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DOI: 10.23952/cot.2023.31
Received November 20, 2022; Accepted March 7, 2023; Published June 20, 2023
Abstract. In this paper, we study the bi-form optimization problem over the generalized spheres. Since bi-form optimization problem includes bi-quadratic optimization as a special case, it is NP-Hard in general. Nevertheless, we characterize the primal and the dual problems associated to the bi-form optimization problem and show that there is no duality gap between them. Based on tensor analysis and the dual characterization, we propose some methods to solve the bi-form optimization problem approximately. Particularly, we analyze the sums of powers of tensor polynomials and give a Positivstellensatz for the bi-form optimization over the spheres. We also show that there is a class of bi-form optimization problems which can be solved in polynomial-time. For a bi-form problem with nonpositive coefficients, we present a globally convergent algorithm which can compute an approximation solution with an explicit approximation ratio in terms of the degree of the bi-form and the orders of the norms.
How to Cite this Article:
S. Hu, Some properties of bi-form optimization over generalized spheres, Commun. Optim. Theory 2023 (2023) 31.