Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Giải pháp toàn cục cho các bài toán phi lồi thông qua sự tiến hóa của các phương trình đạo hàm riêng Hamilton-Jacobi
Communications on Applied Mathematics and Computation - Trang 1-21 - 2023
Tóm tắt
Các nhiệm vụ tính toán thường được đặt ra dưới dạng các bài toán tối ưu hóa. Các hàm mục tiêu cho các kịch bản trong thế giới thực thường không lồi và/hoặc không khả vi. Các phương pháp tiên tiến nhất để giải quyết những vấn đề này thường chỉ đảm bảo hội tụ đến cực tiểu địa phương. Công trình này trình bày phương pháp giảm dần thích ứng Moreau dựa trên Hamilton-Jacobi (HJ-MAD), một thuật toán bậc không có đảm bảo hội tụ đến cực tiểu toàn cục, giả định rằng hàm mục tiêu liên tục. Ý tưởng cốt lõi là tính toán đạo hàm của bao Moreau của mục tiêu (mà là “lồi theo từng mảnh”) với các tham số làm mịn thích ứng. Các đạo hàm của bao Moreau (tức là, các toán tử gần nhất) được xấp xỉ thông qua công thức Hopf-Lax cho phương trình Hamilton-Jacobi nhớt. Các ví dụ số của chúng tôi minh họa sự hội tụ toàn cục.
Từ khóa
#tối ưu hóa #bài toán phi lồi #hội tụ toàn cục #phương trình đạo hàm riêng Hamilton-JacobiTài liệu tham khảo
Almeida, L.B.: A learning rule for asynchronous perceptrons with feedback in a combinatorial environment. In: Diederich, J. (ed) Artificial Neural Networks: Concept Learning, pp 102–111. IEEE Press, Los Alamitos (1990). https://doi.org/10.5555/104134.104145
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)
Beck, A.: First-Order Methods in Optimization. SIAM, Philadelphia (2017)
Berahas, A.S., Byrd, R.H., Nocedal, J.: Derivative-free optimization of noisy functions via quasi-Newton methods. SIAM J. Optim. 29(2), 965–993 (2019)
Bisong, E.: Google colaboratory. In: Building Machine Learning and Deep Learning Models on Google Cloud Platform, pp. 59–64. Springer, New York (2019)
Brooks, S.H.: A discussion of random methods for seeking maxima. Oper. Res. 6(2), 244–251 (1958)
Cai, H., Lou, Y., McKenzie, D., Yin, W.: A zeroth-order block coordinate descent algorithm for huge-scale black-box optimization. In: International Conference on Machine Learning. PMLR, pp 1193–1203 (2021)
Cai, H., Mckenzie, D., Yin, W., Zhang, Z.: A one-bit, comparison-based gradient estimator. Appl. Comput. Harmon. Anal. 60, 242–266 (2022)
Cai, H., Mckenzie, D., Yin, W., Zhang, Z.: Zeroth-order regularized optimization (zoro): approximately sparse gradients and adaptive sampling. SIAM J. Optim. 32(2), 687–714 (2022)
Chaudhari, P., Choromanska, A., Soatto, S., LeCun, Y., Baldassi, C., Borgs, C., Chayes, J., Sagun, L., Zecchina, R.: Entropy-SGD: biasing gradient descent into wide valleys. J. Stat. Mech 2019(12), 124018 (2019)
Chaudhari, P., Oberman, A., Osher, S., Soatto, S., Carlier, G.: Deep relaxation: partial differential equations for optimizing deep neural networks. Res. Math. Sci. 5(3), 1–30 (2018)
Crandall, M.G., Lions, P.-L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43(167), 1–19 (1984)
Davis, D., Drusvyatskiy, D.: Stochastic subgradient method converges at the rate \(o (k^{-1/4})\) on weakly convex functions. arXiv:1802.02988 (2018)
Davis, D., Drusvyatskiy, D.: Stochastic model-based minimization of weakly convex functions. SIAM J. Optim. 29(1), 207–239 (2019)
Davis, D., Drusvyatskiy, D., MacPhee, K.J., Paquette, C.: Subgradient methods for sharp weakly convex functions. J. Optim. Theory Appl. 179(3), 962–982 (2018)
Davis, D., Díaz, M., Drusvyatskiy, D.: Escaping strict saddle points of the Moreau envelope in nonsmooth optimization. SIAM J. Optim. 32(3), 1958–1983 (2022)
