SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Remark 2 (Using a RKHS norm instead of the  $F$ -norm). Note that since  $(\widehat{\lambda}_{\omega})_{\omega}$  sums to 1, the associated kernel is bounded by 1. Hence  $\| u \|_{L_{\infty}} \leq \| u \|_{\mathcal{H}_{\lambda}}$ , and the latter could be used instead of the  $F$ -norm in eq. (2). There are two reasons for taking our approach instead. Firstly, the  $F$ -norm is always tighter than a RKHS norm (see e.g. [Woodworth et al., 2022, Lem. 4]); secondly, we cannot compute  $\| u \|_{\mathcal{H}_{\lambda}}$  efficiently and have to rely instead on another upper bound. However, taking the number of samples  $N = O(\| u \|_{\mathcal{H}_{\lambda}}^{2})$  alleviates this issue.
Exponential concentration bounds with MoM. The scaling in  $1 / \sqrt{\delta}$  in theorem 2 can be prohibitive if one requires a high probability on the result ( $\delta \ll 1$ ). Hopefully, alternative estimator exist for those cases. The Median-of-Mean estimator is an example, illustrated in theorem 3.
Theorem 3 (Certificate with MoM estimator). Let  $(\widehat{\lambda}_{\omega})_{\omega}$  be a probability distribution on  $\mathbb{Z}^d$ , and  $\delta \in (0,1)$ . Draw  $N > 0$  frequencies  $\omega_{i} \sim \widehat{\lambda}_{\omega}$ . Define the MoM estimator with the following: for  $K \in \mathbb{N}$  s.t.  $\delta = e^{-K/8}$  and  $N = Kb$ ,  $b \geq 1$ , write  $B_1, \ldots, B_K$  a partition of  $[N]$ ; then
$$
\operatorname {M o M} _ {\delta} \left(\left| \widehat {u} _ {\omega_ {i}} \right| / \lambda_ {\omega_ {i}}\right) = \operatorname {m e d i a n} \left\{\frac {1}{b} \sum_ {i \in B _ {j}} \frac {\left| \widehat {f} _ {\omega_ {i}} - c \mathbf {1} _ {\omega_ {i} = 0} - \widehat {g} _ {\omega_ {i}} \right|}{\lambda_ {\omega_ {i}}} \right\} _ {1 \leq j \leq K}. \tag {11}
$$
A certificate on  $f_{\star}$  with probability  $1 - \delta$  follows, with
$$
f _ {\star} \geq c - \operatorname {M o M} _ {\delta} \left(\left| \widehat {u} _ {\omega_ {i}} \right| / \lambda_ {\omega_ {i}}\right) - 4 \sqrt {2} \| f - c - g \| _ {\mathcal {H} _ {\lambda}} \sqrt {\frac {\log (1 / \delta)}{N}}. \tag {12}
$$
Proof. Using e.g. Theorem 4.1 from Devroye et al. [2016] we get that the deviation of the MoM estimator from the expectation is bounded by
$$
\left\| u \right\| _ {F} - \operatorname {M o M} _ {\delta} \left(\left| \widehat {u} _ {\omega_ {i}} \right| / \lambda_ {\omega_ {i}}\right) \mid \leq 4 \sqrt {2} \sqrt {\operatorname {V a r} \left(\left| \widehat {u} _ {\omega} \right| / \widehat {\lambda} _ {\omega}\right) \frac {\log (1 / \delta)}{N}} \text {w i t h p r o b a .} 1 - \delta . \tag {13}
$$
Using the upper bound on the variance with the  $\mathcal{H}_{\lambda}$  norm given in eq. (8) and plugging the resulting expression into eq. (2), we obtain the result.
To conclude this section, bounding the  $L_{\infty}$  norm from above with the  $F$ -norm in eq. (3) enables to obtain a certificate on  $f$ , as shown in theorem 1. The  $F$ -norm requires an infinite number of computation in the general case, but can be bounded efficiently with a probabilistic estimate, given by theorem 2 or theorem 3. This is summed up in fig. 1. Note that the difference  $\| \cdot \|_F - \| \cdot \|_{L_\infty}$  is a source of conservatism in the certificate which we do not quantify – yet, the  $F$ -norm is optimal for a class of norms, see [Woodworth et al., 2022, Lemma 3].
