Appendix
Before proving Lemma 1 we need introduce some notation. Given \(k\in \mathbb {N}\) fixed, let \(\tau = \max \{\mathrm{deg}(f), 2k, d\max _{j}\{\mathrm{deg}(g_j)\}\}\). For a sequence \(\mathbf {y}=(y_\alpha )\in \mathbb {N}^n_\tau \), let \(L_\mathbf {y}:\mathbb {R}[x]_\tau \rightarrow \mathbb {R}\) be the Riesz functional:
$$\begin{aligned} f\,\left( :=\sum _{\alpha \in \mathbb {N}^n_\tau }f_\alpha \,x^\alpha \,\right) \quad \mapsto \quad L_\mathbf {y}(f):= \sum _{\alpha \in \mathbb {N}^n_\tau }f_\alpha \,y_\alpha ,\qquad f\in \mathbb {R}[x]_\tau , \end{aligned}$$
and let \(\mathbf {M}_k(\mathbf {y})\) be the moment matrix of order k, associated with \(\mathbf {y}\). If \(q\in \mathbb {R}[x]_k\) with coefficient vector \(\mathbf {q}=(q_\alpha )\), then \(\langle \mathbf {q},\mathbf {M}_k(\mathbf {y})\,\mathbf {q}\rangle =L_\mathbf {y}(q^2)\) and if \(\mathbf {y}\) is the (truncated) moment sequence of a measure \(\mu \),
$$\begin{aligned} \langle \mathbf {q},\mathbf {M}_k(\mathbf {y})\,\mathbf {q}\rangle =L_\mathbf {y}(q^2)=\int q(x)^2\,\mathrm{d}\mu (x). \end{aligned}$$
1.1 Proof of Lemma 1
(a) We first prove that the dual of (7) which is the semidefinite program:
$$\begin{aligned} \rho _d^k := \displaystyle \inf _{\mathbf {y}\in \mathbb {R}^L}\,\{\,L_\mathbf {y}(f)\,:\, \displaystyle \mathbf {M}_k(\mathbf {y})\,\succeq \,0;\, L_\mathbf {y}(1)=1;\quad L_\mathbf {y}(h_{\alpha \beta })\, \ge 0,\quad (\alpha ,\beta )\in \mathbb {N}^{2m}_d\,\}\nonumber \\ \end{aligned}$$
(13)
satisfies Slater’s condition. Recall that K has nonempty interior; so let \(\mathbf {y}\) be the sequence of moments of the Lebesgue measure \(\mu \) on \(\mathbf {K}\), scaled to be a probability measure, so that \(L_\mathbf {y}(1)=1\). Necessarily \(\mathbf {M}_k(\mathbf {y})\succ 0\). Otherwise there would exists \(0\ne q\in \mathbb {R}[x]_k\) such that
$$\begin{aligned} \langle \mathbf {q},\mathbf {M}_k(\mathbf {y})\,\mathbf {q}\rangle = \int _K q(x)^2\,\mathrm{d}\mu (x)=0. \end{aligned}$$
But then q vanishes almost everywhere on K, which implies \(q=0\), a contradiction.
Next, observe that for each \((\alpha ,\beta )\in \mathbb {N}^{2m}_d\), the polynomial \(h_{\alpha \beta }\in \mathbb {R}[x]_\tau \) is nonnegative on K and since there exists \(x_0\in K\) such that \(0<g_j(x_0)<1\) for all \(j=1,\ldots ,m\), there is an open set \(O\subset K\) such that \(h_{\alpha \beta }(x)>0\) on O for all \((\alpha ,\beta )\in \mathbb {N}^{2m}\). Therefore
$$\begin{aligned} L_\mathbf {y}(h_{\alpha \beta })=\int _Kh_{\alpha \beta }\,\mathrm{d}\mu \,\ge \,\int _Oh_{\alpha \beta }\,\mathrm{d}\mu \,>0,\quad \forall \, (\alpha ,\beta )\in \mathbb {N}^{2m}. \end{aligned}$$
Therefore \(\mathbf {y}\) is a strictly feasible solution of (13), that is, Slater’s condition holds true for (13). Hence \(\rho ^k_d=q^k_d\) for all d. It remains to prove that \(q^k_d>-\infty \). But this follows from Theorem 1(b) as soon as d is sufficiently large, say \(d\ge d_0\) for some integer \(d_0\). Indeed then \(-\infty <\theta _d\le q^k_d\le f^*\) for all \(d\ge d_0\) (where \(\theta _d\) is defined in (4)). Finally for each fixed d, (7) and (8) have same optimal value \(q^k_d\) and an optimal solution \((q^k_d,\lambda ^*,Q^*)\) of (7) is also an optimal solution of (8).
(b) Let \(\theta ^*\) be an optimal solution of (9) and let \(\mathbf {y}^*\) be as in (10).
