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I’m trying to find the minimum of a function with 3 variables constrained to a few restrictions with 5 variables applying the penalty method in order to optimise it using the steepest descent method with a backtracking algorithm to compute progressively the alphas. The function and restrictions are the following: min h(A, C, E) =Read more

Consider the following min-max optimization: $ \min_x \max_y V(x,y)$ where V is convex in x and concave in y. I am consider an alternate gradient descent algorithm, which in continuous time has the dynamics: $ \dot{x} = – dV(x,y)/dx$ $ \dot{y} = +dV(x,y)/dy$ Is this approach (under some conditions) globally convergent to the optimal pointRead more

The following proposition is there in Pink’s lecture notes on finite group schemes. Let $ k$ be an algebraically closed field of characteristic $ p$ . The category of finite length $ W(k)$ -modules $ N$ with a $ \sigma$ -linear automorphism ($ \sigma$ is the Witt vector frobenius) $ F: N \to N$ isRead more

One way the Weil pairing for an Abelian Variety $ A/k$ is phrased is the following (for simplicity, let me only deal with the multiplication by $ m$ map ($ [m]$ : A\to A instead of arbitrary isogenies): Define $ P$ to be the group of line bundles $ \mathscr L$ on $ A$ suchRead more