The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.

Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomial-time interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of ever-increasing size. Interior methods...

This paper was conceived in part while the author was sponsored by the visiting scientist program at the IBM T.J. Watson Research Center. Special thanks to Mike Shub, Roy Adler and Shmuel Winograd for their generosity. 1 Introduction

Consider the ane matrix family A(x) = A 0 + k=1 x k A k , mapping a design vector x 2 R into the space of n n real matrices.

TOMLAB is a general purpose, open and integrated MATLAB environment for solving optimization problems on UNIX and PC systems. TOMLAB has meny systems and driver routines for the most common optimization problems and more than 50 algorithms implemented in the toolbox NLPLIB and the toolbox OPERA. NLPLIB TB 1.0 is a MATLAB toolbox for nonlinear programming and parameter estimation and OPERA TB 1.0 is a MATLAB toolbox for operational research, with emphasis on linear and discrete optimization. Of special interest in NLPLIB TB 1.0 are the algorithms for general and separable nonlinear least squares parameter estimation. TOMLAB is using MEX-file interfaces to call solvers written in C/C++ and FORTRAN. Currently MEXfile interfaces have been developed for the commercial solvers MINOS, NPSOL, NPOPT, NLSSOL, LPOPT, QPOPT and LSSOL. From TOMLAB it is also possible to call routines in the MathWorks Optimization Toolbox. Interfaces are available for the model language AMPL and the CUTE (Cons...

TOMLAB is a general purpose, open and integrated MATLAB environment for research and teaching in optimization on UNIX and PC systems. The motivation for TOMLAB is to simplify research on practical optimization problems, giving easy access to all types of solvers; at the same time having full access to the power of MATLAB. By using a simple, but general input format, combined with the ability in MATLAB to evaluate string expressions, it is possible to run internal TOMLAB solvers, MATLAB Optimization Toolbox and commercial solvers written in FORTRAN or C/C++ using MEX-file interfaces. Currently MEX-file interfaces have been developed for MINOS, NPSOL, NPOPT, NLSSOL, LPOPT, QPOPT and LSSOL. TOMLAB may either be used totally parameter driven or menu driven. The basic principles will be discussed. The menu system makes it suitable for teaching. Many standard test problems are included. More test problems are easily added. There are many example and demonstration files. Iterati...

The Matlab toolbox OPERA TB is a set of Matlab m- les, which solves basic linear and discrete optimization problems in operations research and mathematical programming. Included are routines for linear programming (LP), network programming (NP), integer programming (IP) and dynamic programming (DP). OPERA TB, like the nonlinear programming toolbox NLPLIB TB, is a part of TOMLAB � an environment in Matlab for research and teaching in optimization. Linear programs are solved either by direct call to a solver routine or to a multisolver driver routine, or interactively, using the Graphical User Interface (GUI) or a menu system. From OPERA TB it is possible to call solvers in the Math Works Optimization Toolbox and, using a MEX- le interface, general-purpose solvers implemented in Fortran or C. The focus is on dense problems, but sparse linear programs may be solved using the commercial solver MINOS. Presently, OPERA TB implements about thirty algorithms and includes a set of test examples and demonstration les. This paper gives an overview of OPERA TB and presents test results for medium size LP problems. The tests show that the OPERA TB solver converges as fast as commercial Fortran solvers and is at least ve times faster than the simplex LP solver in the Optimization Toolbox 2.0andtwice as fast as the primal-dual interior-pointLP solver in the same toolbox. Running the commercial Fortran solvers using MEX- le interfaces gives a speed-up factor of ve to thirty- ve.

. Much progress has been made in constrained nonlinear optimization in the past ten years, but most large-scale problems still represent a considerable obstacle. In this survey paper we will attempt to give an overview of the current approaches, including interior and exterior methods and algorithms based upon trust regions and line searches. In addition, the importance of software, numerical linear algebra and testing will be addressed. We will try to explain why the difficulties arise, how attempts are being made to overcome them and some of the problems that still remain. Although there will be some emphasis on the LANCELOT and CUTE projects, the intention is to give a broad picture of the state-of-the-art. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA 2 Parallel Algorithms Team, CERFACS, 42 Ave. G. Coriolis, 31057 Toulouse Cedex, France 3 Central Computing Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England ...

The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.

Contents 1 The TOMLAB OpBP-yD-fP0 Environment 7 1.1 Backgrou nd................................................. 8 1.2 TheDesignofTOMLAB.......................................... 8 1.2.1 Global Variables . . ........................................ 22 1.3 SolverRou tinesinTOMLAB ....................................... 24 1.4 Menu ProgramsinTOMLAB....................................... 26 1.5 LowLevelRou tinesandTestProblems.................................. 27 1.5.1 Utility Test RouNEFF ........................................ 34 1.6 TheGraphicalUserInterface ....................................... 34 1.6.1 The Advanced Mode ........................................ 35 2 Solving Linear, Quadratic and Integer Programming Problems 39 2.1 Linear Programming Problems ...................................... 39 2.1.1 AQu0 k Linear Programming SoluxFz .............................. 39 2.1.2 Several Linear Programs ..................................... 39 2.1.