Michael L. Overton

Michael L. Overton
New York University | NYU · Department of Computer Science

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162
Publications
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8,012
Citations
Citations since 2017
23 Research Items
2351 Citations
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20172018201920202021202220230100200300400
20172018201920202021202220230100200300400
20172018201920202021202220230100200300400

Publications

Publications (162)
Preprint
Robust controllers that stabilize dynamical systems even under disturbances and noise are often formulated as solutions of nonsmooth, nonconvex optimization problems. While methods such as gradient sampling can handle the nonconvexity and nonsmoothness, the costs of evaluating the objective function may be substantial, making robust control challen...
Article
Full-text available
More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local...
Article
Given a square matrix A and a polynomial p, the Crouzeix ratio is the norm of the polynomial on the field of values of A divided by the 2-norm of the matrix p(A). Crouzeix’s conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a frequently occ...
Preprint
More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local...
Preprint
Full-text available
Given a square matrix $A$ and a polynomial $p$, the Crouzeix ratio is the norm of the polynomial on the field of values of $A$ divided by the 2-norm of the matrix $p(A)$. Crouzeix's conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a freque...
Article
Full-text available
We consider the problem of optimal placement of concentrated masses along a massless elastic column that is clamped at one end and loaded by a nonconservative follower force at the free end. The goal is to find the largest possible interval such that the variation in the loading parameter within this interval preserves stability of the structure. T...
Chapter
The motivation to study the behavior of limited-memory BFGS (L-BFGS) on nonsmooth optimization problems is based on two empirical observations: the widespread success of L-BFGS in solving large-scale smooth optimization problems, and the remarkable effectiveness of the full BFGS method in solving small to medium-sized nonsmooth optimization problem...
Preprint
Full-text available
We consider the problem of optimal placement of concentrated masses along a massless elastic column that is clamped at one end and loaded by a nonconservative follower force at the free end. The goal is to find the largest possible interval such that the variation in the loading parameter within this interval preserves stability of the structure. T...
Preprint
Full-text available
The motivation to study the behavior of limited-memory BFGS (L-BFGS) on nonsmooth optimization problems is based on two empirical observations: the widespread success of L-BFGS in solving large-scale smooth optimization problems, and the effectiveness of the full BFGS method in solving small to medium-sized nonsmooth optimization problems, based on...
Preprint
We report on our experience with fixed-order H-infinity controller design using the HIFOO toolbox. We applied HIFOO to various benchmark fixed (or reduced) order H-infinity controller design problems in the literature, comparing the results with those published for other methods. The results show that HIFOO can be used as an effective alternative t...
Preprint
We report on our experience with strong stabilization using HIFOO, a toolbox for H-infinity fixed-order controller design. We applied HIFOO to 21 fixed-order stable H-infinity controller design problems in the literature, comparing the results with those published for other methods. The results show that HIFOO often achieves good H-infinity perform...
Chapter
Full-text available
This article reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. We state an intuitively straightforward gradient sampling algorithm and summarize its convergence properties. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for m...
Article
The limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) method is widely used for large-scale unconstrained optimization, but its behavior on nonsmooth problems has received little attention. L-BFGS (limited memory BFGS) can be used with or without ‘scaling’; the use of scaling is normally recommended. A simple special case, when just one BFGS u...
Preprint
We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so wel...
Preprint
Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the num...
Preprint
Full-text available
The limited memory BFGS (L-BFGS) method is widely used for large-scale unconstrained optimization, but its behavior on nonsmooth problems has received little attention. L-BFGS can be used with or without "scaling"; the use of scaling is normally recommended. A simple special case, when just one BFGS update is stored and used at every iteration, is...
Preprint
Full-text available
This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method f...
Article
Full-text available
We consider low-order controller design for large-scale linear time-invariant dynamical systems with inputs and outputs. Model order reduction is a popular technique, but controllers designed for reduced-order models may result in unstable closed-loop plants when applied to the full-order system. We introduce a new method to design a fixed-order co...
