Cone Of Positive Semidefinite Matrices Research Articles

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable ( SC 1 ) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.

Semidefinite programming and arithmetic circuit evaluation

We address the exact semidefinite programming feasibility problem (SDFP) consisting in checking that intersection of the cone of positive semidefinite matrices and some affine subspace of matrices with rational entries is not empty. SDFP is a convex programming problem and is often considered as tractable since some of its approximate versions can be efficiently solved, e.g. by the ellipsoid algorithm. We prove that SDFP can decide comparison of numbers represented by the arithmetic circuits, i.e. circuits that use standard arithmetical operations as gates. Our reduction may give evidence to the intrinsic difficulty of SDFP (contrary to the common expectations) and clarify the complexity status of the exact SDP—an old open problem in the field of mathematical programming.

Bessel convolutions on matrix cones

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known sequential quadratic programming method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.

Clarke Generalized Jacobian of the Projection onto the Cone of Positive Semidefinite Matrices

This paper studies the differentiability properties of the projection onto the cone of positive semidefinite matrices. In particular, the expression of the Clarke generalized Jacobian of the projection at any symmetric matrix is given.

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R 0-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.

Optimization and Applications

This workshop was attended by 47 participants with broad geographic representation from all continents. It was enlightened by this cheerful atmosphere special to Oberwolfach, and which the participants contributed to by their mutual friendship and esteem. The theme was optimization problems, their theory and resolution, and above all applications of them. Some emphasis was put on the still recent subject of optimization over the cone of positive semidefinite matrices (SDP). In the following 20 extended abstracts, one finds We thank Gail Pieper (Argonne National Laboratory) for her careful proofreading of every abstract; her editorial work substantially improved the presentation of this material.

Contracting Optimally an Interval Matrix without Loosing Any Positive Semi-Definite Matrix Is a Tractable Problem

In this paper, we show that the problem of computing the smallest interval submatrix of a given interval matrix [A] which contains all symmetric positive semi-definite (PSD) matrices of [A], is a linear matrix inequality (LMI) problem, a convex optimization problem over the cone of positive semidefinite matrices, that can be solved in polynomial time. From a constraint viewpoint, this problem corresponds to projecting the global constraint PSD (A) over its domain [A]. Projecting such a global constraint, in a constraint propagation process, makes it possible to avoid the decomposition of the PSD constraint into primitive constraints and thus increases the efficiency and the accuracy of the resolution.

A Dual Approach to Semidefinite Least-Squares Problems

In this paper, we study the projection onto the intersection of an affine subspace and a convex set and provide a particular treatment for the cone of positive semidefinite matrices. Among applications of this problem is the calibration of covariance matrices. We propose a Lagrangian dualization of this least-squares problem, which leads us to a convex differentiable dual problem. We propose to solve the latter problem with a quasi-Newton algorithm. We assess this approach with numerical experiments which show that fairly large problems can be solved efficiently.

Linear matrix inequalities for robust strictly positive real design

A necessary and sufficient condition is proposed for the existence of a polynomial p(s) such that the rational function p(s)/q(s) is robustly strictly positive real when q(s) is a given Hurwitz polynomial with polytopic uncertainty. It turns out that the whole set of candidates p(s) is a convex subset of the cone of positive semidefinite matrices, resulting in a straightforward strictly positive real design algorithm based on linear matrix inequalities.

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

A Linesearch Procedure in Barrier Methods for Some Convex Programming Problems

A method is proposed for the linesearch procedure in barrier function methods for convex quadratically constrained quadratic programming problems, which includes linear and quadratic programming as special cases. The same linesearch problem appears in barrier methods for minimization over the cone of positive semidefinite matrices and a similar problem is also obtained when computing the eigenvalues of a matrix perturbed by the addition of a matrix of rank one. The method brackets the minimum of the function obtained in this linesearch by using the roots of successive rational bounds on its derivative and then uses these bounds to generate an approximation to the root. It is globally convergent with a robustness similar to that of the bisection method and it exhibits a superlinear order of convergence. A convergence analysis is presented together with numerical results illustrating the methods.

