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... Sturmfels [23] proposed to use SOS representations of f(x) − γ modulo the gradient ideal of f(x), and show that the minimum value f ∗ can be obtained when f ∗ is attained at some point. Schweighofer =-=[31]-=- proposes to minimize f(x) over a semialgebraic set called the gradient tentacle, and shows that the minimum value f ∗ can be computed when f ∗ > −∞ but not attainable. Jibetean and Laurent [13], and ...

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...emark that the assumption that f achieves a minimum value on R n is restrictive. The minimum value of f on Z(I) need not equal the infimum of f on R n , e.g., consider f(x, y) = x 2 + (xy − 1) 2 . In =-=[23]-=- it is explained how 7the algorithm in [10] can be modified to handle the case where f is bounded from below on R n but does not achieve a minimum value on R n . 3. degree bounds in the compact case ...

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...rrect lower bound. Generally, this case will not happen, as shown by items (d), (e) of Theorem 2.5. In unconstrained optimization, when 23 fmin is not achievable, excellent approaches are proposed in =-=[9, 10, 26]-=-. It is an interesting future work to generalize them to constrained optimization. An important question is for what concrete relaxation order N∗ the SDP relaxation (2.8) is exact for solving (1.1). N...

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...truth values and utilization of quantifier information of input formulas. It significantly reduces the computing time in the lifting phase. Also, many approaches for special types of input formulas have been proposed, for example [30, 32, 33, 40, 5]. Meanwhile, several attempts to develop symbolic-numeric 168 algorithms for solving polynomial optimization problems (POPs) have been done. In [2] an efficient algorithm based on CAD to solve semidefinite programming exactly is studied. Another promising direction is to use generalized critical values for global optimization, which originates with [38] in the scheme of sum of squares. Then within the framework of the critical point method, generalized critical values are first used in [16]. As for sums of squares approach, we note that it provides algebraic certificates to compute lower bounds on global infimum (see [35]). In [26] computation of “symbolic” algebraic certificates (instead of “numerical” ones) is proposed. Moreover, the problem of obtaining theorems on the existence of algebraic certificates based on sums of squares is due to [20]. In this paper, we introduce an efficient approach for solving POPs by using SNCAD. Our approach...

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...equence of upper bounds converges to f∗ even if f∗ is not attained, which when K = Rn, could provide an alternative and/or a complement to the hierarchy of convex relaxations provided in Schweighofer =-=[21]-=- (based on gradient tentacles) and in Hà and Pham [8] (based on the truncated tangency variety), which both provide again a monotone sequence of lower bounds. Finally, we also give a very simple inte...

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...fined by a radical ideal is also analyzed under similar regularity conditions. The case where the minimum value is not reached at points which satisfied the KKT constraints, has also been studied. In =-=[33]-=-, relaxation techniques are studied for functions for which the minimum is not reached and which satisfies some special properties “at infinity”. In [8], tangency constraints are used in a relaxation ...