Article
Crouzeix's conjecture states that for all polynomials p and matrices A, the inequality holds, where the quantity on the left is the 2-norm of the matrix and the norm on the right is the maximum modulus of the polynomial p on , the field of values of A. We report on some extensive numerical experiments investigating the conjecture via nonsmooth mini...
Article
Full-text available
We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximat...
Article
Full-text available
We propose an algorithm for solving nonsmooth, nonconvex, constrained optimization problems as well as a new set of visualization tools for comparing the performance of optimization algorithms. Our algorithm is a sequential quadratic optimization method that employs Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton Hessian approximations and an...
Article
The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optima...
Article
Let W(A) denote the field of values (numerical range) of a matrix A. For any polynomial p and matrix A, define the Crouzeix ratio to have numerator (Formula presented.) and denominator (Formula presented.). Crouzeix’s 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is 1 / 2, over all polynomials p of any degree and...
Article
Let b/a be a strictly proper reduced rational transfer function, with a monic. Consider the problem of designing a controller y/x, with deg(y) < deg(x) < deg(a) – 1 and x monic, subject to lower and upper bounds on the coefficients of y and x, so that the poles of the closed loop transfer function, that is the roots (zeros) of ax + by, are, if poss...
Article
The generalized null space decomposition (GNSD) is a unitary reduction of a general matrix A of order n to a block upper triangular form that reveals the structure of the Jordan blocks of A corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya. It was extended first by Ruhe and then by Golub and Wilkinson, who based the...
Article
The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k consistent affine constraints on the coefficients. We show that there always exists an optimal polynomi...
Article
The spectral bundle SB method was introduced by Helmberg and Rend [A spectral bundle method for semidefinite programming. SIAM J. Optim. 10 2000, pp. 673–696] to solve a class of eigenvalue optimization problems that is equivalent to the class of semidefinite programs with the constant trace property. We investigate the feasibility and effectivenes...
Article
The spectral abscissa is a fundamental map from the set of complex matrices to the real numbers. Denoted α and defined as the maximum of the real parts of the eigenvalues of a matrix X, it has many applications in stability analysis of dynamical systems. The function α is nonconvex and is non-Lipschitz near matrices with multiple eigenvalues. Varia...
Article
Full-text available
Consider the set of monic fourth-order real polynomials transformed so that the constant term is one. In the three-dimensional space of the coefficients describing this set, the domain of asymptotic stability is bounded by a surface with the Whitney umbrella singularity. The maximum of the real parts of the roots of these polynomials is globally mi...
Article
Full-text available
The H∞ norm of a transfer matrix function for a control system is the reciprocal of the largest value of ε such that the associated ε-spectral value set is contained in the stability region for the dynamical system (the left half-plane in the continuous-time case and the unit disk in the discrete-time case). After deriving some fundamental properti...
Article
The combination of ever increasing computational power and new mathematical models has fundamentally changed the field of computational chemistry. One example of this is the use of new algorithms for computing the charge density of a molecular system from which one can predict many physical properties of the system. This thesis presents two new alg...
Article
Full-text available
Given a family of real or complex monic polynomials of fixed degree with one affine constraint on their coefficients, consider the problem of minimizing the root radius (largest modulus of the roots) or root abscissa (largest real part of the roots). We give constructive methods for efficiently computing the globally optimal value as well as an opt...
Article
Full-text available
H ∞ controller design for linear systems is a difficult, nonconvex and typically nonsmooth (nondifferentiable) optimization problem when the order of the controller is fixed to be less than that of the open-loop plant, a typical requirement in e.g. embedded aerospace control systems. In this paper we describe a new matlab package called hifoo, aime...
Article
We discuss two nonsmooth functions on Rn introduced by Nesterov. We show that the first variant is partly smooth in the sense of Lewis and that its only stationary point is the global minimizer. In contrast, we show that the second variant has 2n−12n−1 Clarke stationary points, none of them local minimizers except the global minimizer, but also tha...
