The worst-case linear optimization (WCLO) with uncertainties in the right-hand-side of the constraints often arises from numerous applications such as systemic risk estimate in finance and stochastic optimization, which is known to be NP-hard. In this paper, we investigate the efficient global algorithm for WCLO based on its nonlinear semidefinite relaxation (SDR). We first derive an enhanced nonlinear SDR for WCLO via secant cuts and RLT approaches. A secant search algorithm is then proposed to solve the nonlinear SDR and its global convergence is established. Second, we propose a new global algorithm for WCLO, which integrates the nonlinear SDR with successive convex optimization method, initialization and branch-and-bound, to find a globally optimal solution to the underlying WCLO within a pre-specified \(\epsilon\)-tolerance. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to the WCLO instances.

约束右侧具有不确定性的最坏情况线性优化 (WCLO) 通常来自众多应用，例如金融中的系统性风险估计和随机优化，这被称为 NP-hard。在本文中，我们研究了基于非线性半定松弛 (SDR) 的 WCLO 的有效全局算法。我们首先通过割线切割和 RLT 方法为 WCLO 推导出增强的非线性 SDR。然后提出一种割线搜索算法来求解非线性SDR并建立其全局收敛性。其次，我们为 WCLO 提出了一种新的全局算法，它将非线性 SDR 与连续凸优化方法、初始化和分支定界相结合，以在预先指定的\(\epsilon\ )-宽容。我们建立算法的全局收敛性并估计其复杂度。初步的数值结果表明，所提出的算法可以有效地找到 WCLO 实例的全局最优解。