3 Algorithm and implementation
3.1 Bessel kernel
We now detail the specific choice of kernel we make in order to compute the certificate of theorem 2 or theorem 3 efficiently. Our first observation is to use a kernel stable by product, so that we can easily characterize a Hilbert space the model  $g$  belongs to. This restricts the choice to the exponential family. That's why we define, for a parameter  $s > 0$ ,
$$
\forall x \in \mathbb {T}, q _ {s} (x) = e ^ {s (\cos (2 \pi x) - 1)} = \sum_ {\omega \in \mathbb {Z}} e ^ {- s} I _ {| \omega |} (s) e ^ {2 \pi \mathrm {i} \omega x}, \tag {14}
$$
with  $I_{|\omega|}(\cdot)$  the modified Bessel function of the first kind [Watson, 1922, p.181]. Then, define  $K_s(x,y) = q_s(x - y)$  as in definition 2, and take a tensor product to extend the definition of  $K$  to multiple dimension, i.e.  $K_{\mathbf{s}}(\mathbf{x},\mathbf{y}) = \prod_{\ell=1}^{d} K_{\mathbf{s}_{\ell}}(\mathbf{x}_{\ell},\mathbf{y}_{\ell})$  for any  $\mathbf{x},\mathbf{y} \in \mathbb{T}^d$ . We refer to this kernel as the Bessel kernel, and the associated RKHS as  $\mathcal{H}_{\mathbf{s}}$ . It is stable by product as  $K_{\mathbf{s}}(x,y) = K_{\mathbf{s}/2}(x,y)^2$ . This is key to compute the Fourier transform of the model  $g$ , and in contrast to previous approaches which used the exponential kernel with  $\widehat{q}_{\omega} \propto e^{-s|\omega|}$  Woodworth et al. [2022], Bach and Rudi [2023].
In the following,  $g$  is a K-SoS model defined as in definition 1, with the Bessel kernel of parameter  $\mathbf{s} \in \mathbb{R}_+^d$  defined in eq. (14).
Lemma 1 (Fourier coefficient of the Bessel kernel). For  $\omega \in \mathbb{Z}^d$ , the Fourier coefficient of  $g$  in  $\omega$  can be computed in  $O(drm^2)$  time with
$$
\widehat {g} _ {\omega} = \sum_ {i, j = 1} ^ {m} R _ {i} ^ {\top} R _ {j} \prod_ {\ell = 1} ^ {d} e ^ {- 2 \mathbf {s} _ {\ell}} I _ {| \omega_ {\ell} |} (2 s \cos \pi (\mathbf {z} _ {i \ell} - \mathbf {z} _ {j \ell})) e ^ {- \mathrm {i} \pi \omega_ {\ell} (\mathbf {z} _ {i \ell} + \mathbf {z} _ {j \ell})}. \tag {15}
$$
Proof. From its definition in eq. (5), we rewrite  $g$  as
$$
g (x) = \sum_ {i, j = 1} ^ {m} R _ {i} ^ {\top} R _ {j} \prod_ {\ell = 1} ^ {d} K _ {s} \left(x, \mathbf {z} _ {i \ell}\right) K _ {s} \left(x, \mathbf {z} _ {j \ell}\right). \tag {16}
$$
Now, from the definition of the Bessel kernel in eq. (14), we have that for any  $(x,y,z)\in \mathbb{T}$ $K(x,y)K(x,z) = e^{-2s}e^{2s\cos (2\pi (y - z) / 2)\cos 2\pi (x - (y + z) / 2)}$  . By definition of the modified Bessel function, the Fourier coefficient of this expression are given by  $I_{|\omega |}(2s\cos (2\pi (y - z) / 2))$  . Using this into eq. (16), we get the result.
The second necessary ingredient for using the certificate of theorem 2 is computing a RKHS norm for  $g$ . It relies on the inclusion of  $\mathcal{H}_{2\mathrm{s}}$  into the bigger space of symmetric operator  $S(\mathcal{H}_{\mathrm{s}})$ .
