Eigenvectors and eigenspaces problems in mathematics. Few direct numerical methods are available for solving the polynomial ei genvalue problem pep. Proceedings of a conference held at carnegie mellon. We attempt to solve the following polynomial optimization problem. Here is a compact target set in the complex plane and is a family of matrices depending analytically on.
As a global polynomial optimization problem, the best rankone approximation to higher order tensors has extensive engineering and statistical applications. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. Recently this problem has attracted much attention, due to its widely applications in various engineering problems such as biomedical engi. Third, we study how to solve tensor eigenvalue complementarity problems when b is not not copositive. The coefficients of the algebraic equations are computed directly from the. It plays an auxiliary role in relation to the algorithm for locating global minimum points of the p. The formulation of an eigenvalue problem and its physical meaning 2. For example, in semide nite programmings sdp, many optimization methods need to compute all positive eigenvalues and corresponding eigenvectors of a matrix each iteration in order to. Finally, the paper demonstrates that the polynomial optimization approach extends to principalagent models with multidimensional action sets. Convex relaxation methods for nonconvex polynomial. Approximation methods for inhomogeneous polynomial optimization. For anyone interested in solving nonlinear eigenvalue problems in python, the code i wrote myself to solve this problem can be found here.
We consider the problem of finding eigenvalues and nonzero eigenvectors of a nonlinear eigenvalue problem nlep. Polynomial optimization is a fundamental model in the eld of operations research. Lecture notes on solving large scale eigenvalue problems. Such a nonzero solution x is called an eigenvector corresponding to the eigenvalue the characteristic polynomial of a is the degree n polynomial pt det a. The case when the optimum eigenvalue is multiple, i. The mathematics of eigenvalue optimization received. An algorithm for generalized matrix eigenvalue problems. For such tensors, a ceigenvector xmay not be normalized as bxm 1. This leads to the k dimensional polynomial eigenvalue problem 3. See the descriptions of eig and qz for more information. Indeed, it belongs to the class of npcomplete problems, meaning that it is not yet known whether it can be solved with an algorithm running in polynomial time for general graphs for planar graphs, it has been shown that mc is the dual to the route inspection problem, thus solvable in polynomial time 2. We derive an explicit expression forthe backward errorof an approximate eigenpair ofa. Equation 1 is the eigenvalue equation for the matrix a. A numerical method for polynomial eigenvalue problems.
Thus, we formulate teicps as polynomial optimization in a di. Solving nonhomogeneous pdes eigenfunction expansions. For these abstract boundary eigenvalue problems the notions fundamental matrix function and characteristic matrix function are introduced, generalizing the. Unfortunately, this method requires that both the pde and the bcs be homogeneous. The symmetric eigenvalue problem is in a sweet spot between the two.
Eigenvalue problems an overview sciencedirect topics. But avoid asking for help, clarification, or responding to other answers. So even for m 2 quartics, the polynomial optimization problem is nphard. Backward error and condition of polynomial eigenvalue problems.
We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. Higherdegree eigenvalue complementarity problems for. Proceedings of the 1999 ieee international symposium on computer aided control system design cat. In one example the best we will be able to do is estimate the eigenvalues as that is something that will happen on a fairly regular basis with these kinds of problems. The last third of the article concerns stability, for polynomials, matrices. There are eigenvalues that may be infinite or finite, and possibly zero.
It is interesting to note however that the inverse problem of mc, namely the mincut problem, can be solved in polynomial time by means of network. Many problems encountered in systems theory and system identification require the solution of polynomial optimization problems, which have a. James demmel dissertation committee cochair bernd sturmfels. Polynomial twoparameter eigenvalue problems and matrix pencil. The mathematics of eigenvalue optimization optimization online. Orszag 1971 accurate solution of the orrsommerfeld stability equation, journal of fluid mechanics, 50 pp 689703. Note that for eigenvalue problem, the matrix a can be non symmetric. Global optimization of polynomial functions and applications. Sphere constrained homogeneous polynomial optimization 3 although the aforementioned results do shed some light on the approximability of sphere constrained polynomial optimization problems, they are not entirely satisfactory. Solving fractional polynomial problems by polynomial. In this paper we study constrained eigenvalue optimization of noncommutative nc polynomials, focusing on the polydisc and the ball. Optimization of eigenvalues of nonsymmetric matrices and. By a randomization process, the complementarity eigenvectors are classi. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization.
