jeong, J. An efficient global algorithm for a class of indefinite separable quadratic programs.
*Math. Program.* **158, **143–173 (2016). https://doi.org/10.1007/s10107-015-0918-x

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### Keywords

- Separable quadratic programs
- Indefinite knapsack problems
- Global optimization
- Polynomial complexity

### Mathematics Subject Classification

- 90C20
- 90C26