Lemma 2 (Bound on the RKHS norm of  $g$ ).  $g$  belongs to  $\mathcal{H}_{2s}$ , and  $\| g \|_{\mathcal{H}_{2s}}$  is bounded by the Hilbert-Schmidt norm of  $S(\mathcal{H}_s)$ , which can be computed in  $O(dm^2 + mr^2)$  time, with
$$
\left\| g \right\| _ {\mathcal {H} _ {2 s}} ^ {2} \leq \left\| g \right\| _ {\mathcal {S} \left(\mathcal {H} _ {s}\right)} ^ {2} = \operatorname {T r} \left(R K _ {s, \mathbf {z}} R ^ {\top}\right) ^ {2}. \tag {17}
$$
Proof. Assume that  $d = 1$ ; the reasoning can be extended to multiple dimensions with the tensor product. From the computation of the Fourier coefficient in lemma 1 and the fact that  $I_{|\omega|}(2s\cos(2\pi \cdot)) \leq I_{|\omega|}(2s)$ , we have that  $\widehat{g}_{\omega} = O(I_{|\omega|}(2s))$  hence  $g \in \mathcal{H}_{2s}$ . Finally, since the kernel is stable by product,  $\mathcal{H}_{2s} = \mathcal{H}_s \odot \mathcal{H}_s$ , so we can use e.g. [Paulsen and Raghupathi, 2016, Thm. 5.16], with  $\mathcal{H}_1 = \mathcal{H}_2 = \mathcal{H}_s$  and  $\mathcal{S}(\mathcal{H}_s) = \mathcal{H}_s \otimes \mathcal{H}_s$ , with the operator  $A = (\varphi(\mathbf{z}_1), \ldots, \varphi(\mathbf{z}_m))R^\top R(\varphi(\mathbf{z}_1), \ldots, \varphi(\mathbf{z}_m))^* \in S(\mathcal{H}_s)$ .
With lemma 2, we have that the model  $g$  belongs to  $\mathcal{H}_{2\mathbf{s}}$ , so we will naturally use  $\widehat{\lambda}_{\omega} = \prod_{\ell=1}^{d} e^{-2\mathbf{s}_{\ell}} I_{\omega}(2\mathbf{s}_{\ell})$  in theorem 2; said differently, the space  $\mathcal{H}_{\lambda}$  introduced in eq. (8) is simply  $\mathcal{H}_{2\mathbf{s}}$  defined in eq. (14).
3.2 The algorithm: GloptiNets
We can now describe how GloptiNets yields a certificate on  $f$ . The key observation is that no matter how is obtained our model  $g(R, \mathbf{z})$  from definition 1, we will always be able to compute a certificate with theorems 2 and 3. Thus, even though optimizing eq. (9) w.r.t  $(c, R, \mathbf{z})$  is highly non-convex, we can use any optimization routine and check empirically its efficiency by looking at the certificate. Finally, thanks to its low-rank structure it is cheaper to evaluate  $g$  than evaluating its Fourier coefficient. This is formally shown in proposition 2 in appendix A, where a block-diagonal structure for the model is also introduced. That's why we first optimize  $\sup_{c, g} c - \|f - c - g\|_{\star}$ , where  $\| \cdot \|_{*}$  is a proxy for the  $L_{\infty}$  norm, e.g. the log-sum-exp on a random batch of  $N$  points:
$$
\left\| f - c - g \right\| _ {L _ {\infty}} \approx \max  _ {i \in [ N ]} | f (x _ {i}) - c - g (x _ {i}) | \approx \operatorname {L S E} (f (x _ {i}) - c - g (x _ {i})) _ {i \in [ N ]}. \tag {18}
$$
This optimization can be carried out by any deep learning libraries with automatic differentiation and any flavour of gradient ascent. Only afterwards do we compute the certificate with theorems 2 and 3. This procedure is summed up in alg. 1.
Data: A trigonometric polynomial  $f$ , a candidate  $z$  s.t.  $c = f(z)$ , a model  $g$ , and a probability  $\delta$ .
Result: A certificate  $|f_{\star} - f(z)| \leq \epsilon_{\delta}$  with proba.  $1 - \delta$ .
