A linear programming reformulation of the standard quadratic optimization problem
Journal of Global Optimization
Global minimization of rational functions and the nearest GCDs
Journal of Global Optimization
Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Expressing combinatorial problems by systems of polynomial equations and hilbert's nullstellensatz
Combinatorics, Probability and Computing
Theta Bodies for Polynomial Ideals
SIAM Journal on Optimization
An iterative scheme for valid polynomial inequality generation in binary polynomial programming
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Polynomial Matrix Inequality and Semidefinite Representation
Mathematics of Operations Research
Computing infeasibility certificates for combinatorial problems through Hilbert's Nullstellensatz
Journal of Symbolic Computation
SIAM Journal on Optimization
Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs
SIAM Journal on Optimization
Exploiting equalities in polynomial programming
Operations Research Letters
Moment matrices, border bases and real radical computation
Journal of Symbolic Computation
Exploiting Symmetries in SDP-Relaxations for Polynomial Optimization
Mathematics of Operations Research
A Semidefinite Programming approach for solving Multiobjective Linear Programming
Journal of Global Optimization
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We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x 1, . . . ,xn]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality.