Interwoven SDP in Primal-Dual Proximal Splitting Methods for Adjustable Robust Convex Optimisation with SOS-Convex Polynomial Constraints
We propose a novel methodology for solving a two-stage adjustable robust convex optimisation problem with a general (proximable) convex objective function and constraints defined by sum-of-squares (SOS) convex polynomials. These problems appear in many decision-making applications. However, they are challenging to solve and typically cannot be reformulated as numerically tractable convex optimisation models, such as conic linear programs, that can be solved directly using existing software. We show that the robust problem admits an equivalent representation as a convex composite unconstrained optimisation model that preserves the same objective values, under quadratic decision rules on the adjustable decision variables. Building on this reformulation, we develop a tailored first-order primal-dual proximal splitting method. By leveraging semidefinite programming (SDP) techniques as well as tools from convex analysis and real algebraic geometry, we establish its theoretical properties, including computable SDP-based formulas for projections onto closed convex sets, specified by SOS-convex polynomial inequalities. Numerical experiments on a two-stage lot-sizing model with both linear as well as SOS-convex polynomial storage costs under demand uncertainty demonstrate the effectiveness and applicability of the proposed approach. Our approach enables the incorporation of SDP techniques into a primal-dual proximal splitting framework, thereby broadening the class of problems to which these methods can be effectively applied.