\(\bullet \) If \(\mathrm{rank}\,\mathbf {M}_s(\mathbf {y}^*)=1\) then \(\mathbf {M}_s(\mathbf {y}^*)=v_s(x^*)\,v_s(x^*)^T\) for some \(x^*\in \mathbb {R}^n\); this is due to the Hankel-like structure of the moment matrix combined with the rank-one property. So by definition of the moment matrix \(\mathbf {M}_s(\mathbf {y}^*)\), \(\mathbf {y}^*=(y^*_\alpha )\), \(\alpha \in \mathbb {N}^n_{2s}\), is the vector of moments (up to order 2s) of the Dirac measure \(\delta _{x^*}\) at the point \(x^*\). That is, \(y^*_\alpha =(x^*)^\alpha \) for every \(\alpha \in \mathbb {N}^n_{2s}\). But from (10),
$$\begin{aligned} (x^*)^\alpha =y^*_\alpha =\sum _{p=1}^L\theta ^*_p\,(x^{(p)})^\alpha ,\quad \forall \alpha \in \mathbb {N}^n_{2s}. \end{aligned}$$
In particular, for moments of order 1 we obtain \(x^*=\sum _{p=1}^L\theta ^*_p\,x^{(p)}\). In other words, up to moments of order 2s, one cannot distinguish the Dirac measure \(\delta _{x^*}\) at \(x^*\) from the signed measure \(\mu =\sum _p\theta ^*_p\delta _{x^{(p)}}\) (recall that the \(\theta ^*_p\)’s are not necessarily nonnegative). That is, \((x^*)^\alpha =\int x^\alpha d\delta _{x^*}=\int x^\alpha \mathrm{d}\mu \) for all \(\alpha \in \mathbb {N}^n_{2s}\). This in turn implies that for every \(q\in \mathbb {R}[x]_{2s}\):
$$\begin{aligned} q(x^*)=\langle q,\delta _{x^*}\rangle =\langle q,\mu \rangle = \left\langle q,\sum _{p=1}^L\theta ^*_p\,\delta _{x^{(p)}}\right\rangle = \sum _{p=1}^L\theta ^*_p\,q(x^{(p)}). \end{aligned}$$
Next, as \(\theta ^*\) is feasible for (9) and \(2s\ge \max [\mathrm{deg}(f);\,\mathrm{deg}(g_j)]\),
$$\begin{aligned} 0\le \displaystyle \sum _{p=1}^L \theta ^*_p\,\langle h_{\alpha \beta },\delta _{x^{(p)}}\rangle =\left\langle h_{\alpha \beta },\displaystyle \sum _{p=1}^L \theta ^*_p\,\delta _{x^{(p)}}\right\rangle =h_{\alpha \beta }(x^*), \quad \forall (\alpha ,\beta ):\,\mathrm{deg}(h_{\alpha \beta })\,\le 2s. \end{aligned}$$
In particular, choosing \((\alpha ,\beta )\in \mathbb {N}^{2m}_{2s}\) such that \(h_{\alpha \beta }=g_j\) (i.e. \(\beta =0\), \(\alpha _i=\delta _{i=j}\)), one obtains \(g_j(x^*)\ge 0\), \(j=1,\ldots ,m\), which shows that \(x^*\in K\). In addition,
$$\begin{aligned} f^*\,\ge \,\tilde{q}^k_d= \displaystyle \sum _{p=1}^L \theta ^*_p\,\langle f,\delta _{x^{(p)}}\rangle = \left\langle f,\displaystyle \sum _{p=1}^L \theta ^*_p\,\delta _{x^{(p)}}\right\rangle =f(x^*), \end{aligned}$$
which proves that \(x^*\in K\) is an optimal solution of problem (P).
\(\bullet \) If \(\mathbf {M}_s(\mathbf {y}^*)\succeq 0\), \(\mathbf {M}_{s-r}(g_j\,\mathbf {y}^*)\succeq 0\), \(j=1,\ldots ,m\), and \(\mathrm{rank}\,\mathbf {M}_s(\mathbf {y}^*)=\mathrm{rank}\,\mathbf {M}_{s-r}(\mathbf {y}^*)\) then by Curto and Fialkow (2000, 2005, Theorem 1.1), \(\mathbf {y}^*\) is the vector of moments up to order 2s, of some atomic-probability measure \(\mu \) supported on \(v:=\mathrm{rank}\,\mathbf {M}_s(\mathbf {y}^*)\) points \(z(i)\in K\), \(i=1,\ldots ,v\). That is, there exist positive weights \((w_i)\subset \mathbb {R}_+\) such that