/\*Optimize  $g$  with function values \*/
for epoch  $= 1$  :nepochs do
Algorithm 1: GloptiNets
Sample  $x_{1},\ldots ,x_{N}\in \mathbb{T}^{d}$ $L,\nabla L =$  autodiff(LSE  $(f(x_{i}) - c - g(x_{i}))_{i\in [N]})$  .  $\mathbf{z},R\gets$  optimizer  $(\nabla L)$
/\* Compute a certificate
$\widehat{\lambda}_{\omega}$ : probability distribution on  $\mathbb{Z}^d$  with  $\widehat{\lambda}_{\omega} = \prod_{\ell=1}^{d} e^{-2\mathbf{s}_{\ell}} I_{\omega}(2\mathbf{s}_{\ell})$ ;
Sample  $\Omega = (\omega_{1},\dots ,\omega_{N})\sim \widehat{\lambda}_{\omega}$
Compute  $M = \mathsf{MoM}_{\delta}(|\widehat{f}_{\omega_i} - c\mathbf{1}_{\omega_i = 0} - \widehat{g}_{\omega_i}| / \lambda_{\omega_i})_{i\in [n]}$  and  $\bar{\sigma} = \| g\|_{S(\mathcal{H}_{\mathbf{s}})}$
Returns  $\epsilon_{\delta} = c - M - 4\sqrt{2}\bar{\sigma}\sqrt{\log(1 / \delta) / N}$
Remark 3 (Providing a candidate). In alg. 1, a candidate estimate  $c$  for the minimum value  $f(x_{\star})$  is necessary. However, it is possible to overcome this requirement by incorporating  $c$  as a learnable parameter within the training loop. Moreover,  $x_{\star}$  can be learned using techniques similar to those in Rudi et al. [2020]: by replacing the lower bound  $c$  with a parabola centered at  $z$ ,  $z$  becomes a candidate for  $x_{\star}$  with precision corresponding to the tightness of the certificate. Note however that this method introduces additional hyperparameters.
3.3 Specific implementation for the Chebychev basis
As already observed in Bach and Rudi [2023], a result on trigonometric polynomial on  $\mathbb{T}^d$  directly extends to a real polynomials on  $[-1, 1]^d$ . The reason for that is that minimizing  $h$  on  $[-1, 1]^d$  amounts to minimizing the trigonometric polynomial  $f = h((\cos 2\pi x_1, \ldots, \cos 2\pi x_d))$  on  $\mathbb{T}^d$ . Note however that  $f$  is an even function in all dimension, as for any  $x \in \mathbb{T}^d$ ,  $f(x) = f(x_1, \ldots, -x_i, \ldots, x_d)$ . Thus, approximating  $f - f_\star$  with a K-SoS model of definition 1 is suboptimal, in the sense that we could approximate  $f$  only on  $[0, 1/2]^d$ , which is  $2^{-d}$  smaller. Put differently, the Fourier coefficient of  $f$  are real by design: it would be convenient to enforce this structure in the model  $g$ . This is achieved with proposition 1.
Another detail of practical importance is that this loss can be efficiently backpropagated through; on the other hand, the certificate is not easily vectorized, and the Bessel function involved would require specific approximation to be efficiently backpropagated through.
Proposition 1 (Kernel defined on the Chebychev basis). Let  $q$  be a real, even function on the torus, bounded by 1, as in eq. (6). Let  $K$  be the kernel defined on  $[-1,1]$  by
$$
\forall (u, v) \in (0, 1 / 2), K (\cos 2 \pi u, \cos 2 \pi v) = \frac {1}{2} (q (u + v) + q (u - v)). \tag {19}
$$
Then  $K$  is a symmetric, p.d., hence reproducing kernel, bounded by 1, with explicit feature map given by
$$
\forall (x, y) \in [ - 1, 1 ], K (x, y) = \widehat {q} _ {0} + \sum_ {\omega \in \mathbb {N}} 2 \widehat {q} _ {\omega} H _ {\omega} (x) H _ {\omega} (y). \tag {20}
$$
The proof is available in appendix B. It simply relies on expanding the definition of  $K$  in eq. (19). The resulting expression in eq. (20) exhibits only cosine terms (in the decomposition of  $x \mapsto K(\cos 2\pi x, y)$ ). This enables to directly extend the PSD models from definition 1 with such kernels. Finally, when used with the Bessel kernel of eq. (14), we recover an easy computation of the Chebychev coefficient, as shown in lemma 3, in  $O(drm^2)$  time. This enables to approximate any function expressed on the Chebychev basis. Note that polynomials expressed in other basis can be certified too, by first operating a change of basis.
The code to reproduce these experiments is available at
github.com/gaspardbb/GloptiNets.jl
Settings. Given a function  $h$ , we compute a candidate  $\widehat{x}$  with gradient descent and multiple initializations. The goal is then to certify that  $\widehat{x}$  is indeed a global minimizer of  $h$ . This is a common setup in the Polynomial-SoS literature Wang and Magron [2022]. To illustrate the influence of the number of parameters, the positive model  $g$  defined in definition 1 for GloptiNets designates either a small model GN-small with 1792 parameters, or a bigger model GN-big with 22528 parameters. The latter should have higher expressivity and better interpolate positive functions, leading to tighter certificates. All results for GloptiNets are obtained with confidence  $1 - \delta = 1 - e^{-4} \geq 98\%$ . All other details regarding the experiments are reported in appendix C.
