This paper presents exact Semi-Definite Program (SDP) reformulations for infinite-dimensional moment optimization problems involving a new class of piecewise Sum-of-Squares (SOS)-convex functions and projected spectrahedral support sets. These reformulations show that solving a single SDP finds the optimal value and an optimal probability measure of the original moment problem. This is done by establishing an SOS representation for the non-negativity of a piecewise SOS-convex function over a projected spectrahedron. Finally, as an application and a proof-of-concept illustration, the paper presents numerical results for the Newsvendor and revenue maximization problems with higher-order moments by solving their equivalent SDP reformulations. These reformulations promise a flexible and efficient approach to solving these models. The main novelty of the present work in relation to the recent research lies in finding the solution to moment problems, for the first time, with piecewise SOS-convex functions from their numerically tractable exact SDP reformulations.
B$^3$-PWL: GPU-Batched Branch-and-Bound for Piecewise-Linear Optimization with SOS2 Constraints
Johns Hopkins University（约翰斯·霍普金斯大学）；University of North Carolina, Chapel Hill（北卡罗来纳大学教堂山分校）；City University of Hong Kong（香港城市大学）；The Hong Kong Institute of AI for Science（香港人工智能科学研究所）
Optimal triangulations for piecewise linear approximations of non-convex variable products
Fraunhofer Institute for Integrated Circuits IIS（弗劳恩霍夫集成电路研究所）；Grado Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学格拉多工业与系统工程系）
Volume of quasi-homogeneous sublevel sets: Two linear algebra deterministic algorithms with convergence rates
LAAS-CNRS, University of Toulouse（法国图卢兹大学 LAAS-CNRS）；Faculty of Electrical Engineering, Czech Technical University in Prague（捷克布拉格捷克理工大学电气工程学院）；Toulouse School of Economics（法国图卢兹经济学院）
