An efficient global algorithm for worst-case linear optimization under uncertainties based on nonlinear semidefinite relaxation

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.