Phương pháp Newton tiếp tục với kỹ thuật giảm bậc cho các bài toán tối ưu toàn cầu

Xin-long Luo1, Hang Xiao1, Sen Zhang1
1School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing, China

Tóm tắt

Điểm cực tiểu toàn cầu của một bài toán tối ưu thu hút sự quan tâm trong các lĩnh vực kỹ thuật và rất khó để giải quyết, đặc biệt với những bài toán tối ưu quy mô lớn không lồi. Trong bài viết này, chúng tôi xem xét một thuật toán meme mới cho bài toán này. Cụ thể, chúng tôi sử dụng phương pháp Newton tiếp tục với kỹ thuật giảm bậc để tìm nhiều điểm dừng lại của hàm mục tiêu và sử dụng những điểm dừng đó làm hạt giống ban đầu cho thuật toán tiến hóa, thay vì sử dụng hạt giống ngẫu nhiên như các thuật toán tiến hóa đã biết. Trong khi đó, để giữ cho khả năng sử dụng của phương pháp không cần đạo hàm và sự hội tụ nhanh của phương pháp dựa trên đạo hàm, chúng tôi sử dụng kỹ thuật phân biệt tự động để tính toán gradient và thay thế ma trận Hessian bằng xấp xỉ sai phân hữu hạn của nó. Theo các thí nghiệm số của chúng tôi, thuật toán mới này hoạt động tốt cho các bài toán tối ưu không ràng buộc và tìm ra cực tiểu toàn cầu của chúng một cách hiệu quả, so với các phương pháp tối ưu toàn cầu tiêu biểu khác như các phương pháp khởi động đa lần (tiện ích tích hợp GlobalSearch.m của MATLAB R2021b, GLODS và VRBBO), phương pháp nhánh và ràng buộc (Couenne, một bộ giải nguồn mở tiên tiến cho các vấn đề lập trình phi tuyến hỗn hợp nguyên) và các thuật toán không cần đạo hàm (CMA-ES và MCS).

Từ khóa

#tối ưu toàn cầu #phương pháp Newton #kỹ thuật giảm bậc #thuật toán tiến hóa #gradient #ma trận Hessian

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