System Of Bilinear Equations Research Articles

Numerical algebraic geometry and semidefinite programming

Standard interior point methods in semidefinite programming can be viewed as tracking a solution path for a homotopy defined by a system of bilinear equations. By considering this in the context of numerical algebraic geometry, we employ numerical algebraic geometric techniques such as adaptive precision path tracking, endgames, and projective space to accurately solve semidefinite programs. We develop feasibility tests for both primal and dual problems which can distinguish between the four feasibility types of semidefinite programs. Finally, we couple our feasibility tests with facial reduction to develop a solving approach that can handle every scenario arising in semidefinite programming, including problems with nonzero duality gap. Various examples are used to demonstrate the new methods with comparisons to commonly used semidefinite programming software.

A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus

Formulating a Schubert problem as solutions to a system of equations in either Plücker space or local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale’s $$\alpha $$ -theory.

A convex approach to a class of non-convex building HVAC control problems: Illustration by two case studies

In this paper, a convexification approach is presented for a class of non-convex optimal/model predictive control problems more specifically applied to building HVAC control problems. The original non-convex problems are convexified using a convex envelope approach. The approach is tested on two case studies: a benchmark building HVAC system control problem from the literature and control of a hybrid ground-coupled heat pump (HybGCHP) system. For the first application, convexified model predictive control was used and results were compared with fuzzy and adaptive control results. For the HybGCHP system, convexified optimal control was applied and the results were compared with dynamic programming based optimal control. In the first case superior performance was observed over the corresponding fuzzy and adaptive control results from the literature. For the HybGCHP system the associated convexified optimal control gave almost global optimal results in terms of responses and cost criteria. The suggested method is especially useful for optimal building HVAC control/design problems which include non-convex bilinear or fractional terms. Since a polynomial expression can be recursively expressed as a system of bilinear equations, the proposed method, in principle, can be applied to all systems where polynomial non-convexities exist.

Explicit solutions for a (2 + 1)-dimensional Toda-like chain

We consider a (2 + 1)-dimensional Toda-like chain which can be viewed as a two-dimensional generalization of the Wu–Geng model and which is closely related to the two-dimensional Volterra, two-dimensional Toda and relativistic Toda lattices. In the framework of the Hirota direct approach, we present equations describing this model as a system of bilinear equations that belongs to the Ablowitz–Ladik hierarchy. Using the Jacobi-like determinantal identities and the Fay identity for the theta-functions, we derive its Toeplitz, dark-soliton and quasiperiodic solutions as well as the similar set of solutions for the two-dimensional Volterra chain.

On Feasibility of Interference Alignment in MIMO Interference Networks

The degrees of freedom of MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment solution is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem as a system of bilinear equations and determine its solvability by comparing the number of equations and the number of variables. To this end, we divide interference alignment problems into two classes - proper and improper. An interference alignment problem is called proper if the number of equations does not exceed the number of variables. Otherwise, it is called improper. Examples are presented to support the intuition that for generic channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible.

Explicit Orthonormal Fixed Bases for Spaces of Functions that are Totally Symmetric Under the Rotational Symmetries of a Platonic Solid

Explicit complete orthonormal fixed bases are computed for subspaces of the space of square-integrable functions on the sphere where the subspaces contain functions that are totally symmetric under the rotational symmetries of a Platonic solid. Each function in the fixed basis is a linear combination of spherical harmonics of fixed l. For each symmetry (icosahedral/dodecahedral, octahedral/cubic, tetrahedral), the calculation has three steps: First, a bilinear equation is derived for the coefficients in the linear combination by equating the expansion of a symmetrized δ function in both spherical harmonics and the fixed basis functions for the appropriate subspace. The equation is parameterized by the location (θ0, ϕ0) of the δ function and must be satisfied for all locations. Second, the dependence on the δ-function location is expressed in a Fourier (ϕ0) and a Taylor (θ0) series and thereby a new system of bilinear equations is derived by equating selected coefficients. Third, a recursive solution of the new system is derived and the recursion is solved explicitly with the aid of symbolic computation. The results for the icosahedral case are important for structural studies of small spherical viruses.

Algorithm for reducing the minimal realization problem of two-dimensional systems to a system of bilinear equations

Abstract The problem of the minimal state space realization of two-dimensional transfer functions which are not of any special form such as separable, all pole, all zero, continued fraction expandable, etc. is considered. For this general type of transfer function, an algorithm is presented for the minimal state space realization which is computationally superior over known techniques. The proposed algorithm starts by deriving, prior to and independently of the state space vectors b and c and the scalar d, the matrix A of the space model, nearly by inspection. Subsequently, the vectors b and c and the scalar d are determined on the basis of a bilinear algebraic system of equations.

The Bilinear, Modal State Equations for Age-Dependent Growth Control

There have been previous attempts to model biological processes as bilinear systems [4,9,10]. In these early studies any member of a population was taken to be quite like any other so that the variation of fertility and susceptibility to mortality with age was ignored. In this paper, however, the age-dependent nature of biological growth [5] is accounted for. The modal (eigenfunction) analysis of the basic partial differential equation of age-dependent growth is shown to result in a system of bilinear equations. (The basic mathematical model is a non-self-adjoint operator with a discrete spectrum and the modes are coupled by the control term.) The impulse control of a truncated version of this system of equations is then discussed. It is anticipated that the results presented here will aid planning for optimal amounts of pesticides to agro-ecosystems or for optimal amounts of drugs (or radiations) to unwanted cell populations.