Zeigenvalue methods for a global polynomial optimization problem. In this direction, we consider two di erent variants of moment problem. To get an idea of how hard the problem we are dealing with is, we compare it to integer programming problems. In this paper, we introduce a unified framework of tensor higherdegree eigenvalue complementarity problem thdeicp, which goes beyond the framework of the typical quadratic eigenvalue complementarity problem for matrices. In matrix recovery type problems, people are interested. Lets see how to construct the problem in this form. Constrained polynomial optimization problems with noncommuting variables kristijan cafuta, igor klep1, and janez povh2 abstract.
Approximation methods for complex polynomial optimization. Polynomial optimization and the problem of global nonnegativity of polynomials are active. Polynomial optimization problems are eigenvalue problems. This is a special case of a nonlinear eigenproblem.
Some applications of polynomial optimization in operations. Eigenvalueshave theirgreatest importance in dynamic problems. Combined with the classical fractional programming theory 7, we show how the polynomial optimization theory can be used to globally solve 1 as. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle ones cannot. In section 6, we present numerical experiments for solving tensor eigenvalue complementarity problems. Stability optimization over a polynomial family part i globally optimizing the roots of a monic polynomial subject to one a. Abstract as a global polynomial optimization problem, the best rankone approximation to higher order tensors has extensive engineering and statistical applications. In this thesis, we consider polynomial eigenvalue problems. It has been criticized for its numerical instability.
We also introduce the optimization problems which yield to the eigenvalue and. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization. For the special case of the quadratic eigenvalue problem qep, we show that solving the qep by applying the qz algorithm. The solution of dudt d au is changing with time growing or decaying or oscillating. It is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems. Optimization of polynomial roots, eigenvalues and pseudospectra. One of the mathematical programs is part of the constraints of the other one.
Pdf global optimization with polynomials and the problem. Indeed, generic solution methods based on nonlinear. Polynomial optimization problems are eigenvalue problems 9 where the number o f columns in k 6 corresponds to the number of monomials in the bases of mono mials w 0 to w 4. It is particularly effective when it is brought into the socalled matrix condensed form. Zeigenvalue methods for a global optimization polynomial. Global optimization and the problem of moments 797 same. Convex extensions and envelopes are of primary importance to the efficiency of global optimization methods. This algorithm has the advantage of preserving pseudosymmetric tridiagonal forms. Sep 29, 2007 as a global polynomial optimization problem, the best rankone approximation to higher order tensors has extensive engineering and statistical applications. This problem has interesting applications in the optimum design of constructions 16. Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge.
Hopkins university press, baltimore, md, usa, third edition, 1996. We say that a polynomial pis a sum of squares sos, if it can be written as p p i q 2 for some other polynomials q i. The rst variant is the global polynomial optimization problem, i. The determinant is a hyperbolic polynomial on sn relative to the identity matrix i. A popular approach for solving such problems is to approximate by a polynomial or rational eigenvalue problem of the form. We propose an implementation of the hz algorithm that allows stability in most cases and comparable results with other classical algorithms for ill conditioned problems. Pdf polynomial optimization problems are eigenvalue problems. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem gep. This paper studies tensor eigenvalue complementarity problems. Two point boundary value and periodic eigenvalue problems. For example, in semide nite programmings sdp, many optimization methods need to compute all positive eigenvalues and corresponding eigenvectors of a matrix each iteration in order to preserve the semide nite structure. The polynomial eigenvalue problem pep 1,4,7 involves. Many problems encountered in systems theory and system identification require the solution of polynomial optimization problems, which have a polynomial objective function and polynomial constraints.
Thanks for contributing an answer to mathematics stack exchange. This matrix with polynomial coefficients across the top is a companion matrix. A similar conclusion also holds for the constrained optimization problem pk in 1. Introduction to eigenvalues and eigenvectors problems in. In this section, we lay out a path from polynomial optimization problems to eigenvalue problems. We first propose a direct z eigenvalue method for this problem when the dimension is two. Tensor eigenvalue complementarity problems springerlink. Based upon the symmetry assumptions on the underlying. We studied the applications in shape optimization of transfer functions. More on pseudospectra for polynomial eigenvalue problems and applications in control theory. Polynomial eigenvalue problems arise in many applications some applications are.
