We consider the Riemannian geometry defined on a convex set by the Hessian of a self-concordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the … Read more
Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the … Read more
The central path in linear optimization always converges to the analytic center of the optimal set. This result was extended to semidefinite programming by Goldfarb and Scheinberg (SIAM J. Optim. 8: 871-886, 1998). In this paper we show that this latter result is not correct in the absence of strict complementarity. We provide a counterexample, … Read more
Let $\Pcal=\{x | Ax\le b\}$, where $A$ is an $m\times n$ matrix. We assume that $\Pcal$ contains a ball of radius one centered at the origin, and is contained in a ball of radius $R$ centered at the origin. We consider the problem of approximating the maximum volume ellipsoid inscribed in $\Pcal$. Such ellipsoids have … Read more
We consider spectral functions $f \circ \lambda$, where $f$ is any permutation-invariant mapping from $\Cx^n$ to $\Rl$, and $\lambda$ is the eigenvalue map from the set of $n \times n$ complex matrices to $\Cx^n$, ordering the eigenvalues lexicographically. For example, if $f$ is the function “maximum real part CitationMath. Programming 90 (2001), pp. 317-352
The abscissa mapping on the affine variety $M_n$ of monic polynomials of degree $n$ 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 … Read more
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 perturbations of the … Read more
Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to … Read more
Many interesting real functions on Euclidean space are differentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a non-Lipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdifferential of such a function … Read more
Consider the affine matrix family $A(x) = A_0 + \sum_{k=1}^m x_k A_k$, mapping a design vector $x\in\Rl^m$ into the space of $n \times n$ real matrices. Consider the affine matrix family $A(x) = A_0 + \sum_{k=1}^m x_k A_k$, mapping a design vector $x\in\Rl^m$ into the space of $n \times n$ real matrices. We are interested … Read more