This paper primarily focuses on solving the Heilbronn problem of convex polygons,which involves minimizing the area of a convex polygon P_(1)P_(2)···P_(n) while satisfying the condition that the areas o...This paper primarily focuses on solving the Heilbronn problem of convex polygons,which involves minimizing the area of a convex polygon P_(1)P_(2)···P_(n) while satisfying the condition that the areas of all triangles formed by consecutive vertices are equal to 1/2.The problem is reformulated as a polynomial optimization problem with a bilinear objective function and bilinear constraints.A new method is presented to verify the upper and lower bounds for the optimization problem.The upper bound is obtained by the affine regular decagon.Then Bilinear Matrix Inequalities(BMI)theory and the branch-and-bound technique are used to verify the lower bound of the problem.The paper concludes by proving that the lower bound for the area minimization problem of a convex polygon with 10 vertices is 13.076548.The relative error compared to the global optimum is 0.104%.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.12171159the National Natural Science Fund of China Research Fund for International Scientists under Grant No.12350410363。
文摘This paper primarily focuses on solving the Heilbronn problem of convex polygons,which involves minimizing the area of a convex polygon P_(1)P_(2)···P_(n) while satisfying the condition that the areas of all triangles formed by consecutive vertices are equal to 1/2.The problem is reformulated as a polynomial optimization problem with a bilinear objective function and bilinear constraints.A new method is presented to verify the upper and lower bounds for the optimization problem.The upper bound is obtained by the affine regular decagon.Then Bilinear Matrix Inequalities(BMI)theory and the branch-and-bound technique are used to verify the lower bound of the problem.The paper concludes by proving that the lower bound for the area minimization problem of a convex polygon with 10 vertices is 13.076548.The relative error compared to the global optimum is 0.104%.
