In this paper we formulate and solve a mean-field game described by a linear stochastic dynamics and a quadratic or exponential-quadratic cost functional for each generic player. The optimal strategies for the players are given explicitly using a simple and direct method based on square completion and a Girsanov-type change of measure, suggested in Duncan et al. in e.g. [3, 4] for the mean-fiel...

We give some new canonical representations for forms over C. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in (x1, . . . , xn) can be written uniquely as a sum of the cubes of linear forms `ij(xi, . . . , xj), 1 ≤ i ≤ j ≤ n. A general ternary quartic form is the sum of the square of a quad...

We propose a general method to combine several estimators of the same quantity in order to produce a better estimate. In the spirit of model and forecast averaging, the final estimator is computed as a weighted average of the initial ones, where the weights are constrained to sum to one. In this framework, the optimal weights, minimizing the quadratic loss, are entirely determined by the mean s...

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were i...

We present solutions of the Schrodinger equation with superposition Manning-Rosen plus inversely Mobius square quadratic Yukawa potentials using parametric Nikiforov Uvarov method along an approximation to centrifugal term. The bound state energy eigenvalues for any angular momentum quantum number l and corresponding un-normalized wave functions are calculated. mixed potential which in some par...

We consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping. This is a sufficiently rich class that includes max-functions, largest eigenvalue functions, and norm-1 regularized functions. The bundle method uses an oracle that is able to compute separately the function and subgradient information for...

An extension of the proximal minimization algorithm is considered where only some of the minimization variables appear in the quadratic proximal term. The resulting iterates are interpreted in terms of the iterates of the standard algorithm, and a uniform descent property is shown that holds independently of the proximal terms used. This property is used to give simple convergence proofs of par...

Non-convex and non-smooth optimization plays an important role in machine learning. Proximal gradient method is one of the most important methods for solving the nonconvex and non-smooth problems, where a proximal operator need to be solved exactly for each step. However, in a lot of problems the proximal operator does not have an analytic solution, or is expensive to obtain an exact solution. ...

We consider the inversion of a linear operator with centered Gaussian white noise by MAP estimation with a Gaussian prior distribution on the solution. The actual estimator is computed approximately by a numerical method. We propose a relation between the stationarity measure of this approximate solution to the mean square error of the exact solution. This relation enables the formulation of a ...