Article
Full-text available
We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f , not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals con-taining a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthe...
Article
Full-text available
The ε-pseudospectral abscissa αε and radius ρε of an n × n matrix are respectively the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang. Variational analysis of pseudospectra. SIAM Journal on Optimization, 19:1048-1072, 2008] that for fixed ε >...
Article
Full-text available
We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are loca...
Article
Full-text available
The $\varepsilon$-pseudospectral abscissa and radius of an $n\times n$ matrix are, respectively, the maximal real part and the maximal modulus of points in its $\varepsilon$-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue...
Chapter
For semidefinite programming (SDP) problems, traditional primal-dual interior-point methods based on conventional matrix operations have an upper limit on the problem size that the computer can handle due to memory constraints. But for a special kind of SDP problem, which is called the banded symmetric semidefinite programming (BSDP) problem, a mem...
Article
Let A be a matrix with distinct eigenvalues and let w(A)w(A) be the distance from A to the set of defective matrices (using either the 2-norm or the Frobenius norm). Define ΛϵΛϵ, the ϵϵ-pseudospectrum of A, to be the set of points in the complex plane which are eigenvalues of matrices A+EA+E with ‖E‖<ϵ‖E‖<ϵ, and let c(A)c(A) be the supremum of all...
Article
Full-text available
Given a family of real or complex monic polynomials of fixed degree with one affine constraint on their coefficients, consider the problem of minimizing the root radius (largest modulus of the roots) or root abscissa (largest real part of the roots). We give constructive methods for efficiently computing the globally optimal value as well as an opt...
Conference Paper
Full-text available
Given a family of real or complex monic polynomials of fixed degree with one fixed affine constraint on their coefficients, consider the problem of minimizing the root radius (largest modulus of the roots) or abscissa (largest real part of the roots). We give constructive methods for finding globally optimal solutions to these problems. In the real...
Article
We characterize the spectral behavior of a primal Schur-complement-based block diagonal precondi-tioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned ma...
Article
Multiobjective control design is known to be a difficult problem both in theory and practice. Our approach is to search for locally optimal solutions of a nonsmooth optimization problem that is built to incorporate minimization objectives and constraints for multiple plants. We report on the success of this approach using our public-domain Matlab t...
Conference Paper
Full-text available
We report on our experience with strong stabilization using HIFOO, a toolbox for H<sup>∞</sup> fixed-order controller design. We applied HIFOO to 21 fixed-order stable H<sup>∞</sup> controller design problems in the literature, comparing the results with those published for other methods. The results show that HIFOO often achieves good H<sup>∞</sup...
Conference Paper
We report on our experience with fixed-order Hfr<sup>infin</sup> controller design using the HIFOO toolbox. We applied HIFOO to various benchmark fixed (or reduced) order Hfr<sup>infin</sup> controller design problems in the literature, comparing the results with those published for other methods. The results show that HIFOO can be used as an effec...
Article
Full-text available
N. Z. Shor’s r-algorithm [Minimization methods for non-differentiable functions, Berlin: Springer-Verlag (1985; Zbl 0561.90058)] is an iterative method for unconstrained optimization, designed for minimizing nonsmooth functions, for which its reported success has been considerable. Although some limited convergence results are known, nothing seems...
Article
Full-text available
We investigate the BFGS algorithm with an inexact line search when applied to non-smooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, as-suming only absolute continuity. We also prove that the line sear...
Article
Full-text available
Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the dis...
Article
Full-text available
It has been a long-time dream in electronic structure theory in phys- ical chemistry/chemical physics to compute ground state energies of atomic and molecular systems by employing a variational approach in which the two-body reduced density matrix (RDM) is the unknown variable. Realization of the RDM approach has benefited greatly from recent devel...
Article
We consider the problem of optimizing the asymptotic convergence rate of a parameter-dependent nonreversible Markov chain. We begin with a single-parameter case studied by Diaconis, Holmes and Neal and then introduce multiple parameters. We use nonsmooth analysis to investigate whether the presence of multiple parameters allows a faster asymptotic...