Figure 2: Certificate vs. number of parameters in  $g$  for a given function  $h$ . The higher the RKHS norm of  $h$ , the more difficult it is to approximate uniformly and the looser the certificate, independently of the function type. The more parameters in the k-SoS model, the tighter the certificates obtained with theorem 3.
Polynomials. We first consider the case where  $h$  is a random trigonometric polynomial. Note that this is a restrictive analysis, as GloptiNets can handle any smooth functions (i.e. with infinite non-zero Fourier coefficients). Polynomials have various dimension  $d$ , degree  $p$ , number of coefficients  $n$ , but a constant RKHS norm  $\mathcal{H}_{21d}$ . We compare the performances of GloptiNets to TSSOS, in its complex polynomial variant Wang and Magron [2022]. The latter is used with parameters such that it executes the fastest, but without guarantees of convergence to the global minimum  $f_{\star}$ . Table 1 shows the certificates  $h(x_{\star}) - h(\widehat{x})$  and the execution times (lower is better,  $t$  in seconds) for TSSOS, GN-small and GN-big. Figure 2 provides certificate on a random polynomial, function of the number of parameters in  $g$ .
Kernel mixtures. While polynomials provide ground for comparison with existing work, GloptiNets is not confined to this function class. This is evidenced by experiments on kernel mixtures, where our approach stands as the only viable alternative we are aware of. The function we certify are of the form  $h(x) = \sum_{i=1}^{n} \alpha_i K(x_i, x)$ , where  $K$  is the Bessel kernel of eq. (14). Kernel mixtures are ubiquitous in machine learning and arise e.g. when performing kernel ridge regression. Certificates obtained on mixtures are compared with those obtained on polynomials in fig. 2, function of the model size  $g$ .
Table 1: GloptiNets and TSSOS on random trigonometric polynomials. While TSSOS provides machine-precision certificates, its running time grows exponentially with the problem size, and eventually fails on problems 3 and 6. On the other hand, GloptiNets has constant running time no matter the problem size, and its certificates can be tightened by increasing the model size.
d	p	n	TSSOS	GN-small	GN-big
Certif.	t	Certif.	t	Certif.	t
3	5	85	5.3·10-11	3	8.35·10-4	6·103	2.64·10-4	9·103
7	231	4.7·10-13	120	9.51·10-4	6·103	2.90·10-4	9·103
9	489	out of memory!	-	1.18·10-3	6·103	3.34·10-4	9·103
4	3	33	3.1·10-10	0.1	2.46·10-2	1·104	3.45·10-3	2·104
5	225	4.8·10-12	53	3.71·10-2	1·104	3.59·10-3	2·104
7	833	out of memory!	-	4.76·10-2	1·104	4.85·10-3	2·104
Results. There are two key hindsight about the performances of GloptiNets. Firstly, its certificate does not depend on the structure of the function to optimize. Thus, although GloptiNets does not match the performances of TSSOS on small polynomials, it can tackle polynomials which cannot be handled by competitors, with arbitrarily as many coefficients  $(n = \infty)$ . For instance, TSSOS cannot handle problems with  $n \in \{489, 833\}$  in table 1. More importantly, GloptiNets can certify a richer class of functions than polynomials, among which kernel mixtures. The performances of GloptiNets mostly depends on the complexity of the function to certify, as measured with its RKHS norm.
Secondly, note that a bigger model yields tighter certificate. This is detailed in fig. 2, where the same function  $f$  is optimized with various models. The dependency of the certificate on the norm of  $f$  is shown in fig. 3 in appendix C, along with experiments with Chebychev polynomials.
One limitation of GloptiNets is the trade-off resulting from its high flexibility for obtaining a certificate as in alg. 1. While this flexibility offers numerous advantages, it also introduces the need for an extensive hyperparameter search. Although we have identified a set of hyperparameters that align with deep learning practices – utilizing a Momentum optimizer with cosine decay and a large initial learning rate – the optimal settings may vary depending on the specific problem at hand.
