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Bernstein Polynomial Relaxations for Polynomial Optimization Problems

In this paper, we examine linear programming (LP) relaxations based on Bernstein polynomials for polynomial optimization problems (POPs). We present a progression of increasingly more precise LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. The well-known bounds on Bernstein polynomials over the unit box combined with linear inter-relationships between Bernstein polynomials help us formulate "Bernstein inequalities" which yield tighter lower bounds for POPs in bounded rectangular domains. The results can be easily extended to optimization over polyhedral and semi-algebraic domains. We also examine techniques to increase the precision of these relaxations by considering higher degree relaxations, and a branch-and-cut scheme