Article
Full-text available
Nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed-order controller design. We support this claim by applying nonsmooth analysis and methods to a challenging "Belgian chocolate" stabilization problem pose...
Article
Full-text available
1 Abstract The -pseudospectrum of a matrix A is the subset of the com-plex plane consisting of all eigenvalues of complex matrices within a distance of A, measured by the operator 2-norm. Given a nonderoga-tory matrix A 0 , for small > 0, we show that the -pseudospectrum of any matrix A near A 0 consists of compact convex neighborhoods of the eigen...
Article
Full-text available
The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke-Lewis-Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an impro...
Article
We consider the following problem: find a fixed-order linear controller that maximizes the closed-loop asymptotic decay rate for the classical two-mass-spring system. This can be formulated as the problem of minimizing the abscissa (maximum of the real parts of the roots) of a polynomial whose coefficients depend linearly on the controller paramete...
Conference Paper
Full-text available
We characterize the complex passivity radius of a rational transfer matrix G(s):=C(sI n −A)<sup>−1</sup>B+D and propose an approach to compute it. The method depends on computing the smallest structured indefinite perturbation to a Hermitian matrix that makes it singular. We consider both additive and multiplicative perturbations, giving details fo...
Article
The following problem is addressed: given square matrices $A$ and $B$, compute the smallest $\eps$ such that $A {\hspace{.75pt}} + {\hspace{.75pt}} E$ and $B {\hspace{.75pt}} + {\hspace{.75pt}} F$ have a common eigenvalue for some $E$, $F$ with $\max(\|E\|_2,\|F\|_2) \leq \eps$. An algorithm to compute this quantity to any prescribed accuracy is pr...
Article
Full-text available
The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis...
Article
Two useful measures of the robust stability of the discrete-time dynamical system x k &plus;1 &equals; Ax k are the &epsi;-pseudospectral radius and the numerical radius of A . The &epsi;-pseudospectral radius of A is the largest of the moduli of the points in the &epsi;-pseudospectrum of A , while the numerical radius is the largest of the moduli...
Article
To identify potential risk factors for embryonic loss before 35 to 42 days of gestation in dairy cattle. Prospective observational study. 381 cows. Body condition score was determined at the time of artificial insemination (AI; day 0) and on days 20, 23, and 27 and between days 35 and 41; serum progesterone concentration was measured on days 0; 20...
Article
For nonrelativistic electrons in an external potential the ground state energy depends only upon the two-body reduced density matrix (2-RDM) and a lower-bound approximation may be obtained by minimizing the energy with respect to the 2-RDM subject to some representability conditions. Work going back to the 1970s and the recent work [1] showed that...
Article
Full-text available
Let f be a continuous function on R,, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but is a function whose gradie...
Article
Full-text available
We give a detailed numerical and theoretical analysis of a stabilization problem posed by V. Blondel in 1994. Our approach illustrates the effectiveness of a new "gradient sampling" algorithm for finding local optimizers of nonsmooth, nonconvex optimization problems arising in control, as well as the power of nonsmooth analysis for understanding va...
Article
Objectives were to determine factors associated with conception rate (CR) and pregnancy loss (PL) in high producing lactating Holstein cows. In Study 1, CR was evaluated in 7633 artificial inseminations (AI) of 3161 dairy cows in two dairy farms. Pregnancy diagnosis was performed by palpation per rectum 39+/-3 days after AI. Environmental temperatu...
Article
We consider spectral functions f #,where f is any permutation-invariant mapping from C to R,and# is the eigenvalue map from the set of n n complex matrices to C , ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f # is the spectral abscissa, while if f is "maximum modulus", then f # is the spec...
Article
Full-text available
Consider the linear space n of polynomials of degree n or less over the complex field. The abscissa mapping on n is the mapping that takes a polynomial to the maximum real part of its roots. This mapping plays a key role in the study of stability properties for linear systems. Burke and Overton have shown that the abscissa mapping is everywhere s...
