SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To evaluate the effectiveness of the proposed method, we carried out a series of 20 numerical experiments for various formulations of model problems. The results are presented in Table 1, where we report the approximation to the minimum value for each model problem (P-1, P-2, ..., P-20) and all considered optimization methods (PROTES, BS-1, BS-2, ..., BS-7). Taking into account the analysis of discrete optimization methods in the previous section as baselines we consider two tensor-based optimization methods: TTOpt $^5$  (BS1) and Optima-TT $^6$  (BS2), and five popular gradient-free optimization algorithms from the nevergrad framework [3]: OnePlusOne (BS3), PSO (BS4), NoisyBandit (BS5), SPSA (BS6), and Portfolio approach (BS7), which is based on the combination of CMA-ES, DE, and scr-Hammersley methods. The model problems and obtained results will be discussed in detail below in this section.
For all the considered optimization problems, we used the default set of parameters for baselines, and for PROTES we fixed parameters as  $K = 100$ ,  $k = 10$ ,  $k_{gd} = 1$ ,  $\lambda = 0.05$ ,  $R = 5$  (the description
We used implementation of the method from https://github.com/AndreiChertkov/ttopt.
<sup>6</sup>We used implementation from https://github.com/AndreiChertkov/teneva. The TT-tensor for optimization was generated by the TT-cross method.
See https://github.com/facebookresearch/nevergrad.
Table 1: Minimization results for all selected benchmarks (P-01 – P-20). The values obtained by the proposed method PROTES and by all considered baselines (BS1 – BS7) are reported.
PROTES	BS-1	BS-2	BS-3	BS-4	BS-5	BS-6	BS-7
ANALYTIC FUNCTIONS	P-01	1.3E+01	1.3E+01	1.3E+01	1.3E+01	1.3E+01	2.1E+01	1.3E+01	1.3E+01
P-02	6.5E+00	6.5E+00	6.5E+00	6.9E+00	6.8E+00	1.5E+01	7.5E+00	6.8E+00
P-03	-9.4E-01	-9.4E-01	-9.4E-01	-9.4E-01	-9.4E-01	-3.5E-01	-9.4E-01	-9.4E-01
P-04	1.3E+00	1.3E+00	1.3E+00	1.3E+00	1.3E+00	6.3E+00	1.3E+00	1.3E+00
P-05	-3.7E+00	-3.7E+00	-3.7E+00	-2.6E+00	-3.0E+00	-1.8E+00	-1.2E+00	-3.7E+00
P-06	1.2E-01	1.2E-01	1.2E-01	1.2E-01	1.2E-01	1.3E-01	4.2E-01	1.2E-01
P-07	6.2E+06	6.2E+06	6.2E+06	6.3E+06	1.7E+07	2.2E+10	3.1E+08	6.2E+06
P-08	6.0E+01	6.0E+01	6.0E+01	6.0E+01	6.0E+01	1.2E+02	1.0E+02	6.0E+01
P-09	2.7E+00	2.7E+00	2.7E+00	3.0E+00	2.7E+00	2.9E+00	3.4E+00	2.7E+00
P-10	-8.7E+02	-8.7E+02	-8.7E+02	-6.1E+02	-6.9E+02	7.0E+02	2.6E+03	-8.5E+02
QUBO	P-11	-3.6E+02	-3.5E+02	-3.4E+02	-3.2E+02	-3.4E+02	-3.2E+02	-3.3E+02	-3.6E+02
P-12	-5.9E+03	-5.9E+03	-5.9E+03	-5.6E+03	-5.9E+03	-5.3E+03	-5.9E+03	-5.9E+03
P-13	-3.1E+00	-3.0E+00	-2.8E+00	0.0E+00	1.5E+01	2.8E+02	-2.9E+00	-3.0E+00
P-14	-3.1E+03	-2.8E+03	-3.0E+03	-2.6E+03	-3.0E+03	-2.7E+03	-3.0E+03	-3.0E+03
CONTROL	P-15	6.7E-03	7.4E-03	2.3E-02	8.4E-03	8.9E-03	3.1E-02	8.7E-02	7.3E-03
P-16	1.4E-02	2.6E-02	3.5E-02	1.7E-02	1.7E-02	5.3E-02	5.2E-02	1.4E-02
