Copositive reformulations can be used to derive convex underestimators of optimal value functions of QCQPs, which can be dualized to derive piecewise affine approximations. We apply this machinery to the so-called radial OPF problem. First, we reformulate the problem via indicator functions, which can be cast as optimal value functions. Using the copositivity-based convex underestimators we can derive a family of convex relaxations that are at least as strong as the popular positive semidefinite relaxation. We use a Benders Decomposition method for solving the resulting relaxations which require solving copositive subproblems. These are tackled via a cutting plane algorithm where a copositivity test with respect to a special quadratic cone has to be performed in each iteration. Due to the structure of said quadratic cone, this test can be performed in polynomial time.

The standard quadratic optimization problem (StQP) consists of minimizing a quadratic form over the standard simplex. Without convexity or concavity of the quadratic form, the StQP is NP-hard. This problem has many interesting applications. Sometimes, the data matrix is uncertain. We investigate models where the distribution of the data matrix is known but where both the StQP after realization of the data matrix and the here-and-now problem are indefinite.

In this talk, we present exact Semi-Definite Program (SDP) reformulations for a class of moment optimisation problems involving piecewise Sums-of-Squares (SOS) convex functions and projected spectrahedral support sets. These moment problems cover real-world applications such as the newsvendor and revenue maximisation problems with higher-order moments. Generally, solving moment problems is computationally intractable because evaluating a multi-dimensional integral for the expectations and searching through the infinite-dimensional space of probability distributions is numerically hard.

Our approach involves establishing an SOS representation for the non-negativity of a piecewise SOS-convex function on a projected spectrahedron and employing conic program duality. We show that both the optimal value and an optimal probability measure of the original moment problem can be found by solving a single SDP, which can be solved efficiently using commonly available software.

This talk is based on the articles by Huang, Q. Y., Jeyakumar, V., and Li, G. (2024). Exact Semi-Definite Programs for Piecewise SOS-Convex Moment Optimisation and Applications. arXiv preprint: https://arxiv.org/abs/2402.07064
Huang, Q. Y., and Jeyakumar, V. (2024). A distributional Farkas’ lemma and moment optimisation problems with no-gap dual semi-definite programs. Optimisation Letters. DOI: https://doi.org/10.1007/s11590-024-02097-x