Article
Full-text available
The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J. Quantum Chem. 13, 697 (1978)] and extend the well-kn...
Article
The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q and G conditions. The additional conditions (called T1 and T2 here) are implicit in work of R. M. Erdahl and extend the well-known three-index diagonal conditions al...
Article
Full-text available
There are four different levels of continuing education program evaluation: participant perceptions of the program or course; participant competence with new skills, knowledge, and abilities; participant performance or change in behavior; and health care or client outcomes, such as resultant changes in patient care or herd/flock production performa...
Article
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We show that 2-norm pseudospectra of m-by-n matrices have no more than 2m(4m - 1) connected components. Such bounds are pertinent for computing the distance to uncontrollability of a control system, since this distance is the minimum value of a function whose level sets are pseudospectra. We also discuss algorithms for computing this distance, incl...
Article
To determine whether 4 mg of estradiol cypionate (ECP) administered prophylactically to high-risk postparturient dairy cows decreases incidence of postpartum metritis. Randomized, placebo-controlled, triple-masked clinical trial. 250 postparturient dairy cows in a herd with postparturient hypocalcemia, retained fetal membranes, dystocia, stillbirth...
Article
Full-text available
We show that pseudospectra of m-by-n matrices (for m n) have no more than 2m(4m 1) connected components. Such bounds are pertinent for computing the distance to uncontrollability of a control system, since this distance is the minimum value of a function whose level sets are pseudospectra.
Article
Full-text available
A dynamical system x &dot; &equals; Ax is robustly stable when all eigenvalues of complex matrices within a given distance of the square matrix A lie in the left half&hyphen;plane. The ‘pseudospectral abscissa’, which is the largest real part of such an eigenvalue, measures the robust stability of A . We present an algorithm for computing the pseud...
Article
Full-text available
Stabilization by static output feedback (SOF) is a long-standing open problem in control: given an n by n matrix A and rectangular matrices В and C, find a p by q matrix К such that A + BKC is stable. Low-order controller design is a practically important problem that can be cast in the same framework, with (p+k)(q+k) design parameters instead of p...
Article
Full-text available
Stabilization by static output feedback (SOF) is a long-standing open problem in control: given an n by n matrix A and rectangular matrices B and C, find a p by q matrix K such that A + BKC is stable. Low-order controller design is a practically important problem that can be cast in the same framework, with (p+k)(q+k) design parameters instead of p...
Article
Full-text available
We present a method for constructing multivariate refinable Hermite interpolants and their associated subdivision algorithms based on a combination of analytical and numerical approaches. Being the limit of a linear iterative procedure, the critical L Sobolev smoothness of a refinable Hermite interpolant is given by the spectral radius of a matrix...
Article
The ground state energy and other important observables of a many-fermion system with one- and two-body interactions only can all be obtained from the first order and second order Reduced Density Matrices (RDM's) of the system. Using these density matrices and a family of associated representability conditions one may obtain an approximation method...
Article
Full-text available
The -pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance of A. We are interested in two aspects of optimization and pseudospectra". The rst concerns maximizing the function eal part" over an -pseudospectrum of a xed matrx: this de nes a function known as the -pseudo...
Article
Consider the affine matrix family A(x)=A0+∑k=1mxkAk, mapping a design vector into the space of n×n real matrices. We are interested in the question of how to choose x to optimize the stability of the matrix A(x). A typical motivation is that one wishes to control the stability of the dynamical system . A classic example is stabilization by output f...
Article
The abscissa mapping on the aflqne variety 2k4n of monic polynomials of degree tz is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping is continuous on 2k4, but not Lipschitz...
Article
We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbatio...
Article
Full-text available
Many interesting real functions on Euclidean space are dierentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a nonLipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdi erential of su...
Article
Full-text available
Many interesting real functions on Euclidean space are dierentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a nonLipschitz function). In practice, the gradient is often easy to compute.
Article
We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbatio...