## Abstract

Non-convex knapsack separable quadratic optimization with compact box constraints is an NP-hard problem. We present tight lower and upper bounding procedures that are computationally-efficient as the problem size grows. The lower bound is based on Lagrangian relaxation, and it is computed in linear-time. When the bound is not an exact global solution, a worst-case bound-quality measure is developed. Moreover, the lower bounding (LB) solution is improved to construct a feasible solution, leading to an upper bound (UB) on the given problem. The UB is based on fixing the convex variables and a subset of non-convex and linear variables at the LB solution, and considering the remaining problem, which is always feasible. The UB is computable with linear-time complexity, and it is the global optimum in certain verifiable cases when the duality gap is zero. In our limited computational experiments, the UB has very small relative gap with an exact global optimum solution when there is a nonzero duality gap. Performance of the bounds is demonstrated through a broad range of randomly generated problem instances and comparisons with existing global and non-global solvers. The proposed method on these indefinite problems of extremely large size is an order of magnitude faster than the alternative solvers, including IBM’s commercial software Cplex12.6.

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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: \(\min \{0.5d_kx^2_k-c_kx_k\;:\; l_k\le x_k \le u_k \}\).

- 2.
The code was compiled into MATLAB with Mex codes shared at http://www.mathworks.com/matlabcentral/fileexchange/29453-nth-element by Peter Li. The ISO International Standard ISO/IEC 14882 guarantees the linear time median search of

*nth-element*function.

## References

- 1.
Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Network Flows, 4th edn. Wiley, New York (2009)

- 2.
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Applications, 3rd edn. Wiley, New York (2006)

- 3.
Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput.

**4**(1), 33–52 (2012) - 4.
Cominetti, R., Mascarenhas, W., Silva, P.: A Newton’s method for the continuous quadratic knapsack problem. Math. Program. Comput.

**6**(2), 151–169 (2014) - 5.
Dai, Y.H., Fletcher, R.: New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds. Math. Program.

**106**(3), 403–421 (2006) - 6.
Edirisinghe, C., Jeong, J.: An efficient global algorithm for a class of indefinite separable quadratic programs. Math. Program.

**158**(1), 143–173 (2016) - 7.
Floyd, R.W., Rivest, R.L.: Algorithm 489: the algorithm select for finding the i th smallest of n elements. Commun. ACM

**18**(3), 173 (1975) - 8.
Martello, S., Pisinger, D., Toth, P.: Dynamic programming and strong bounds for the 0–1 knapsack problem. Manag. Sci.

**45**(3), 414–424 (1999) - 9.
Moré, J.J., Vavasis, S.A.: On the solution of concave knapsack problems. Math. Program.

**49**(1), 397–411 (1991) - 10.
Nakagawa, Y., James, R.J.W., Rego, C., Edirisinghe, C.: Entropy-based optimization of nonlinear separable discrete decision models. Manag. Sci.

**60**(3), 695–707 (2014) - 11.
Patriksson, M., Strömberg, C.: Algorithms for the continuous nonlinear resource allocation problem—new implementations and numerical studies. Eur. J. Oper. Res.

**243**(3), 703–722 (2015) - 12.
Pisinger, D.: Where are the hard knapsack problems? Comput. Oper. Res.

**32**(9), 2271–2284 (2005) - 13.
Robinson, A.G., Jiang, N., Lerme, C.S.: On the continuous quadratic knapsack-problem. Math. Program.

**55**(1), 99–108 (1992) - 14.
Vavasis, S.A.: Local minima for indefinite quadratic knapsack-problems. Math. Program.

**54**(2), 127–153 (1992)

## Acknowledgements

We sincerely thank the anonymous referees for their thorough and insightful reviews and suggestions to improve the exposition of the paper.

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Edirisinghe, C., Jeong, J. Tight bounds on indefinite separable singly-constrained quadratic programs in linear-time.
*Math. Program.* **164, **193–227 (2017). https://doi.org/10.1007/s10107-016-1082-7

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### Keywords

- Separable quadratic optimization
- Indefinite knapsack problems
- Lower and upper bounds
- Linear-time complexity

### Mathematics Subject Classification

- 90C20
- 90C26