P-17	3.0E-02	5.7E-01	1.5E-01	4.8E-02	3.6E-02	7.7E-02	5.3E-02	3.7E-02
CONTROL +CONSTR.	P-18	1.4E-02	1.1E-02	1.4E-02	3.4E-02	6.2E-02	2.8E-01	6.4E-02	2.1E-02
P-19	6.4E-02	5.7E-01	6.7E-02	FAIL	FAIL	FAIL	FAIL	FAIL
P-20	1.5E-01	FAIL	2.0E-01	FAIL	FAIL	FAIL	FAIL	FAIL
of these parameters was presented in Algorithm 1). For all methods, the limit on the number of requests to the objective function was fixed at the value  $M = 10^{4}$ . As seen from Table 1, PROTES, in contrast to alternative approaches, gives a consistently top result for almost all model problems (the best result for 19 of the 20 problems considered).
6.1 Multivariable analytic functions
First, we consider the optimization task for various tensors arising from the discretization of multivariable analytic functions. We select 10 popular benchmarks: Ackley (P-01), Alpine (P-02), Exponential (P-03), Griewank (P-04), Michalewicz (P-05), Piston $^{8}$  (P-06), Qing (P-07), Rastrigin (P-08), Schaffer (P-09) and Schwefel (P-10). These functions have a complex landscape and are often used in problems to evaluate the effectiveness of optimization algorithms [8, 16], including tensor-based optimizers [4, 39]. We consider the 7-dimensional case (since this is the dimension of the Piston function) and discretization on a uniform grid with 16 nodes.
As follows from Table 1 (benchmarks P-1, P-2, ..., P-10), our method, like the other two tensor approaches (BS-1 and BS-2), gave the most accurate solution for all model problems. The most sophisticated approach from the nevergrad package (BS-7) was the next in accuracy (the method did not converge only in two cases out of ten).
6.2 Quadratic unconstrained binary optimization
QUBO is a widely known NP-hard problem [12] which unifies a wide variety of combinatorial optimization problems from finance and economics applications to machine learning and quantum computing. QUBO formulation in a very natural manner utilizes penalty functions, yielding exact model representations in contrast to the approximate representations produced by customary uses of penalty functions. The standard QUBO problem can be formulated as follows
$$
\mathsf {f} (\pmb {x}) = \pmb {x} ^ {T} Q \pmb {x} \to \underset {\pmb {x}} {\min}, \quad s. t. \pmb {x} \in \{0, 1 \} ^ {d},
$$
where  $\pmb{x}$  is a vector of binary decision variables of the length  $d$  and  $Q \in \mathbb{R}^{d \times d}$  is a square matrix of constants. In all our experiments, we fixed the number of dimensions as  $d = 50$ .
This function corresponds to the problem of modeling the time that takes a piston to complete one cycle within a cylinder; the description of its parameters can be found in [44, 4].
Table 2: Minimization results for benchmark P-12 for the 1000 dimensional case with different random initializations. Results for each value of  $p$  are averaged over 5 runs of optimizers.
