This paper presents exact dual semi-definite programs (SDPs) for robust SOS-convex polynomial optimization problems with affinely adjustable variables in the sense that the optimal values of the robust problem and its associated dual SDP are equal with the solution attainment of the dual problem. This class of robust convex optimization problems includes the corresponding quadratically constrained convex quadratic optimization problems and separable convex polynomial optimization problems, and it employs a general bounded spectrahedron uncertainty set that covers the most commonly used uncertainty sets of numerically solvable robust optimization models, such as boxes, balls and ellipsoids. As special cases, it also demonstrates that explicit exact dual SDP and second-order cone programming (SOCP) in terms of original data hold for the robust two-stage convex quadratic programs with quadratic constraints and the robust two-stage separable convex quadratic programs under an ellipsoidal uncertainty set, respectively. Finally, the paper illustrates the results via numerical implementations of the developed SDP duality scheme on adjustable robust lot-sizing problems with nonlinear costs under demand uncertainty.

本文提出了具有仿射可调变量的稳健 SOS 凸多项式优化问题的精确对偶半定规划 (SDP)，即稳健问题及其相关对偶 SDP 的最优值等于对偶问题的解获得. 这类鲁棒凸优化问题包括对应的二次约束凸二次优化问题和可分离凸多项式优化问题，它采用了一个通用的有界谱面体不确定集，涵盖了数值可解鲁棒优化模型中最常用的不确定集，例如框、球和椭球。作为特殊情况，它还表明，就原始数据而言，显式精确对偶 SDP 和二阶锥规划（SOCP）适用于具有二次约束的稳健两阶段凸二次规划和椭球不确定集下的稳健两阶段可分离凸二次规划， 分别。最后，本文通过对具有非线性成本的可调节鲁棒批量问题在需求不确定性下的 SDP 对偶方案的数值实现来说明结果。