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...uares (SOS) representations, was the motivation for providing other SOS certificates and yielded the sufficient conditions of [4] and subsequently of [2, 5]. And in a recent work Ghasemi and Marshall =-=[6]-=- have shown how to compute a lower bound on the global optimum of a multivariate polynomial on R n , by solving a certain geometric program. This formulation as a geometric program is based on the suf...

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...mial nonnegativity that are perhaps stronger than a SOS decompostion, but still provide useful solutions for various control applications. A previous approach that has a similar spirit is the work in =-=[18]-=-, which tackles the specific task of finding a lower bound on the minimum of a polynomial using geometric programming (GP). However, these GP-based conditions seem to be too strong and we show in [1] ...

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...e in some sense, these techniques exploit special sparsity structure of POPs to reduce the number of variables and the size of the matrix variable in the SDP (2). Recent work in this direction, e.g., =-=[6, 7]-=- also exploit special structure of POPs to solve larger sized problems. Nie and Wang [24] proposes a use of regularization method for solving SDP relaxation problems instead of primal-dual interior-po...

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... problem with generalized lm-type constraint was studied in [51] and a geometric programming relaxation problem was proposed to calculate the lower bound of the optimal value of the problem (see also =-=[52]-=-). It was demonstrated that the lower bound provided by the geometric programming relaxation is a lower bound of the sumof-squares relaxation and can be more efficient from the computational point of ...

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...(f0 − r)x 2d 0 + n∑ j=1 fα(j)x α(j)x 2d−|α(j)| 0 + ∑ α∈∆(f) fαx αx 2d−|α| 0 . Then f − r is a SONC resp. a sum of binomial squares if only only if f − r is a SONC resp. a sum of binomial squares (see =-=[6]-=-). Our sufficient conditions in Theorem 3.1 now read as following. (1’) |fα| = ∏n j=0 ( aα,j λ (α) j )λ(α)j = ( aα,0 λ (α) 0 )λ(α)0 ∏n j=1 ( aα,j λ (α) j )λ(α)j , (2’) fα(j) ≥ ∑ α∈∆(f) aα,j for all 1 ...

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...ffective certificates of global nonnegativity for certain “sparse” polynomials of large degree, which are sometimes referred to as “fewnomials” in the literature [11]; this point is also exploited in =-=[10]-=-, where the authors employ GPs to compute lower bounds on certain multivariate polynomials more efficiently than SDPs based on sum-of-squares techniques. In fact one can combine the SDP and the relati...

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...te programming. Here, g0 := 1 and ∑ R[x]2 denotes the set of elements of R[x] which are sums of squares. Denote by d the least even integer ≥ max{2,deg(f),deg(gj) : j = 1, . . . ,m}. The algorithm in =-=[5]-=- deals with the case m = 0, producing a lower bound fgp for f on Rn computable by geometric programming. See [2], [4] and [7] for precursors of [5]. The algorithm 2000 Mathematics Subject Classificati...

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...e in some sense, these techniques exploit special sparsity structure of POPs to reduce the number of variables and the size of the matrix variable in the SDP (2). Recent work in this direction, e.g., =-=[6, 7]-=- also exploit special structure of POPs to solve larger sized problems. Nie and Wang [24] proposes a use of regularization method for solving SDP relaxation problems instead of primal-dual interior-po...