Michael L. Overton

Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results

Primal-dual interior-point path-following methods for semidefinite programming are considered. Several variants are discussed, based on Newtons method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called the XZ, XZ+ZX, and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well defined and its Jacobian is nonsingular at the solution, under nondegeneracy assumptions. The associated Schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption. Practical aspects are discussed, including Mehrotra predictor-corrector variants and issues of numerical stability. Compared to the other methods considered, the XZ+ZX method is more robust with respect to its ability to step close to the boundary, converges more rapidly, and achieves higher accuracy.

A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization

Let f be a continuous function on

Large-Scale Optimization of Eigenvalues

\Rl^n

On minimizing the maximum eigenvalue of a symmetric matrix

, 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 gradient is easily computed where it is defined. We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. No subgradient information is required by the algorithm. When f is locally Lipschitz and has bounded level sets, and the sampling radius

Complementarity and nondegeneracy in semidefinite programming

\eps

HIFOO - A MATLAB PACKAGE FOR FIXED-ORDER CONTROLLER DESIGN AND H∞ OPTIMIZATION

is fixed, we show that, with probability 1, the algorithm generates a sequence with a cluster point that is Clarke

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

\eps

Thr formulation and analysis of numerical methods for inverse Eigenvalue problems

-stationary. Furthermore, we show that if f has a unique Clarke stationary point

Nonsmooth optimization via quasi-Newton methods

\bar x

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

, then the set of all cluster points generated by the algorithm converges to