In the same vein, the certificates given by GloptiNets are of moderate accuracy. While adding more parameters into the k-SoS model certainly helps (as shown in fig. 2), alternative optimization schemes to interpolate  $h - h(\hat{x})$  with  $g$  might provide easier improvement. For instance, we found that using approximate second-order scheme in alg. 1 is key to obtaining good certificates.
In the specific settings of polynomial optimization, we highlight that our model is not competitive on problems which exhibits some algebraic structure, as for instance term sparsity or the constant trace property. Typically, problems with coefficients of low degrees (less or equal than 2), which encompass notably the OPF problem, are really well handled by the family of solvers TSSOS belongs to. Finally, GloptiNets does not handle constraints yet.
The GloptiNets algorithm presented in this work lays the foundation for a new family of solvers which provide certificates to non-convex problems. While our approach does not aim to replace the well-established Lasserre's hierarchy for sparse polynomials, it offers a fresh perspective on tackling a new set of problems at scale. Through demonstrations on synthetic examples, we have showcased the potential of our approach. Further research directions include extensive parameter tuning to obtain tighter certificates, with the possibility of leveraging second-order optimization schemes, along with warm-restart schemes for application which requires solving multiple similar problems sequentially.
Acknowledgments. AR acknowledges support of the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). AR acknowledges support of the European Research Council (grant REAL 947908). JM was supported by the ERC grant number 101087696 (APHE-LAIA project) and by ANR 3IA MIAI@Grenoble Alpe (ANR-19-P3IA-0003).
Francis Bach and Alessandro Rudi. Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials. SIAM Journal on Optimization, 33(3):2137-2159, 2023.
Eloise Berthier, Justin Carpentier, Alessandro Rudi, and Francis Bach. Infinite-Dimensional Sums-of-Squares for Optimal Control. In 2022 IEEE 61st Conference on Decision and Control (CDC), pages 577-582, December 2022. doi: 10.1109/CDC51059.2022.9992396.
Stephen P Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
Luc Devroye, Matthieu Lerasle, Gabor Lugosi, and Roberto I. Oliveira. Sub-Gaussian Mean Estimators. The Annals of Statistics, 44(6):2695-2725, 2016.
Dinh Dūng, Vladimir N. Temlyakov, and Tino Ullrich. Hyperbolic Cross Approximation. arXiv:2211.04889, April 2017.
Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. MIT press, 2016.
Didier Henrion, Milan Korda, and Jean-Bernard Lasserre. The Moment-SOS Hierarchy, volume 4 of Optimization and Its Applications. World Scientific Publishing Europe Ltd., December 2020. doi: 10.1142/q0252.
Joseph J. Hilling and Anthony Sudbery. The geometric measure of multipartite entanglement and the singular values of a hypermatrix. Journal of Mathematical Physics, 51(7):072102, July 2010.
Reiner Horst and Panos M Pardalos. Handbook of global optimization, volume 2. Springer Science & Business Media, 2013.
Cédric Josz and Daniel K. Molzahn. Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables. SIAM Journal on Optimization, 28(2):1017-1048, January 2018.
Jean B. Lasserre. Global Optimization with Polynomials and the Problem of Moments. SIAM Journal on Optimization, 11(3):796-817, January 2001. doi: 10.1137/S1052623400366802.
Jean Bernard Lasserre. Moments, Positive Polynomials and Their Applications, volume 1 of Series on Optimization and Its Applications. October 2009. doi: 10.1142/p665.
Monique Laurent and Lucas Slot. An effective version of schmudgen's positivstellensatz for the hypercube. Optimization Letters, September 2022. doi: 10.1007/s11590-022-01922-5.
Ngoc Hoang Anh Mai, J. B. Lasserre, Victor Magron, and Jie Wang. Exploiting Constant Trace Property in Large-scale Polynomial Optimization. ACM Transactions on Mathematical Software, 48(4):40:1-40:39, December 2022.
Ulysse Marteau-Ferey, Francis Bach, and Alessandro Rudi. Non-parametric Models for Non-negative Functions. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 12816-12826. Curran Associates, Inc., 2020.
Pablo Moscato et al. On evolution, search, optimization, genetic algorithms and martial arts: Towards memetic algorithms. Caltech concurrent computation program, C3P Report, 826(1989):37, 1989.
Boris Muzellec, Adrien Vacher, Francis Bach, François-Xavier Vialard, and Alessandro Rudi. Near-optimal estimation of smooth transport maps with kernel sums-of-squares. arXiv:2112.01907, December 2021.