A new Farkas lemma for positive semidefinite matrices

We consider a general linear system over the cone of positive semidefinite matrices. To check consistency, one may not invoke the standard Farkas lemma because the cone of positive semidefinite matrices is not polyhedral, and thus an additional closedness assumption is required. In this note, we provide a modified Farkas lemma with no closedness assumption.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Max-min eigenvalue problems, primal-dual Interior point algorithms, and Trust region subproblemst

Two Primal-dual interior point algorithms are presented for the problem of maximizing the smallest eigenvalue of a symmetric matrix over diagonal perturbations. These algorithms prove to be simple, robust, and efficient. Both algorithms are based on transforming the problem to one with constraints over the cone of positive semidefinite matrices, i.e. Lowner order constraints. One of the algorithms does this transformation through an intermediate transformation to a trust region subproblem. This allows the removal of a dense row

Extermal positive semidefinite matrices with given sparsity pattern

We prove a conjecture of Helton, Pierce and Rodman about the connection between the ranks of matrices generating extreme rays in the cone of positive semidefinite matrices with a given sparsity pattern and the amount of fill-in for the specific sparsity pattern. We also present a necessary condition on the given sparsity pattern for the existence of a rank k extermal in the cone.

The equivalence of two partial orders on a convex cone of positive semidefinite matrices

The Loewner partial order ≧ is defined on the space of Hermitian matrices by A ≧ B if A − B is positive semidefinite. Given a strictly increasing function f: ( a, b) → R, we define the partial order ≧ f on the set of Hermitian matrices with spectrum contained in ( a, b) by A ≧ fB if f(A) ≧ f(B). We say that the partial orders ≧ and ≧ f are equivalent ona set S of Hermitian matrices if A ≧ B if and only if A ≧ fB for all A, B ϵ S . It is clear that if the cone C is commutative, i.e., AB = BA for all A, B ϵ C , then the two partial orders are equivalent. Stepniak conjectured the converse for the function f( t) = t 2, and proved it for n ⩽ 3. We provide a counterexample to Stepniak's conjecture for n ≧ 4, and we characterize the convex cones C of positive semidefinite matrices on which ≧ and ≧ f are equivalent for a class of functions that includes f( t) = t p , p > 1, and f( t) = e t . We introduce the class of strongly monotone matrix functions and prove the following result of independent interest: Let ƒ be a strongly monotone matrix function of order n, and suppose that A and B are n-by- n Hermitian matrices such that A − B is positive semidefinite and that A and B have no common eigenvectors. Then f( A) − f( B) is positive definite. We also show that the functions f( t) = t p , 0 < p < 1, and f( t) = log t are strongly monotone of all orders. We also consider the partial order ≧ t 2. In this special case it is possible to obtain the results using more elementary techniques. We prove some results about the partial orders ≧ t p and ≧ exp. We conclude with two open questions.

Positive semidefinite matrices with a given sparsity pattern

This paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries. One of the main purposes is to obtain information and, if possible, describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros. In particular, we study those patterns for which the ranks of extremal matrices are less than a given number.

On the cone of positive semidefinite matrices

A survey of some general properties of the cone of positive semidefinite matrices, its faces, two isometric isomorphisms, and linear transformations on it is given.

Nonnegative Minimum Biased Invariant Estimation in Variance Component Models

In a general variance component model, quadratic estimators of the components of variance are considered which are invariant with respect to mean value translations and have minimum bias, analogously to estimation theory of mean value parameters. Here the minimum is taken over an appropriate cone of positive semidefinite matrices, after having made a reduction by invariance. Among these estimators, which always exist, the one of minimum norm is characterized. This characterization is achieved by systems of necessary and sufficient conditions, and by a nonlinear cone-restricted pseudoinverse. A representation of this pseudoinverse is given, that allows computation without consideration of the boundary. In models where the decomposing covariance matrices span a commutative quadratic subspace, a representation of the considered estimator is derived that requires merely to solve an ordinary convex quadratic optimization problem. As an example, we present the two-way nested classification random model. In the case that unbiased quadratic estimation is possible, this estimator automatically becomes the nonnegative MINQUE. Besides this, a general representation of the MINQUE is given, that involves just one matrix pseudoinversion in the reduced model.

Extreme points of a convex subset of the cone of positive semidefinite matrices

Extreme points of a convex subset of the cone of positive semidefinite matrices