In this paper we consider a system of quadratic equations |〈zj ,x〉|2 = bj , j = 1, ...,m, where x ∈ R is unknown while normal random vectors zj ∈ R and quadratic measurements bj ∈ R are known. The system is assumed to be underdetermined, i.e., m < n. We prove that if there exists a sparse solution x i.e., at most k components of x are non-zero, then by solving a convex optimization program, we ...

In this paper we consider the standard linear SDP problem, and its low rank nonlinear programming reformulation, based on a Gramian representation of a positive semidefinite matrix. For this nonconvex quadratic problem with quadratic equality constraints, we give necessary and sufficient conditions of global optimality expressed in terms of the Lagrangian function.

In this paper we propose a structured doubling algorithm (SDA) for the computation of the minimal nonnegative solutions to the nonsymmetric algebraic Riccati equation (NARE) and its dual equation simultaneously, for a singular case. Similar to the Newton’s method we establish a global and linear convergence for SDA under the singular condition, using only elementary matrix theory. Numerical exp...

Abstract. We extend Sklyanin’s method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are a...

We consider the problem of the perturbation of a class of linear-quadratic systems where the change from one structure (for the dynamics and costs) to another are governed by a nite-state Markov process where the transition are perturbed. The problem above lead to the analysis of some perturbed linearly coupled set of quadratic equations (Riccati Equation). We show that the matrix obtained as t...

This paper is concerned with the estimation of tra-jectories for accelerated motion from image sequences. Unlike in many other approaches, that assume linear trajectories, we propose a quadratic model that incorporates both velocity and acceleration. This model corresponds better to practical applications especially when the estimation is performed over several images, e.g., in motion-compensat...

We develop the inverse scattering transform method for the Novikov equation ut − utxx + 4uux = 3uuxuxx + uuxxx considered on the line x ∈ (−∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann–Hilbert (RH) problem, which in this case is a 3×3 matrix problem. The structure of this RH problem shares many common features with the case of ...

This paper presents a systematic and computable method for choosing the regularization parameter appearing in Tikhonov-type regularization based on non-quadratic regularizers. First, we extend the notion of the L-curve, originally defined for quadratically regularized problems, to the case of non-quadratic functions. We then associate the optimal value of the regularization parameter for these ...

A new approach for distributed power load flow calculation using nonlinear equations is presented. This new approach, which is similar to the Newton Raphson’s simple method, uses an inverse Jacobian matrix of initial states for the iteration process. Moreover, nonlinear quadratic equations have been used as they are more appropriate for the distributed power load flow calculation. This paper de...