Piecewise SOS-Convex Moment Optimization and Applications via Exact Semi-Definite Programs
arxiv(2024)
摘要
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 also 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.
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