We consider the chance-constrained binary knapsack problem (CKP), where the item weights are independent and normally distributed. We introduce a continuous relaxation for the CKP, represented as a non-convex optimization problem, which we call the non-convex relaxation. A comparative study shows that the non-convex relaxation provides an upper bound for the CKP, at least as tight as those obtained from other continuous relaxations for the CKP. Furthermore, the quality of the obtained upper bound is guaranteed to be at most twice the optimal objective value of the CKP. Despite its non-convex nature, we show that the non-convex relaxation can be solved in polynomial time. Subsequently, we proposed a polynomial-time 1/2-approximation algorithm for the CKP based on this relaxation, providing a lower bound for the CKP. Computational test results demonstrate that the non-convex relaxation and the proposed approximation algorithm yields tight lower and upper bounds for the CKP within a short computation time, ensuring the quality of the obtained bounds.

翻译：本文考虑物品重量服从独立正态分布的机会约束二元背包问题（CKP）。我们针对该问题提出一种连续松弛方法，将其转化为非凸优化问题并命名为非凸松弛。对比研究表明，该非凸松弛为CKP提供的上界，其紧性至少不弱于其他连续松弛方法。此外，该上界质量有理论保证：其值不超过CKP最优目标值的两倍。尽管具有非凸特性，我们证明该非凸松弛可在多项式时间内求解。进一步地，基于该松弛方法，我们提出了一个多项式时间的1/2近似算法，为CKP提供下界。计算实验结果表明，该非凸松弛与所提出的近似算法能在短时间内为CKP提供紧致的上下界，有效保证了所得界的质量。