Monomial asymptotic polynomials and applications to polynomial optimization problems
Email:
dutrang1986@utc.edu.vn
Từ khóa:
polynomial optimization, semi-algebraic, asymptotic, sum of squares
Tóm tắt
The problem of polynomial optimization plays an important role in many fields such as physics, chemistry, and economics. This problem has received research attention from many mathematicians recently. In this paper, we study the polynomial optimization problem over a non-compact semi-algebraic set, for which its constraint set of polynomials G is asymptotic with a finite family of monomials. By changing variables via a suitable monomial mapping, we transform the problem under consideration into the polynomial optimization problem over a compact semi-algebraic feasible set. We then apply the well-known result that the optimal value of a polynomial over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semi-definite programs and the convergence is finite generically, to obtain results in the general case when the cone C(G) is unimodular. In particular, in the case of polynomials in two variables, we solve the problem quite completely without requiring C(G) to be unimodularTài liệu tham khảo
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[10]. D. T. T. Trang, H. M. Toan, Polynomial optimization on some unbounded closed semi-algebraic sets, J. Optim. Theory Appl., 183 (2019) 352-363. https://doi.org/10.1007/s10957-019-01544-5
[11]. D. T. T. Trang, H. M. Toan, N. T. Hong, Algebra of polynomials bounded on a semi-algebraic set [f≤r], Acta Mathematica Vietnamica, 46 (2021) 821-838. https://doi.org/10.1007/s40306-021-00422-5
[2]. Y. Nesterov, Squared functional systems and optimization problems, in: H. Frenk (Eds.), High Performance Optimization. Applied Optimization, first ed., Springer, Boston, 2000, pp. 405-440.
[3]. J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001) 796-817. https://doi.org/10.1137/S1052623400366802
[4]. K. Schmüdgen, The K-moment problem for compact semi-algebraic sets, Math. Ann., 289 (1991) 203-206. https://doi.org/10.1007/BF01446568
[5]. M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42 (1993) 969-984. https://iumj.org/article/3638/
[6]. J. B. Lasserre, Moments, Positive Polynomials and their Applications, first ed., Imperial College Press, London, 2010.
[7]. M. Marshall, Positive Polynomials and Sum of Squares. Mathematical Surveys and Monographs, first ed., American Mathematical Society, Providence, 2008.
[8]. H. V. Ha, T. S. Pham, Genericity in Polynomial Optimization, first ed., World Scientific Publishing Europe Ltd., London, 2017.
[9]. V. Jeyakumar, J. B. Lasserre, G. Li, On polynomial optimization over non-compact semi-algebraic sets, J. Optim. Theory Appl., 163 (2014) 707-718. https://doi.org/10.1007/s10957-014-0545-3
[10]. D. T. T. Trang, H. M. Toan, Polynomial optimization on some unbounded closed semi-algebraic sets, J. Optim. Theory Appl., 183 (2019) 352-363. https://doi.org/10.1007/s10957-019-01544-5
[11]. D. T. T. Trang, H. M. Toan, N. T. Hong, Algebra of polynomials bounded on a semi-algebraic set [f≤r], Acta Mathematica Vietnamica, 46 (2021) 821-838. https://doi.org/10.1007/s40306-021-00422-5
Tải xuống
Chưa có dữ liệu thống kê
Nhận bài
05/06/2024
Nhận bài sửa
04/09/2024
Chấp nhận đăng
10/09/2024
Xuất bản
15/09/2024
Chuyên mục
Công trình khoa học
Kiểu trích dẫn
Du Thi Thu, T. (1726333200). Monomial asymptotic polynomials and applications to polynomial optimization problems. Tạp Chí Khoa Học Giao Thông Vận Tải, 75(7), 2093-2103. https://doi.org/10.47869/tcsj.75.7.7
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