Linear algebraeigenvalues and eigenvectorssolutions. More generally, could be a linear map, but most commonly it is a finitedimensional. Global optimization with polynomials and the problem of. The main steps are phrasing the task at hand as a set of polynomial equations by applying the method of lagrange multipliers, casting the problem of. This gives us a hierarchy of sdp problems, converging to the value of the original polynomial optimization problem.
It is intended as a first introduction to solving these simple problems with a spectral method. A nonlinear eigenproblem is a generalization of an ordinary eigenproblem to equations that depend nonlinearly on the eigenvalue. Some preliminaries in polynomial optimization and moment problems are given in section 2. Basic properties of standard and complementarity tensor eigenvalues are discussed. Solution of odes and eigenvalue problems with a chebyshev. First, we study some topological properties of higherdegree cone eigenvalues of tensors.
Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. The polyeig function uses the qz factorization to find intermediate results in the computation of generalized eigenvalues. Megretski mit the root radius and the root abscissa stability optimization over a polynomial family optimizing the root radius, real case algorithm. A is singular if and only if 0 is an eigenvalue of a. Stability optimization for polynomials and matrices in. Adhikari, b backward perturbation and sensitivity analysis of structured polynomial eigenvalue problems. Zeigenvalue methods for a global polynomial optimization. A polynomial optimization approach to principalagent. Deterministic approximation algorithms for sphere constrained. Eigenvalue decompositions evd are commonly used in large varieties of matrix optimization problems with spectral or lowrank structures. Note that i deal with the more general case of a nonlinear eigenvalue problem in the sense that it has a nonlinear dependence on lambda. Solving nonhomogeneous pdes eigenfunction expansions 12. Multiple eigenvalues in optimization problems sciencedirect.
Such problems arise often in theory and practice, particularly. A chart of backward errors for singly and doubly structured eigenvalue problems. Optimization problems involving the eigenvalues of symmetric and nonsymmetric. Since k n, this eigenvalue problem can easily be solved by the qz algorithm after linearization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization. Introduction to concepts and advances in polynomial. In this section we will define eigenvalues and eigenfunctions for boundary value problems. Boundary eigenvalue problems as considered in later chapters have an underlying abstract operator theoretic structure, which is investigated in section 1. We extend results on eigenvalue and eigenvector condition numbers of matrix polynomials to condition numbers with perturbations measured with a weighted frobenius norm. Polynomial optimization problems are eigenvalue problems 3. Applying the method of lagrange multipliers yields a set of multivariate polynomial equations.
Note that obtaining a feasible solution to az b, z. Different from traditional optimization solution methods, in this paper, we propose some z eigenvalue methods for solving this problem. We consider the problem of finding the unconstrained global minimum of a realvalued polynomial px. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of. Methods for eigenvalue problems with applications in model order. Eigenvalues and nonsmooth optimization cornell university. Request pdf z eigenvalue methods for a global optimization polynomial optimization problem as a global polynomial optimization problem, the best rankone approximation to higher order tensors. Overton, variational analysis of the spectral abscissa at a matrix with a nongeneric multiple eigenvalue. Eigenvalue optimization acta numerica cambridge core. Seeking structure eigenvalue problems and optimization cornell. R r, as well as the global minimum of px, in a compact set k defined by polynomial inequalities. The polynomial optimization problems are typically nonconvex, highly nonlinear, and nphard in general. A popular relaxation scheme for this problem is through the machinery of the socalled sum of squares optimization.
On one hand, the approximation results developed in 14. I v 0, \displaystyle a\lambda iv0, 2 where i is the n by n identity matrix and 0 is the zero vector. The polynomial eigenvalue problem school of mathematics. A numerical method for polynomial eigenvalue problems using. Backward error and condition of polynomial eigenvalue. This is the form of a generalized eigenvalue problem. The coefficients of the algebraic equations are computed directly from the orthogonality property of the chebyshev polynomials using integration. Siam journal on matrix analysis and applications 35. This paper studies how to compute all real eigenvalues, associated to real eigenvectors, of a symmetric tensor.
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