Ermoliev, Y.M., Wets, R.-B.: Numerical Techniques for Stochastic Optimization. Springer-Verlag, New York (1988)
Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence, RI (2010)
Evans, L.C.: Envelopes and nonconvex Hamilton-Jacobi equations. Calc. Var. Partial. Differ. Equ. 50(1), 257–282 (2014)
Hastie, T., Tibshirani, R., Friedman, J.H., Friedman, J.H.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, vol. 2. Springer, New York (2009)
Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)
Jones, E., Oliphant, T., Peterson, P., et al. SciPy: open source scientific tools for Python. http://www.scipy.org/ (2001)
Jourani, A., Thibault, L., Zagrodny, D.: Differential properties of the Moreau envelope. J. Funct. Anal. 266(3), 1185–1237 (2014)
Kim, B., Cai, H., McKenzie, D., Yin, W.: Curvature-aware derivative-free optimization. arXiv:2109.13391 (2021)
Kingma , D.P., Ba, J.: Adam: a method for stochastic optimization. In: ICLR 2015, San Diego, CA (2015)
Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)
Kozak, D., Becker, S., Doostan, A., Tenorio, L.: Stochastic subspace descent. arXiv:1904.01145 (2019)
Kozak, D., Becker, S., Doostan, A., Tenorio, L.: A stochastic subspace approach to gradient-free optimization in high dimensions. Comput. Optim. Appl. 79(2), 339–368 (2021)
Kozak, D., Molinari, C., Rosasco, L., Tenorio, L., Villa, S.: Zeroth order optimization with orthogonal random directions. arXiv:2107.03941 (2021)
Larson, J., Menickelly, M., Wild, S.M.: Derivative-free optimization methods. Acta Numer. 28, 287–404 (2019)
Le Digabel, S.: Algorithm 909: NOMAD: nonlinear optimization with the mads algorithm. ACM Transact. Math. Softw. (TOMS) 37(4), 1–15 (2011)
Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)
Moré, J., Wild, S.: Benchmarking derivative-free optimization algorithms. SIAM J. Optim. 20(1), 172–191 (2009)
Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Hebd. Seances Acad. Sci. 255, 238–240 (1962)
Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)
Olson, B., Hashmi, I., Molloy, K., Shehu, A.: Basin hopping as a general and versatile optimization framework for the characterization of biological macromolecules. Adv. Artif. Intell. 2012, 674832 (2012)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)
Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)
Rastrigin, L.: The convergence of the random search method in the extremal control of a many parameter system. Automat. Remote Control 24, 1337–1342 (1963)
Rockafellar, R.T.: Convex Analysis, vol. 18. Princeton University Press, Princeton (1970)
Scaman, K., Dos Santos, L., Barlier, M., Colin, I.: A simple and efficient smoothing method for faster optimization and local exploration. Adv. Neural. Inf. Process. Syst. 33, 6503–6513 (2020)
Shi, H.-J. M., Xie,Y., Xuan, M. Q., Nocedal, J.: Adaptive finite-difference interval estimation for noisy derivative-free optimization. arXiv:2110.06380 (2021)
Shi, H.-J.M., Xuan, M.Q., Oztoprak, F., Nocedal, J.: On the numerical performance of derivative-free optimization methods based on finite-difference approximations. arXiv:2102.09762 (2021)
Storn, R., Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997)
Surjanovic, S., Bingham, D.: Virtual library of simulation experiments: test functions and datasets. Retrieved June 23. http://www.sfu.ca/~ssurjano (2021)
Wales, D.J., Doye, J.P.: Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. J. Phys. Chem. A 101(28), 5111–5116 (1997)