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Semidefinite representations for finite varieties. (English) Zbl 1152.90007

This paper studies the following problem: \[ \text{inf\,}f(x)\text{ such that }h_1(x)= 0,\dots, h_n(x)= 0,\;h_{n+1}(x)\geq 0,\dots, h_{n+k}(x)\geq 0, \] where \(f,h_1,\dots, h_{n+k}\in R[x_1,x_2,\dots, x_n]\). That is the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. It is a central problem in combinational optimization and optimization of a linear or polynomial function over a basic closed semi-algebraic set. When the polynomial equations have a finite set of complex solutions, this problem can be reformulated as a semidefinite problem.
In this paper, the semidefinite represent action involve combinational moment matrix, which are matrix induced by a basic of quotient vector space \(R[x_1,x_2,\dots, x_n]/I\), where \(I\) is the ideal generated by polynomial equations in the problem. This paper proves the finite convergence of a hierarchy of semidefinite relaxation introduced by Lasserre. Semidefinite approximation can be constructed by considering truncated combinatorial, moment, matrices, rank conditions are given in a grid case that ensures the approximation solve the original problem is optimality.

MSC:

90C30 Nonlinear programming
90C22 Semidefinite programming

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References:

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