Global optimization with polynomials and the problem of moments. (English) Zbl 1010.90061

Summary: We consider the problem of finding the unconstrained global minimum of a real-valued polynomial \(p(x): \mathbb R^n\to\mathbb R\), as well as the global minimum of \(p(x)\), in a compact set \(K\) defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality problems. A notion of Karush–Kuhn–Tucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided.


90C26 Nonconvex programming, global optimization
90C22 Semidefinite programming
13J30 Real algebra
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