PROTES	BS-1	BS-2	BS-3	BS-4	BS-5	BS-6	BS-7
P-12	p = 0.1	-4.86E+05	-4.40E+05	FAIL	-4.10E+05	-4.35E+05	-3.94E+05	-3.84E+05	-4.71E+05
p = 0.2	-9.70E+05	-8.78E+05	FAIL	-8.24E+05	-8.71E+05	-7.87E+05	-7.64E+05	-9.38E+05
p = 0.3	-1.46E+06	-1.32E+06	FAIL	-1.23E+06	-1.30E+06	-1.18E+06	-1.14E+06	-1.41E+06
p = 0.4	-1.94E+06	-1.75E+06	FAIL	-1.64E+06	-1.72E+06	-1.57E+06	-1.52E+06	-1.88E+06
p = 0.5	-2.43E+06	-2.19E+06	FAIL	-2.04E+06	-2.16E+06	-1.96E+06	-1.90E+06	-2.35E+06
We consider the following QUBO problems from the qubogen package: Max-Cut Problem (P-11; which refers to finding a partition of an undirected graph into two sets such that the number of edges between the two sets is as large as possible), Minimum Vertex Cover Problem (P-12; which refers to finding a cover with a minimum number of vertices in the subset of the graph vertices such that each edge in the graph is incident) and Quadratic Knapsack Problem (P-13; which refers to finding a subset of maximum profit that satisfies the budget limitations from a set of potential projects with specified interactions between pairs of projects).
We also consider one more benchmark (P-14) from the work [11] (problem  $k_{3}$ ;  $d = 50$ ), where angle-modulated bat algorithm (AMBA) was proposed for high-dimensional QUBO problems with engineering application to antenna topology optimization. This is the ordinary binary knapsack problem with fixed weights  $w_{i} \in [5,20]$ , profits  $p_i \in [50,100]$  ( $i = 1,2,\dots,d$ ), and the maximum capacity  $C = 1000$ . In experiments, we used the same values of the weights and profits as in [11].
For all four considered problems (P-11, P-12, P-13, P-14), the proposed method PROTES gives the best result, as seen from Table 1, and the baseline BS-7 again turned out to be the next in accuracy. We also note that several optimization methods were compared in [11] for the P-14 problem: BPSO (with the result  $-2854$ ), BBA (with the result  $-2976$ ), AMBA (with the result  $-2956$ ), A-AMBA (with the result  $-2961$ ), P-AMBA (with the result  $-2989$ ), and the solution obtained using the PROTES method (the result  $-3079$ ) turns out to be more accurate.
To study the stability of the proposed optimization method in the essentially multidimensional case, we also consider  $d = 1000$  for the benchmark P-12. Additionally, we vary the value of the vertex connection probability in the generated random graph (i.e., the parameter in the corresponding function from the well-known networkx package), getting 5 different optimization problems with  $p = 0.1$ ,  $p = 0.2$ ,  $p = 0.3$ ,  $p = 0.4$ , and  $p = 0.5$  (note that in Table 1, we used the value  $p = 0.5$ ). We performed the calculations with PROTES and baselines BS1 - BS7 with the same settings as above. In Table 2, we report the results averaged over 5 runs of each optimizer for each value of  $p$ . In all cases, PROTES demonstrates the best result. We note that BS2 failed for this problem because the selected dimension value is too large to construct the TT-approximation of the tensor.
In Figure 3, we plotted the convergence curves for  $p = 0.1$ ,  $p = 0.3$ ,  $p = 0.5$ , and all optimizers (except the BS2, since it failed), showing the average result and the spread of values over 5 restarts for P-12 ( $d = 1000$ ). Our optimizer PROTES has a slight variance of the solution and consistently shows the best result when the number of iterations exceeds 2000.
6.3 Optimal control
Suppose we have a state variable  $z \in \mathbb{R}$  controlled by a binary variable  $x$  called control (i.e., it is just a switch with modes "off" = 0 and "on" = 1) over some discrete interval of time  $[0, T]$ . The state  $z(t + 1)$  at time  $t + 1$  depends on the control  $x(t)$  at time  $t$  and is obtained from the solution of the following differential equation  $\dot{z}(\tau) = \mathsf{g}(z(\tau), x(t))$ ,  $t \leq \tau < t + 1$ , where the function  $\mathsf{g}$  is called an equation function. The optimal control problem is to find such a sequence of controls  $\boldsymbol{x}^* = [x^*(0), x^*(1), \dots, x^*(T)]$  (optimal solution) over the given time interval  $[0, T]$  that minimizes the given objective function  $\mathsf{G}$ .
See https://github.com/tamukey/qubogen.
(a)  $p = 0.1$  (scaled  $\times 10^{-5}$ )
(b)  $p = 0.3$  (scaled  $\times 10^{-6}$ )
(c)  $p = 0.5$  (scaled  $\times 10^{-6}$ )
Figure 3: Minimization results for benchmark P-12 in the 1000-dimensional case. The results for various values of the vertex connection probability  $p$  of the generated random graph are presented. For each of the optimizers PROTES, BS1, and BS3 - BS7 (BS2 failed for this problem), we plot the value of the solution averaged over 5 runs with a solid line and fill in the area between the worst and best result with the same color.
Formulating the problem mathematically, we need to find such a solution
$$
\mathsf {G} (\boldsymbol {z}, \boldsymbol {x}) \to \min  _ {\boldsymbol {z}, \boldsymbol {x}} \quad \text {s . t .} \left\{ \begin{array}{l} z (0) = z _ {0}, \\ \dot {z} (\tau) = \mathsf {g} (z (\tau), \boldsymbol {x} (t)), t \leq \tau <   t + 1, \\ \boldsymbol {x} (t) \in \{0, 1 \}, t = 0, 1, \ldots , T, \end{array} \right.
$$
where  $\boldsymbol{z} = [z(0), z(1), \ldots, z(T)]$  is a state variable path. In numerical experiments, we consider the nonlinear equation function  $\mathsf{g}(z, x) = z^3 - x$ , and since it is nonlinear, finding an optimal solution raises a lot of difficulties. We take the objective function  $\mathsf{G}$  in the form  $\mathsf{G}(\boldsymbol{z}, \boldsymbol{x}) = \frac{1}{2} \sum_{t=0}^{T} (z(t) - z_{\mathrm{ref}})^2$ . The initial and the reference state are fixed at values  $z_0 = 0.8$ ,  $z_{\mathrm{ref}} = 0.7$ . For a fixed initial value  $z_0$  and fixed equation function  $\mathsf{g}$ , the objective function  $\mathsf{G}$  can be represented as a binary multidimensional tensor, whose elements are calculated using the following function:  $\mathsf{f}(\boldsymbol{x}) = \mathsf{G}(\boldsymbol{z}(\boldsymbol{x}), \boldsymbol{x})$ , hence we can apply discrete optimization methods to find  $\boldsymbol{x}_{\min}$ , which approximates the optimal solution  $\boldsymbol{x}^*$ .
We considered several values for variable  $T$ , such as 25, 50, and 100 (benchmarks P-15, P-16, and P-17, respectively). As follows from the results presented in Table 1, PROTES gives the most accurate solution in all three cases. Note that a result comparable in accuracy to our method is obtained only in one case when using baselines (i.e., P-16, BS-7).
6.4 Optimal control with constraints
In practical applications, some conditions or constraints may be imposed on the solution of the optimal control problem. In this work, we consider the following control constraint  $P$ : the control variable  $x \in \{0,1\}^N$  can take the value "1" no less than 3 times in a row during the whole time interval. Formally, this can be written as follows:
$$
P = \left\{\boldsymbol {x} \left| \begin{array}{l} x [ t ] \geq x [ t - 1 ] - x [ t - 2 ] \\ x [ t ] \geq x [ t - 1 ] - x [ t - 3 ] \end{array} , \forall t: 1 \leq t \leq N + 2; \text {w e l e t} x [ t ] := 0 \text {f o r} t <   1 \text {a n d} t > N \right. \right\}.
$$
To account for this condition in the PROTES algorithm, we constructively build the initial distribution in the form of an indicator tensor as described in Section 4 in the constrained optimization subsection. The details of this construction are presented in the Appendix. The numerical results for  $T = 25$  (P-18),  $T = 50$  (P-19), and  $T = 100$  (P-20) are reported in Table 1. In two cases out of three (P-19, P-20), our method showed the best result, and in one case (P-18) slightly yielding to the TTOpt method (BS-1), which, however, in two other cases, gave a significantly worse result.
Figure 4: Relative error of the optimization result with PROTES method for the P-14 model problem with a known exact minimum for different values of the hyperparameters  $K$ ,  $k$  and  $R$ .
6.5 Robustness and performance of the PROTES
The results in Table 1 relate to the "intuitive" selection of the hyperparameters for the PROTES method (as was mentioned above, we have used the values:  $K = 100$ ,  $k = 10$ ,  $k_{gd} = 1$ ,  $\lambda = 0.05$ , and  $R = 5$ ). In Figure 4, we present an analysis of the dependence of the optimization result for the benchmark P-14 on the choice of hyperparameters  $K$ ,  $k$ , and  $R$ , with fixed  $k_{gd} = 1$  and  $\lambda = 0.05$ . We report the relative error of the result for all combinations  $K = 50, 100, 150, 200, 250$ ,  $k = 5, 10, 15, 20, 25$ ;  $R = 3, 5, 7$ . As we can see from the plots, the hyperparameters used in the main calculations are not optimal for this particular problem, that is, additional fine-tuning of the method for specific problems or classes of problems is possible. At the same time, according to the results in Figure 4, the method remains stable over a wide range of hyperparameter values. We also note that all computations were carried out on a regular laptop, while the operating time of the considered optimizers was commensurate; for example, for the benchmark P-17, the measured operating time (in seconds) is: PROTES (641), BS-1 (607), BS-2 (4245), BS-7 (780).
A more detailed analysis of the PROTES performance and the dependence of optimization results on the values of hyperparameters are presented in the Appendix. Also, in the Appendix, we consider another promising application of our optimizer for several popular reinforcement learning problems from Mujoco / OpenAI-GYM collection.
In this work, we presented an optimization algorithm PROTES based on sampling from the probability density defined in the tensor train format. We used the same set of hyperparameters for all considered numerical experiments, so our algorithm is rather universal. To take into account the constraints, as in the problem of optimal control with constraints, we only considered them in the form of a specially selected initial approximation (a special form of an indicator tensor in the tensor train format); further on, the algorithm did not consider the constraints explicitly. This approach allows us to extend the algorithm's capabilities by using the properties of the tensor train representation. Numerical experiments show that we outperform many popular optimization methods.
The main direction in our future work is scaling the method to large dimensions. For  $d \geq 1000$ , we have encountered numerous technical difficulties, which can be alleviated by other tensor formats (such as hierarchical Tucker, which can be parallelized over  $d$ ) and more efficient implementations of the optimization method (now we use standard automatic differentiation without special tensor optimization methods such as Riemannian optimization).
The work is supported by the Ministry of Science and Higher Education of the Russian Federation (Grant 075-10-2021-068).
A1 Stability of the PROTES
To check the stability of the optimization result, a series of 10 calculations were performed for each method (PROTES, BS1 - BS7) with random initializations. We consider the binary knapsack problem from the work [11] (benchmark P-14, described in detail in the main text of the work) with the known exact minimum  $-3103$ . For all methods, the limit on the number of requests was fixed at the value  $M = 10^5$ . All other parameters were the same as in the computations from the main text, i.e., the PROTES parameters are  $K = 100$ ,  $k = 10$ ,  $k_{gd} = 1$ ,  $\lambda = 0.05$ ,  $R = 5$ . In Table 3 we present the average and best results over 10 runs for each optimization method. As follows from the reported results, only PROTES and Portfolio method (BS-7) managed to successfully find the exact optimum, while the average result for PROTES is significantly better than that of all the baselines.
Table 3: Average and best result for 10 independent runs for the P-14 benchmark.
PROTES	BS-1	BS-2	BS-3	BS-4	BS-5	BS-6	BS-7
P-14	MEAN	-3095	-2992	-3048	-2650	-2937	-2701	-3064	-3075
BEST	-3103	-3074	-3084	-2825	-2996	-2752	-3094	-3103
A2 Reinforcement learning with PROTES
As in the work [37], we consider maximization of the cumulative reward of the episode for several popular reinforcement learning problems from Mujoco / OpenAI-GYM collection: InvertedPendulum (P-21), Swimmer (P-22), Lunar Lander (P-23), and Half Cheetah (P-24). The policy is represented by a neural network with three layers and tanh activations. Parameters of the neural network are discretized on a grid with limits from  $-1$  to  $+1$  and 256 nodes (i.e., mode size  $N$ ). The total number of neural network parameters for the problems under consideration turns out to be from 50 to 68 (i.e., dimension  $d$ ). Thus, as in the work [37], we obtain a discrete on-policy learning problem (search for the values of the neural network parameters that lead to the maximum reward). To speed up the calculations, we took only two baselines (BS1, and BS7) for comparison, which showed the best results in the main experiments presented in the paper. We used the same optimizer settings as in the main calculations described in the text of the paper; however, in this case, we chose the budget value equal to  $M = 10^5$ , as in [37]. For PROTES, BS1, and BS7, we repeated the calculations three times and reported the average result for found maximum and computation time in Table 4. PROTES shows the best result in terms of accuracy in 3 cases out of 4, and once it is in second place, slightly inferior to the BS7.
Table 4: Maximization results for InvertedPendulum (P-21), Swimmer (P-22), Lunar Lander (P-23) and Half Cheetah (P-24) benchmarks from Mujoco collection. We report the reward and calculation time (seconds), averaged over 3 runs, for the proposed method PROTES and baselines BS1 and BS7.
REWARD	TIME
PROTES	BS-1	BS-7	PROTES	BS-1	BS-7
RL AGENTS	P-21	1.00E+03	7.28E+02	3.49E+02	8.1E+02	6.3E+02	1.3E+04
P-22	3.61E+02	3.43E+02	3.55E+02	9.7E+03	7.7E+03	2.3E+04
P-23	2.80E+02	-1.87E+01	2.64E+02	3.5E+03	8.1E+03	1.5E+04
P-24	2.26E+03	7.78E+02	2.66E+03	8.5E+03	5.5E+03	2.2E+04
A3 Choice of hyperparameters for the PROTES
We conduct additional experiments with varying the values of the hyperparameters of the PROTES method for benchmarks P-01 (Figure 5 and Figure 6; we report the optimization result) and P-14 (Figure 4 in the main text and Figure 7 below; we report the relative error of the optimization result). In the first series of experiments (Figure 5 below and Figure 4 in the main text), we fixed the values  $k_{gd} = 1$  and  $\lambda = 0.05$  and tried all combinations of the remaining hyperparameters:
Figure 5: Optimization result (approximation of the minimum value) with PROTES method for the P-01 for different values of the hyperparameters:  $K$ ,  $k$  and  $R$ .
Figure 6: Optimization result (approximation of the minimum value) with PROTES method for the P-01 model problem for different values of the hyperparameters: learning rate (LR),  $k_{gd}$ ,  $R$ .
Figure 7: Relative error of the optimization result with PROTES method for the P-14 model problem with a known exact minimum for different values of the hyperparameters: learning rate (LR),  $k_{gd}$ ,  $R$ .
$K = 50,100,150,200,250; k = 5,10,15,20,25; R = 3,5,7$ . In the second series of experiments (Figure 6 and 7), we fixed the values  $K = 100$  and  $k = 10$ , and tried all combinations:  $k_{gd} = 1,10,25,50,100, \lambda = 0.005,0.01,0.05,0.1,0.5; R = 3,5,7$ .
