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Let f be a continuous function on R n , 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 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 ǫ is fixed, we show that, with probability one, the algorithm generates a sequence with a cluster point that is Clarke ǫ-stationary. Furthermore, we show that if f has a unique Clarke stationary pointx, then the set of all cluster points generated by the algorithm converges tox as ǫ is reduced to zero.

Primal-dual interior-point path-following methods for semide nite programming (SDP) are considered. Several variants are discussed, based on Newton's method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called respectively the XZ, XZ+ZX and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well-de ned 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.

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-known three-index diagonal conditions also known as the Weinhold-Wilson inequalities. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Formulating the reduced density matrix computation using the standard dual formulation of semidefinite programming, as opposed to the primal one, results in substantial computational savings and makes it possible to study larger systems than was done previously. Calculations of the ground state energy and the dipole moment are reported for 47 different systems, in each case using an STO-6G basis set and comparing with Hartree-Fock, singly and doubly substituted configuration interaction, Brueckner doubles (with triples), coupled cluster singles and doubles with perturbational treatment of triples, and full configuration interaction calculations. It is found that the use of the T1 and T2 conditions gives a significant improvement over just the P, Q, and G conditions, and provides in all cases that we have studied more accurate results than the other mentioned approximations.

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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 containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.

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, aimed at solving fixed-order stabilization and local optimization problems. It depends on a new hybrid algorithm for nonsmooth, nonconvex optimization based on several techniques, namely quasiNewton updating, bundling and gradient sampling. The user may request hifoo to optimize one of several objectives, including H ∞ norm, which requires either the Control System Toolbox for matlab or, for much better performance, the linorm function in the slicot package. No other external package is required, but the quadratic programming code quadprog from either mosek or the Optimization Toolbox for matlab is recommended. Numerical experiments on benchmark problem instances from the COMPl e ib database indicate that hifoo could be an efficient and reliable computer-aided control system design (CACSD) tool, with a potential for realistic industrial applications.

Abstract. We consider the formulation and local analysis of various quadratically convergent methods for solving the symmetric matrix inverse eigenvalue problem. One of these methods is new. We study the case where multiple eigenvalues are given: we show how to state the problem so that it is not overdetermined, and describe how to modify the numerical methods to retain quadratic convergence on the modified problem. We give a general convergence analysis, which covers both the distinct and the multiple eigenvalue cases. We also present numerical experiments which illustrate our results.