Deciding reachability of the infimum of a multivariate polynomial

Let <i>f</i> ∈ Q[<i>X</i>_1,...,<i>X<sub>n</sub></i>] be of degree <i>D</i>. Algorithms for solving the unconstrained global optimization problem <i>f</i>*=inf<sub><i>x</i></sub>∈<i>R<sup>n</sup></i> <i>f</i>(<i>x</i>) are of first importance since this problem appears frequently in numerous applications in engineering sciences. This can be tackled by either designing appropriaten quantifier elimination algorithms or by certifying lower bounds on <i>f</i>* by means of sums of squares decompositions but there is no efficient algorithm for deciding if <i>f</i>* is a minimum. This paper is dedicated to this important problem. We design a probabilistic algorithm that decides, for a given <i>f</i> and the corresponding <i>f</i>*, if <i>f</i>* is reached over <i>R<sup>n</sup></i> and computes a point <i>x</i>* ∈ <i>R<sup>n</sup></i> such that <i>f</i>(<i>x</i>*)=<i>f</i>* if such a point exists. This algorithm makes use of algebraic elimination algorithms and real root isolation. If <i>L</i> is the length of a straight-line program evaluating <i>f</i>, algebraic elimination steps run in <i>O</i>(log(<i>D</i>-1)<i>n</i><sup>6</sup>(<i>nL</i>+<i>n</i><sup>4</sup>)<i>U</i>((<i>D</i>-1)<sup><i>n</i>+1</sup>)<sup>3</sup>) arithmetic operations in <i>Q</i> where <i>D</i>=deg(<i>f</i>) and <i>U</i>(<i>x</i>)=<i>x</i>(log (<i>x</i>))<sup>2</sup> log log (<i>x</i>). Experiments show its practical efficiency.

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