As follows from the presented results, for problem P-01, the dependence on the choice of hyperparameters turns out to be extremely weak, except for outliers for the small values of the learning rate  $\lambda = 0.005$  and  $\lambda = 0.01$ . For more complex problem P-14, the dependence of the result on the
Table 5: Computation time in seconds for all selected benchmarks (P-01 – P-20) and for all used optimization methods (PROTES, BS1 – BS7).
PROTES	BS-1	BS-2	BS-3	BS-4	BS-5	BS-6	BS-7
ANALYTIC FUNCTIONS	P-01	3.28	0.06	0.11	23.45	25.31	22.78	22.03	67.23
P-02	2.25	0.05	0.06	22.42	28.68	20.6	19.78	77.01
P-03	2.36	0.05	0.03	26.11	23.62	20.55	19.71	65.62
P-04	2.32	0.05	0.07	22.04	24.04	20.67	19.9	65.51
P-05	2.33	0.05	0.06	26.47	26.56	20.78	20.1	78.42
P-06	2.34	0.05	0.1	23.51	28.32	22.42	21.58	84.1
P-07	2.25	0.06	0.03	26.72	34.95	21.66	21.26	83.54
P-08	2.25	0.05	0.07	22.7	24.03	21.29	20.28	65.47
P-09	2.31	0.06	0.11	22.98	24.11	21.14	20.56	65.55
P-10	2.35	0.05	0.03	23.56	33.66	21.25	22.61	81.94
QUBO	P-11	2.7	0.06	0.44	16.32	22.54	17.65	17.98	67.61
P-12	2.16	0.05	0.36	17.06	21.95	17.28	16.77	79.74
P-13	2.29	0.05	0.34	18.78	21.09	20.51	17.97	77.22
P-14	2.33	0.13	0.4	16.41	27.42	17.61	17.59	74.77
CONTROL	P-15	513.6	1256.0	1839.0	550.0	545.0	620.3	530.4	707.5
P-16	542.4	969.4	3007.0	578.0	595.0	570.7	573.7	697.5
P-17	640.7	607.3	4245.0	673.6	687.6	661.0	687.1	779.5
CONTROL +CONSTR.	P-18	328.1	92.9	588.1	319.1	516.2	202.9	533.9	474.5
P-19	8.74	69.49	912.0	17.13	17.45	16.16	16.84	47.43
P-20	9.23	0.53	931.8	21.16	21.42	20.28	20.84	62.26
choice of hyperparameters is more complex, but it can be seen that the method remains stable over a wide range of hyperparameter values.
A4 Performance comparison of the optimization methods
All the results described in the main text and presented there in Table 1 were obtained on a regular laptop. In Table 5 we report the related computation time for each method (PROTES, BS1 - BS7) and each model problem (P01 - P20). As follows from the results, the PROTES works much faster than classical optimization methods (BS3 - BS7), as well as faster than tensor-based methods (BS1 and BS2) for several optimal control problems (P-15 - P-20). However, the TTOpt (BS-1) and Optima-TT (BS-2) methods are faster for simpler analytic (P-01 - P-10) and QUBO (P-11 - P-14) problems. This is due to the fact that within the framework of the dynamic TT-rank refinement procedure in these methods, the TT-rank, and hence the computation time, turn out to be significantly higher for the optimal control problems. We note that the very short running time of the TTOpt method for P-20 is because most of the requests of the method did not satisfy the imposed constraints, and in this case, the differential equation was not solved.
A5 Derivative functions in the optimal control problem
We use the following derivative functions for the constructive building of the tensor described in [32]
$$
f _ {0} ^ {k} (x) = \left\{ \begin{array}{c c} 0, & x = 0 \text {o r} x = l \\ \text {N o n e}, & \text {o t h e r w i s e}, \end{array} \right.
$$
$$
f _ {1} ^ {k} (x) = \min  (l, x + 1),
$$