Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briey on semide nite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.

In this paper, we describe our implementation of a primal-dual infeasible-interior-point algorithm for large-scale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL -- Linear-programming Interior-Point SOLvers. LIPSOL is designed to take the advantages of MATLAB's sparse-matrix functions and external interface facilities, and of existing Fortran sparse Cholesky codes. Under the MATLAB environment, LIPSOL inherits a high degree of simplicity and versatility in comparison to its counterparts in Fortran or C language. More importantly, our extensive computational results demonstrate that LIPSOL also attains an impressive performance comparable with that of efficient Fortran or C codes in solving large-scale problems. In addition, we discuss in detail a technique for overcoming numerical instability in Cholesky factorization at the end-stage of iterations in interior-point algorithms. Keywords: Linear programming, Primal-Dual infeasible-interior-p...

This work concerns primal-dual interior-point methods for semidefinite programming (SDP) that use a search direction originally proposed by Helmberg-Rendl-Vanderbei-Wolkowicz [5] and Kojima-Shindoh-Hara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [11] and also in [15] through different means and in different forms. In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variable-metric measures of centrality. These results provide convenient tools for deriving polynomiality results for primal-dual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples...

This work concerns primal-dual interior-point methods for semidefinite programming (SDP) that use a linearized complementarity equation originally proposed by Kojima, Shindoh and Hara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [11] and also in [15] through different means and in different forms. In this paper, we give a very short derivation of the key equalities and inequalities along the exact line used in linear programming (LP), producing basic relationships that have highly compact forms almost identical to their counterparts in LP. We also introduce a new definition of the central path and variable-metric measures of centrality. These results provide convenient tools for extending existing polynomiality results for many, if not most, algorithms from LP to SDP with little complication. We present examples of such extensions, including the long-step infeasible-...

We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path HP (XS) j [P XSP \Gamma1 + (PXSP \Gamma1 ) T ]=2 = ¯I, introduced by Zhang. At an iterate (X; S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP \Gamma1 is symmetric. This class of matrices includes the three well-known choices, namely: P = S 1=2 and P = X \Gamma1=2 proposed by Monteiro, and the matrix P corresponding to the Nesterov-Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov-Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear...

We consider solving a sequence of weighted linear least squares problems where the changes from one problem to the next are the weights and the right hand side (or data). This is the case for primaldual interior-point methods. We derive a class of preconditioners based on a low rank correction to a Cholesky factorization of a weighted normal equation coefficient matrix with the previous weight. Key Words. Weighted linear least squares, Preconditioners, Preconditioned conjugate gradient for least squares, Linear programming, Primaldual infeasible-interior-point algorithms. 1 Introduction In this paper, we present a class of preconditioners based on low rank corrections to the Cholesky factorization of a weighted normal equation coefficient matrix. This class of preconditioners leads to good performance for interiorpoint methods for linear programming. Particularly, we have implemented primal-dual Newton method to test this class of preconditioners. The numerical results on large scale...

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...

Semidefinite programming (SDP) is a convex optimization problem in the space of symmetric matrices. Primal-dual interior-point methods for SDP are discussed. These generate primal and dual matrices X and Z which commute only in the limit. A new method is proposed which iterates in the space of commuting matrices. Let S! n\Thetan denote the set of real symmetric n \Theta n matrices. The standard inner product on this space is A ffl B = tr AB = P i;j a ij b ij : By X 0, where X 2 S! n\Thetan , we mean that X is positive semidefinite. Consider the semidefinite programming problem (SDP) min C ffl X (1) s:t: A i ffl X = b i i = 1; . . . ; m; X 0: (2) Here C and A i , i = 1; . . . ; m, are all fixed matrices in S! n\Thetan , and the unknown variable X also lies in S! n\Thetan . The semidefinite constraint on X is said to be nonsmooth, since it is equivalent to a bound constraint on the least eigenvalue of X , which is not a differentiable function of X . The constraint is, how...

We deal with the primal-dual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for so-called large-update methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of large-update method before. Our new analysis not only provides a unified way for the analysis of both large-update and small-update methods, but also improves the known iteration bounds. Keywords: Linear optimization, interior-point method, primal-dual method, proximity measure, polynomial complexity. AMS Subject Classification: 9...

The computation of the analytic center of the solution set can be important in linear programming applications where it is desirable to obtain a solution that is not near the relative boundary of the solution set. In this work we discuss the effective computation of the analytic center solution by the use of primal-dual interior-point methods. A primal-dual interior-point algorithm designed for effectively computing the analytic-center solution is proposed and numerical results are presented. This research was partially supported by NSF Cooperative Agreement No. CCR-88- 09615, NSF Grant DMS 9305760, ARO Grant 9DAAL03-90-G-0093 DOE Grant DEFG05-86-ER25017, and AFOSR Grant 89-0363. y Department of Computational and Applied Mathematics and Center for Research on Parallel Computation, Rice University, Houston, TX 77251-1892. Partially supported by Fulbright/LASPAU. z Department of Computational and Applied Mathematics and Center for Research on Parallel Computation, Rice University,...