$$\begin{aligned} \mu =\sum _{i=1}^v w_i\,\delta _{z(i)};\quad \sum _{i=1}^vw_i=1;\quad w_i\,>\,0,\,i=1,\ldots ,v. \end{aligned}$$
Therefore,
$$\begin{aligned} f^*\,\ge \,\tilde{q}^k_d=\sum _{p=1}^L\theta ^*_p\,\langle f,\delta _{x(p)}\rangle =\sum _{\alpha \in \mathbb {N}^n} f_\alpha \,y^*_\alpha =\int _K f\,\mathrm{d}\mu \,\ge \,f^*, \end{aligned}$$
which shows that \(\tilde{q}^k_d=f^*\). In addition
$$\begin{aligned}0=f^*- \int _K f\,\mathrm{d}\mu =\int _K (f^*-f)\,\mathrm{d}\mu = \sum _{i=1}^v \underbrace{w_i}_{>0}\,(\underbrace{f^*-f(z(i))}_{\le 0}, \end{aligned}$$
which implies \(f(z(i))=f^*\) for every \(i=1,\ldots ,v\). Finally, the v global minimizers can be extracted from the moment matrix \(\mathbf {M}_s(\mathbf {y}^*)\) by the simple linear algebra procedure described in Henrion and Lasserre (2005). \(\square \)
1.2 Test functions for BSOS and GloptiPoly in Table 1
Example P4_2 (4 variables, degree 2):
$$\begin{aligned} f= & {} x_{1}^2-x_{2}^2+x_{3}^2-x_{4}^2+x_1-x_2; \\ g_1= & {} 2x_{1}^2+3x_{2}^2+2x_{1}x_{2}+2x_{3}^2+3x_{4}^2+2x_{3}x_{4}; \\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}; \\ g_4= & {} x_{1}^2+4x_{2}^2-3x_{1}x_{2}+x_{3}^2+4x_{4}^2-3x_{3}x_{4};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}; \quad x\ge 0. \end{aligned}$$
Example P4_4 (4 variables, degree 4):
$$\begin{aligned} f= & {} x_{1}^4-x_{2}^4+x_{3}^4-x_{4}^4; \\ g_1= & {} 2x_{1}^4+3x_{2}^2+2x_{1}x_{2}+2x_{3}^4+3x_{4}^2+2x_{3}x_{4}; \\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}; \\ g_4= & {} x_{1}^2+4x_{2}^4-3x_{1}x_{2}+x_{3}^2+4x_{4}^4-3x_{3}x_{4};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}; \quad x\ge 0. \end{aligned}$$
Example P4_6 (4 variables, degree 6):
$$\begin{aligned} f= & {} x_{1}^4x_{2}^2+x_{1}^2x_{2}^4-x_{1}^2x_{2}^2+x_{3}^4x_{4}^2+x_{3}^2x_{4}^4-x_{3}^2x_{4}^2; \\ g_1= & {} x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2; \\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4};\\ g_3= & {} x_{1}^2+6x_{2}^4-8x_{1}x_{2}+x_{3}^2+6x_{4}^4-8x_{3}x_{4}+2.5; \\ g_4= & {} x_{1}^4+3x_{2}^4+x_{3}^4+3x_{4}^4;\quad g_5=x_{1}^2+x_{2}^3+x_{3}^2+x_{4}^3; \quad x\ge 0. \end{aligned}$$
Example P4_8 (4 variables, degree 8):
$$\begin{aligned} f= & {} x_{1}^4x_{2}^2+x_{1}^2x_{2}^6-x_{1}^2x_{2}^2 +x_{3}^4x_{4}^2+x_{3}^2x_{4}^6-x_{3}^2x_{4}^2;\quad g_1=x_{1}^2+x_{2}^2+x_{3}^2+x_{4}^2; \\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4};\\ g_3= & {} x_{1}^2+6x_{2}^4-8x_{1}x_{2}+x_{3}^2+6x_{4}^4-8x_{3}x_{4}+2.5; \\ g_4= & {} x_{1}^4+3x_{2}^4+x_{3}^4+3x_{4}^4;\quad g_5=x_{1}^2+x_{2}^3+x_{3}^2+x_{4}^3; \quad x\ge 0. \end{aligned}$$
Example P6_2 (6 variables, degree 2):
$$\begin{aligned} f= & {} x_{1}^2-x_{2}^2+x_{3}^2-x_{4}^2+x_{5}^2-x_{6}^2 + x_1-x_2; \\ g_1= & {} 2x_{1}^2+3x_{2}^2+2x_{1}x_{2}+2x_{3}^2+3x_{4}^2+2x_{3}x_{4} +2x_{5}^2+3x_{6}^2+2x_{5}x_{6}; \\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4} +3x_{5}^2+2x_{6}^2-4x_{5}x_{6}; \\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4} +x_{5}^2+6x_{6}^2-4x_{5}x_{6}; \\ g_4= & {} x_{1}^2+4x_{2}^2-3x_{1}x_{2}+x_{3}^2+4x_{4}^2-3x_{3}x_{4}+x_{5}^2 +4x_{6}^2-3x_{5}x_{6}; \\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4} +2x_{5}^2+5x_{6}^2+3x_{5}x_{6}; \quad x\ge 0. \end{aligned}$$
Example P6_4 (6 variables, degree 4):
$$\begin{aligned} f= & {} x_{1}^4-x_{2}^2+x_{3}^4-x_{4}^2+x_{5}^4-x_{6}^2 + x_1 - x_2; \\ g_1= & {} 2x_{1}^4+x_{2}^2+2x_{1}x_{2}+2x_{3}^4+x_{4}^2+2x_{3}x_{4}+2x_{5}^4+x_{6}^2+2x_{5}x_{6};\\ g_2= & {} 3x_{1}^2+x_{2}^2-4x_{1}x_{2}+3x_{3}^2+x_{4}^2-4x_{3}x_{4}+3x_{5}^2+x_{6}^2-4x_{5}x_{6};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2+6x_{6}^2-4x_{5}x_{6};\\ g_4= & {} x_{1}^2+3x_{2}^4-3x_{1}x_{2}+x_{3}^2+3x_{4}^4-3x_{3}x_{4}+x_{5}^2+3x_{6}^4-3x_{5}x_{6};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4} +2x_{5}^2+5x_{6}^2+3x_{5}x_{6}, \quad x\ge 0. \end{aligned}$$
Example P6_6 (6 variables, degree 6):
$$\begin{aligned} f= & {} x_{1}^6-x_{2}^6+x_{3}^6-x_{4}^6+x_{5}^6-x_{6}^6 +x_1-x_2; \\ g_1= & {} 2x_{1}^6+3x_{2}^2+2x_{1}x_{2}+2x_{3}^6+3x_{4}^2+2x_{3}x_{4}+2x_{5}^6+3x_{6}^2+2x_{5}x_{6};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2+2x_{6}^2-4x_{5}x_{6};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2+6x_{6}^2-4x_{5}x_{6};\\ g_4= & {} x_{1}^2+4x_{2}^6-3x_{1}x_{2}+x_{3}^2+4x_{4}^6-3x_{3}x_{4}+x_{5}^2+4x_{6}^6-3x_{5}x_{6};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4} +2x_{5}^2+5x_{6}^2+3x_{5}x_{6}, \quad x\ge 0. \end{aligned}$$
Example P6_8 (6 variables, degree 8):
$$\begin{aligned} f= & {} x_{1}^8-x_{2}^8+x_{3}^8-x_{4}^8+x_{5}^8-x_{6}^8 + x_1-x_2; \\ g_1= & {} 2x_{1}^8+3x_{2}^2+2x_{1}x_{2}+2x_{3}^8+3x_{4}^2+2x_{3}x_{4}+2x_{5}^8+3x_{6}^2+2x_{5}x_{6};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2+2x_{6}^2-4x_{5}x_{6};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2+6x_{6}^2-4x_{5}x_{6};\\ g_4= & {} x_{1}^2+4x_{2}^8-3x_{1}x_{2}+x_{3}^2+4x_{4}^8-3x_{3}x_{4}+x_{5}^2+4x_{6}^8-3x_{5}x_{6};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}+2x_{5}^2+5x_{6}^2+3x_{5}x_{6}, \quad x\ge 0. \end{aligned}$$
Example P8_2 (8 variables, degree 2):
$$\begin{aligned} f= & {} x_{1}^2-x_{2}^2+x_{3}^2-x_{4}^2+x_{5}^2-x_{6}^2+x_{7}^2-x_{8}^2 + x_1-x_2; \\ g_1= & {} 2x_{1}^2+3x_{2}^2+2x_{1}x_{2}+2x_{3}^2+3x_{4}^2+2x_{3}x_{4}+2x_{5}^2\\&+\,3x_{6}^2+2x_{5}x_{6}+2x_{7}^2+3x_{8}^2+2x_{7}x_{8};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2\\&+\,2x_{6}^2-4x_{5}x_{6}+3x_{7}^2+2x_{8}^2-4x_{7}x_{8};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2\\&+\,6x_{6}^2-4x_{5}x_{6}+x_{7}^2+6x_{8}^2-4x_{7}x_{8};\\ g_4= & {} x_{1}^2+4x_{2}^2-3x_{1}x_{2}+x_{3}^2+4x_{4}^2-3x_{3}x_{4}+x_{5}^2\\&\ +\,4x_{6}^2-3x_{5}x_{6}+x_{7}^2+4x_{8}^2-3x_{7}x_{8};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}\\&\ +\,2x_{5}^2+5x_{6}^2+3x_{5}x_{6}+2x_{7}^2+5x_{8}^2+3x_{7}x_{8}; \quad x\ge 0. \end{aligned}$$
Example P8_4 (8 variables, degree 4):
$$\begin{aligned} f= & {} x_{1}^4-x_{2}^4+x_{3}^4-x_{4}^4+x_{5}^4-x_{6}^4+x_{7}^4-x_{8}^4 + x_1-x_2;\\ g_1= & {} 2x_{1}^4+3x_{2}^2+2x_{1}x_{2}+2x_{3}^4+3x_{4}^2+2x_{3}x_{4}+2x_{5}^4\\&\ +\,3x_{6}^2+2x_{5}x_{6}+2x_{7}^4+3x_{8}^2+2x_{7}x_{8};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2\\&\ +\,2x_{6}^2-4x_{5}x_{6}+3x_{7}^2+2x_{8}^2-4x_{7}x_{8};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2\\&\ +\,6x_{6}^2-4x_{5}x_{6}+x_{7}^2+6x_{8}^2-4x_{7}x_{8};\\ g_4= & {} x_{1}^2+4x_{2}^4-3x_{1}x_{2}+x_{3}^2+4x_{4}^4-3x_{3}x_{4}+x_{5}^2\\&\ +\,4x_{6}^4-3x_{5}x_{6}+x_{7}^2+4x_{8}^4-3x_{7}x_{8};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}+2x_{5}^2\\&\ +\,5x_{6}^2+3x_{5}x_{6}+2x_{7}^2+5x_{8}^2+3x_{7}x_{8}, \quad x\ge 0. \end{aligned}$$
Example P8_6 (8 variables, degree 6):
$$\begin{aligned} f= & {} x_{1}^6-x_{2}^6+x_{3}^6-x_{4}^6+x_{5}^6-x_{6}^6+x_{7}^6-x_{8}^6 + x_1-x_2;\\ g_1= & {} 2x_{1}^6+3x_{2}^2+2x_{1}x_{2}+2x_{3}^6+3x_{4}^2+2x_{3}x_{4}+2x_{5}^6\\&\ +\,3x_{6}^2+2x_{5}x_{6}+2x_{7}^6+3x_{8}^2+2x_{7}x_{8};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2\\&\ +\,2x_{6}^2-4x_{5}x_{6}+3x_{7}^2+2x_{8}^2-4x_{7}x_{8};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2\\&\ +\,6x_{6}^2-4x_{5}x_{6}+x_{7}^2+6x_{8}^2-4x_{7}x_{8};\\ g_4= & {} x_{1}^2+4x_{2}^6-3x_{1}x_{2}+x_{3}^2+4x_{4}^6-3x_{3}x_{4}+x_{5}^2\\&\ +\,4x_{6}^6-3x_{5}x_{6}+x_{7}^2+4x_{8}^6-3x_{7}x_{8};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}+2x_{5}^2\\&\ +\,5x_{6}^2+3x_{5}x_{6}+2x_{7}^2+5x_{8}^2+3x_{7}x_{8}, \quad x\ge 0. \end{aligned}$$
Example P10_2 (10 variables, degree 2):
$$\begin{aligned} f= & {} x_{1}^2-x_{2}^2+x_{3}^2-x_{4}^2+x_{5}^2-x_{6}^2+x_{7}^2-x_{8}^2+x_{9}^2-x_{10}^2+x_1-x_2;\\ g_1= & {} 2x_{1}^2+3x_{2}^2+2x_{1}x_{2}+2x_{3}^2+3x_{4}^2+2x_{3}x_{4}+2x_{5}^2+3x_{6}^2+2x_{5}x_{6}\\&\ +\,2x_{7}^2+3x_{8}^2+2x_{7}x_{8}+2x_{9}^2+3x_{10}^2+2x_{9}x_{10};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2+2x_{6}^2-4x_{5}x_{6}\\&\ +\,3x_{7}^2+2x_{8}^2-4x_{7}x_{8}+3x_{9}^2+2x_{10}^2-4x_{9}x_{10};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2+6x_{6}^2-4x_{5}x_{6}\\&\ +\,x_{7}^2+6x_{8}^2-4x_{7}x_{8}+x_{9}^2+6x_{10}^2-4x_{9}x_{10};\\ g_4= & {} x_{1}^2+4x_{2}^2-3x_{1}x_{2}+x_{3}^2+4x_{4}^2-3x_{3}x_{4}+x_{5}^2+4x_{6}^2-3x_{5}x_{6}\\&\ +\,x_{7}^2+4x_{8}^2-3x_{7}x_{8}+x_{9}^2+4x_{10}^2-3x_{9}x_{10};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}+2x_{5}^2+5x_{6}^2+3x_{5}x_{6}\\&\ +\,2x_{7}^2+5x_{8}^2+3x_{7}x_{8}+2x_{9}^2+5x_{10}^2+3x_{9}x_{10}; \quad x \ge 0. \end{aligned}$$
Example P10_4 (10 variables, degree 4):
$$\begin{aligned} f= & {} x_{1}^4-x_{2}^4+x_{3}^4-x_{4}^4+x_{5}^4-x_{6}^4+x_{7}^4-x_{8}^4+x_{9}^4-x_{10}^4 + x_1-x_2;\\ g_1= & {} 2x_{1}^4+3x_{2}^2+2x_{1}x_{2}+2x_{3}^4+3x_{4}^2+2x_{3}x_{4}+2x_{5}^4+3x_{6}^2+2x_{5}x_{6}\\&\ +\,2x_{7}^4+3x_{8}^2+2x_{7}x_{8}+2x_{9}^4+3x_{11}^2+2x_{9}x_{10};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2+2x_{6}^2-4x_{5}x_{6}\\&\ +\,3x_{7}^2+2x_{8}^2-4x_{7}x_{8}+3x_{9}^2+2x_{10}^2-4x_{9}x_{10};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2+6x_{6}^2-4x_{5}x_{6}\\&\ +\,x_{7}^2+6x_{8}^2-4x_{7}x_{8}+x_{9}^2+6x_{10}^2-4x_{9}x_{10};\\ g_4= & {} x_{1}^2+4x_{2}^4-3x_{1}x_{2}+x_{3}^2+4x_{4}^4-3x_{3}x_{4}+x_{5}^2+4x_{6}^4-3x_{5}x_{6}\\&\ +\,x_{7}^2+4x_{8}^4-3x_{7}x_{8}+x_{9}^2+4x_{10}^4-3x_{9}x_{10};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}+2x_{5}^2+5x_{6}^2+3x_{5}x_{6}\\&\ +\,2x_{7}^2+5x_{8}^2+3x_{7}x_{8}+2x_{9}^2+5x_{10}^2+3x_{9}x_{10}; \quad x \ge 0. \end{aligned}$$
Example P20_2 (20 variables, degree 2):
$$\begin{aligned} f= & {} x_{1}^2-x_{2}^2+x_{3}^2-x_{4}^2+x_{5}^2-x_{6}^2+x_{7}^2-x_{8}^2+x_{9}^2-x_{10}^2+x_{11}^2-x_{12}^2 + x_1-x_2\\&+\,x_{13}^2-x_{14}^2+x_{15}^2-x_{16}^2+x_{17}^2-x_{18}^2+x_{19}^2-x_{20}^2;\\ g_1= & {} 2x_{1}^2+3x_{2}^2+2x_{1}x_{2}+2x_{3}^2+3x_{4}^2+2x_{3}x_{4}+2x_{5}^2+3x_{6}^2+2x_{5}x_{6}+2x_{7}^2+3x_{8}^2\\&+\,2x_{7}x_{8}+2x_{9}^2+3x_{10}^2+2x_{9}x_{10}+2x_{11}^2+3x_{12}^2+2x_{11}x_{12}+2x_{13}^2+3x_{14}^2\\&+\,2x_{13}x_{14}+2x_{15}^2+3x_{16}^2+2x_{15}x_{16}+2x_{17}^2+3x_{18}^2+2x_{17}x_{18}+2x_{19}^2+3x_{10}^2\\&+\,2x_{20}x_{20};\\ g_2= & {} 3x_{1}^2+2x_{2}^2-4x_{1}x_{2}+3x_{3}^2+2x_{4}^2-4x_{3}x_{4}+3x_{5}^2+2x_{6}^2-4x_{5}x_{6}+3x_{7}^2+2x_{8}^2\\&-\,4x_{7}x_{8}+3x_{9}^2+2x_{10}^2-4x_{9}x_{10}+3x_{11}^2+2x_{12}^2-4x_{11}x_{12}+3x_{13}^2+2x_{14}^2\\&-\,4x_{13}x_{14}+3x_{15}^2+2x_{16}^2-4x_{15}x_{16}+3x_{17}^2+2x_{19}^2-4x_{18}x_{18}+3x_{19}^2+2x_{20}^2\\&-\,4x_{19}x_{20};\\ g_3= & {} x_{1}^2+6x_{2}^2-4x_{1}x_{2}+x_{3}^2+6x_{4}^2-4x_{3}x_{4}+x_{5}^2+6x_{6}^2-4x_{5}x_{6}+x_{7}^2+6x_{8}^2-4x_{7}x_{8}\\&+\,x_{9}^2+6x_{10}^2-4x_{9}x_{10}+x_{11}^2+6x_{12}^2-4x_{11}x_{12}+x_{13}^2+6x_{14}^2-4x_{13}x_{14}\\&+\,x_{15}^2+6x_{17}^2-4x_{16}x_{16}+x_{17}^2+6x_{18}^2-4x_{17}x_{18}+x_{19}^2+6x_{20}^2-4x_{19}x_{20};\\ g_4= & {} x_{1}^2+4x_{2}^2-3x_{1}x_{2}+x_{3}^2+4x_{4}^2-3x_{3}x_{4}+x_{5}^2+4x_{6}^2-3x_{5}x_{6}+x_{7}^2+4x_{8}^2-3x_{7}x_{8}\\&+\,x_{9}^2+4x_{10}^2-3x_{9}x_{10}+x_{1}^2+4x_{12}^2-3x_{11}x_{12}+x_{13}^2+4x_{14}^2-3x_{15}x_{14}\\&+\,x_{15}^2+4x_{16}^2-3x_{15}x_{16}+x_{17}^2+4x_{18}^2-3x_{17}x_{18}+x_{19}^2+4x_{20}^2-3x_{19}x_{20};\\ g_5= & {} 2x_{1}^2+5x_{2}^2+3x_{1}x_{2}+2x_{3}^2+5x_{4}^2+3x_{3}x_{4}+2x_{5}^2+5x_{6}^2+3x_{5}x_{6}+2x_{7}^2+5x_{8}^2\\&+\,3x_{7}x_{8}+2x_{9}^2+5x_{10}^2+3x_{9}x_{10}+2x_{11}^2+5x_{13}^2+3x_{12}x_{12}+2x_{13}^2+5x_{14}^2\\&+\,3x_{13}x_{14}+2x_{15}^2+5x_{16}^2+3x_{15}x_{16}+2x_{17}^2+5x_{18}^2+3x_{17}x_{18}+2x_{19}^2+5x_{20}^2\\&+\,3x_{19}x_{20}; \quad x \ge 0. \end{aligned}$$
Example P20_4 (20 variables, degree 4): same as P20_2 except that f is replaced by
$$\begin{aligned} f= & {} x_{1}^4-x_{2}^4+x_{3}^2-x_{4}^2+x_{5}^2-x_{6}^2+x_{7}^2-x_{8}^2+x_{9}^2-x_{10}^2+x_{11}^2-x_{12}^2 + x_1-x_2\\&+x_{13}^2-x_{14}^2+x_{15}^2-x_{16}^2+x_{17}^2-x_{18}^2+x_{19}^2-x_{20}^2; \end{aligned}$$
1.3 Test functions for BSOS versus LP relaxations of Krivine-Stengle on convex problems in Table 2
Example C4_2 (4 variables, degree 2):
$$\begin{aligned} f= & {} x_1^2+x_2^2+x_3^2+x_4^2 + 2x_1x_2-x_1-x_2;\\ g_1= & {} -x_1^2-2x_2^2-x_3^2-2x_4^2+1;\\ g_2= & {} -2x_1^2-x_2^2-2x_3^2-x_4^2+1; \\ g_3= & {} -x_1^2-4x_2^2-x_3^2-4x_4^2+1.25;\\ g_4= & {} -4x_1^2-x_2^2-4x_3^2-x_4^2+1.25; \\ g_5= & {} -2x_1^2-3x_2^2-2x_3^2-3x_4^2+1.1; \quad x\ge 0. \end{aligned}$$
Example C4_4 (4 variables, degree 4):
$$\begin{aligned} f= & {} x_1^4+x_2^4+x_3^4+x_4^4 + 3x_1^2x_2^2-x_1-x_2; \\ g_1= & {} -x_1^4-2x_2^4-x_3^4-2x_4^4+1;\\ g_2= & {} -2x_1^4-x_2^4-2x_3^4-x_4^4+1; \\ g_3= & {} -x_1^4-4x_2^4-x_3^4-4x_4^4+1.25;\\ g_4= & {} -4x_1^4-x_2^4-4x_3^4-x_4^4+1.25; \\ g_5= & {} -2x_1^4-3x_2^2-2x_3^4-3x_4^2+1.1; \quad x\ge 0. \end{aligned}$$
Example C4_6 (4 variables, degree 6):
$$\begin{aligned} f= & {} x_1^6+x_2^6+x_3^6+x_4^6 + \frac{10}{3} x_1^3x_2^3-x_1-x_2; \\ g_1= & {} -x_1^6-2x_2^6-x_3^6-2x_4^6+1;\\ g_2= & {} -2x_1^6-x_2^6-2x_3^6-x_4^6+1;\\ g_3= & {} -x_1^6-4x_2^2-x_3^6-4x_4^2+1.25;\\ g_4= & {} -4x_1^6-x_2^2-4x_3^6-x_4^2+1.25;\nonumber \\ g_5= & {} -2x_1^2-3x_2^6-2x_3^2-3x_4^6+1.1; \quad x\ge 0. \end{aligned}$$
Example C6_2 (6 variables, degree 2):
$$\begin{aligned} f= & {} x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2 + 2x_1x_2-x_1-x_2;\nonumber \\ g_1= & {} -x_1^2-2x_2^2-x_3^2-2x_4^2-x_5^2-2x_6^2+1;\\ g_2= & {} -2x_1^2-x_2^2-2x_3^2-x_4^2-2x_5^2-x_6^2+1;\nonumber \\ g_3= & {} -x_1^2-4x_2^2-x_3^2-4x_4^2-x_5^2-4x_6^2+1.25;\\ g_4= & {} -4x_1^2-x_2^2-4x_3^2-x_4^2-4x_5^2-x_6^2+1.25;\nonumber \\ g_5= & {} -2x_1^2-3x_2^2-2x_3^2-3x_4^2-2x_5^2-3x_6^2+1.1; \quad x\ge 0. \end{aligned}$$
Example C6_4 (6 variables, degree 4):
$$\begin{aligned} f= & {} x_1^4+x_2^4+x_3^4+x_4^4+x_5^4+x_6^4+3x_1^2x_2^2-x_1-x_2;\nonumber \\ g_1= & {} -x_1^4-2x_2^4-x_3^4-2x_4^4-x_5^4-2x_6^4+1;\\ g_2= & {} -2x_1^4-x_2^4-2x_3^4-x_4^4-2x_5^4-x_6^4+1;\nonumber \\ g_3= & {} -x_1^4-4x_2^4-x_3^4-4x_4^4-x_5^4-4x_6^4+1.25;\\ g_4= & {} -4x_1^4-x_2^4-4x_3^4-x_4^4-4x_5^4-x_6^4+1.25;\\ g_5= & {} -2x_1^4-3x_2^2-2x_3^4-3x_4^2-2x_5^4-3x_6^2+1.1; \quad x\ge 0. \end{aligned}$$
Example C6_6 (6 variables, degree 6):
$$\begin{aligned} f= & {} x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6+\frac{10}{3}x_1^2x_2^3-x_1-x_2;\nonumber \\ g_1= & {} -x_1^6-2x_2^6-x_3^6-2x_4^6-x_5^6-2x_6^6+1;\\ g_2= & {} -2x_1^6-x_2^6-2x_3^6-x_4^6-2x_5^6-x_6^6+1;\nonumber \\ g_3= & {} -x_1^6-4x_2^2-x_3^6-4x_4^2-x_5^6-4x_6^2+1.25;\\ g_4= & {} -4x_1^6-x_2^2-4x_3^6-x_4^2-4x_5^6-x_6^2+1.25;\nonumber \\ g_5= & {} -2x_1^2-3x_2^6-2x_3^2-3x_4^6-2x_5^2-3x_6^6+1.1; \quad x\ge 0. \end{aligned}$$
Example C8_2 (8 variables, degree 2):
$$\begin{aligned} f= & {} x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2+2x_1x_2-x_1-x_2;\nonumber \\ g_1= & {} -x_1^2-2x_2^2-x_3^2-2x_4^2-x_5^2-2x_6^2-x_7^2-2x_8^2+1;\\ g_2= & {} -2x_1^2-x_2^2-2x_3^2-x_4^2-2x_5^2-x_6^2-2x_7^2-x_8^2+1; \nonumber \\ g_3= & {} -x_1^2-4x_2^2-x_3^2-4x_4^2-x_5^2-4x_6^2-x_7^2-4x_8^2+1.25;\\ g_4= & {} -4x_1^2-x_2^2-4x_3^2-x_4^2-4x_5^2-x_6^2-4x_7^2-x_8^2+1.25;\nonumber \\ g_5= & {} -2x_1^2-3x_2^2-2x_3^2-3x_4^2-2x_5^2-3x_6^2-2x_7^2-3x_8^2+1.1; \quad x \ge 0. \end{aligned}$$
Example C8_4 (8 variables, degree 4):
$$\begin{aligned} f= & {} x_1^4+x_2^4+x_3^4+x_4^4+x_5^4+x_6^4+x_7^4+x_8^4+3x_1^2x_2^2-x_1-x_2;\nonumber \\ g_1= & {} -x_1^4-2x_2^4-x_3^4-2x_4^4-x_5^2-2x_6^4-x_7^4-2x_8^4+1;\\ g_2= & {} -2x_1^4-x_2^4-2x_3^4-x_4^4-2x_5^2-x_6^4-2x_7^4-x_8^4+1;\nonumber \\ g_3= & {} -x_1^4-4x_2^4-x_3^4-4x_4^4-x_5^4-4x_6^4-x_7^4-4x_8^4+1.25;\\ g_4= & {} -4x_1^4-x_2^4-4x_3^4-x_4^4-4x_5^4-x_6^4-4x_7^4-x_8^4+1.25;\nonumber \\ g_5= & {} -2x_1^4-3x_2^2-2x_3^4-3x_4^2-2x_5^4-3x_6^2-2x_7^4-3x_8^2+1.1; \quad x \ge 0. \end{aligned}$$
Example C10_2 (10 variables, degree 2):
$$\begin{aligned} f= & {} x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2+x_9^2+x_{10}^2 +2x_1x_2-x_1-x_2; \\ g_1= & {} -x_1^2-2x_2^2-x_3^2-2x_4^2-x_5^2-2x_6^2-x_7^2-2x_8^2-x_9^2-2x_{10}^2+1; \\ g_2= & {} -2x_1^2-x_2^2-2x_3^2-x_4^2-2x_5^2-x_6^2-2x_7^2-x_8^2-2x_9^2-x_{10}^2+1; \\ g_3= & {} -x_1^2-4x_2^2-x_3^2-4x_4^2-x_5^2-4x_6^2-x_7^2-4x_8^2-x_9^2-4x_{10}^2+1.25;\\ g_4= & {} -4x_1^2-x_2^2-4x_3^2-x_4^2-4x_5^2-x_6^2-4x_7^2-x_8^2-4x_9^2-x_{10}^2+1.25;\\ g_5= & {} -2x_1^2-3x_2^2-2x_3^2-3x_4^2-2x_5^2-3x_6^2-2x_7^2-3x_8^2-2x_9^2 \\&-3x_{10}^2+1.1; \quad x \ge 0. \end{aligned}$$
Example C10_4 (10 variables, degree 4):
$$\begin{aligned} f= & {} x_1^4+x_2^4+x_3^4+x_4^4+x_5^4+x_6^4+x_7^4+x_8^4+x_9^4+x_{10}^4 +3x_1^2x_2^2-x_1-x_2; \\ g_1= & {} -x_1^4-2x_2^4-x_3^4-2x_4^4-x_5^4-2x_6^4-x_7^4-2x_8^4-x_9^4-2x_{10}^4+1; \\ g_2= & {} -2x_1^4-x_2^4-2x_3^4-x_4^4-2x_5^4-x_6^4-2x_7^4-x_8^4-2x_9^4-x_{10}^4+1; \\ g_3= & {} -x_1^4-4x_2^4-x_3^4-4x_4^4-x_5^4-4x_6^4-x_7^4-4x_8^4-x_9^4-4x_{10}^4+1.25;\\ g_4= & {} -4x_1^4-x_2^4-4x_3^4-x_4^4-4x_5^4-x_6^4-4x_7^4-x_8^4-4x_9^4-x_{10}^4+1.25;\\ g_5= & {} -2x_1^4-3x_2^2-2x_3^4-3x_4^2-2x_5^4-3x_6^2-2x_7^4-3x_8^2-2x_9^4 \\&-3x_{10}^2+1.1; \quad x \ge 0. \end{aligned}$$
Example C20_2 (20 variables, degree 2):
$$\begin{aligned} f= & {} x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2+x_9^2+x_{10}^2 + 2x_1x_2-x_1-x_2 \\&+\,x_{11}^2+x_{12}^2+x_{13}^2+x_{14}^2+x_{15}^2+x_{16}^2+x_{17}^2+x_{18}^2+x_{19}^2+x_{20}^2;\\ g_1= & {} -x_1^2-2x_2^2-x_3^2-2x_4^2-x_5^2-2x_6^2-x_7^2-2x_8^2-x_9^2-2x_{10}^2 \\&-\,x_{11}^2-2x_{12}^2-x_{13}^2-2x_{14}^2-x_{15}^2-2x_{16}^2-x_{17}^2-2x_{18}^2-x_{19}^2-2x_{20}^2+1; \\ g_2= & {} -2x_1^2-x_2^2-2x_3^2-x_4^2-2x_5^2-x_6^2-2x_7^2-x_8^2-2x_9^2-x_{10}^2 \\&-\,2x_{11}^2-x_{12}^2-2x_{13}^2-x_{14}^2-2x_{15}^2-x_{16}^2-2x_{17}^2-x_{18}^2-2x_{19}^2-x_{20}^2+1;\\ g_3= & {} -x_1^2-4x_2^2-x_3^2-4x_4^2-x_5^2-4x_6^2-x_7^2-4x_8^2-x_9^2-4x_{10}^2 \\&-\,x_{11}^2-4x_{12}^2-x_{13}^2-4x_{14}^2-x_{15}^2-4x_{16}^2-x_{17}^2-4x_{18}^2-x_{19}^2-4x_{20}^2+1.25;\\ g_4= & {} -4x_1^2-x_2^2-4x_3^2-x_4^2-4x_5^2-x_6^2-4x_7^2-x_8^2-4x_9^2-x_{10}^2\\&-\,4x_{11}^2-x_{12}^2-4x_{13}^2-x_{14}^2-4x_{15}^2-x_{16}^2-4x_{17}^2-x_{18}^2-4x_{19}^2-x_{20}^2+1.25;\\ g_5= & {} -2x_1^2-3x_2^2-2x_3^2-3x_4^2-2x_5^2-3x_6^2-2x_7^2-3x_8^2-2x_9^2-3x_{10}^2\\&-\,2x_{11}^2-3x_{12}^2-2x_{13}^2-3x_{14}^2-2x_{15}^2-3x_{16}^2-2x_{17}^2-3x_{18}^2-2x_{19}^2\\&-\,3x_{20}^2+1.1; \quad x \ge 0. \end{aligned}$$
