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Amir Ali Ahmadi (a_a_aprinceton.edu) Abstract: In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellens\"atze from the late 20th century (e.g., due to Stengle, Putinar, or Schm\"udgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellens\"atze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programmingbased hierarchy that relies solely on Artin's solution to Hilbert's 17th problem. We also use a classical result of Poly\'a on global positivity of even forms to construct an ``optimizationfree'' converging hierarchy for general polynomial minimization problems with compact feasible sets. This hierarchy only requires polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming and secondorder cone programmingbased hierarchies for polynomial minimization problems that rely on the recently introduced concepts of dsos (diagonally dominant sum of squares) and sdsos (scaled diagonally dominant sum of squares) polynomials. We remark that the scope of this paper is theoretical at this stage as our hierarchiesthough they involve at most two sum of squares constraints or only elementary arithmetic at each levelrequire the use of bisection and increase the number of variables (resp. degree) of the problem by the number of inequality constraints plus three (resp. by a factor of two). Keywords: Positivstellensatze, polynomial optimization, convex optimization Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Linear, Cone and Semidefinite Programming Category 3: Global Optimization Citation: Download: [PDF] Entry Submitted: 10/09/2017 Modify/Update this entry  
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