Interior-point methods

  title={Interior-point methods},
  author={Stephen Boyd and Lieven Vandenberghe},
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, semide nite 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… 

Stable interior-point method for convex quadratic programming with strict error bounds

A short step interior point method for solving a class of nonlinear programming problems with quadratic objective function and is shown to have weak polynomial time complexity.

Convergence Analysis of an Inexact Infeasible Interior Point Method for Semidefinite Programming

An extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima, Megiddo and Mizuno allows the use of inexact search directions and proves the global convergence of the method.

On Generalized Primal-Dual Interior-Point Methods with Non-uniform Complementarity Perturbations for Quadratic Programming

This technical note discusses convergence conditions of a generalized variant of primal-dual interior point methods. The generalization arises due to the permitted case of having a non-uniform

Primal-dual interior-point Methods for Semidefinite Programming from an algebraic point of view, or: Using Noncommutativity for Optimization.

Since more than three decades, interior-point methods proved very useful for optimization, from linear over semidefinite to conic (and partly beyond non-convex) programming; despite the fact that

A geodesic interior-point method for linear optimization over symmetric cones

A new interior-point method for symmetric-cone optimization is developed, a common generalization of linear, second-order-cone, and semidefinite programming, that yields a primal-dual-symmetric, scale-invariant, and line-search-free algorithm that uses just half the variables of a standard primal- dual method.


This paper presents new possibilities offered by the interior-point methods, which appears from practical necessity, from the need of efficient means of solving large-scale problems.

Lagrangian Transformation and Interior Ellipsoid Methods in Convex Optimization

  • R. Polyak
  • Mathematics
    J. Optim. Theory Appl.
  • 2015
It is shown that the primal exterior LT method is equivalent to the dual interior ellipsoid method (IEM), and the complexity bound for the IEM is established assuming boundedness of both the primal and the dual optimal sets.

Lagrangian Transformation and Interior Ellipsoid Methods in Convex Optimization

  • R. Polyak
  • Mathematics
    Journal of Optimization Theory and Applications
  • 2014
The rediscovery of the affine scaling method in the late 1980s was one of the turning points which led to a new chapter in Modern Optimization—the interior point methods (IPMs). Simultaneously and

A Quadratically Convergent ( ) O n Interior-Point Algorithm for the P * ( κ )-Matrix Horizontal Linear Complementarity Problem

The currently best known iteration bound for the algorithm with small-update method is obtained, namely, (1 ) log n O n κ ε   +     , which is as good as the linear analogue.

Interior point method for dynamic constrained optimization in continuous time

An interior point method is developed that asymptotically succeeds in tracking the optimal point in nonstationary settings using a time-varying constraint slack and a prediction-correction structure that relies on time derivatives of functions and constraints and Newton steps in the spatial domain.



Recent Developments in Interior-Point Methods

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming, and has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear programs.

Interior-point polynomial algorithms in convex programming

This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.

Self-Scaled Barriers and Interior-Point Methods for Convex Programming

This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled, with long-step and symmetric primal-dual methods.

Potential-reduction methods in mathematical programming

  • M. Todd
  • Mathematics
    Math. Program.
  • 1996
A survey of interior-point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration, focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis.

Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

It is argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion.

Primal-Dual Interior-Point Methods for Self-Scaled Cones

This paper presents efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods for convex programming problems expressed in conic form when the cone and its associated barrier are self-scaled.

Potential Reduction and Primal-Dual Methods

This chapter will be content with the bounds on the number of iterations of the interior-point algorithms with polynomial iteration complexity that have been studied extensively since 1984.

Interior Point Methods of Mathematical Programming

This chapter discusses Interior Point Approaches for the VLSI Placement Problem, as well as implementation of Interior-Point Methods for Large Scale Linear Programs, and Semidefinite Programming.

Interior methods for constrained optimization

A self-contained survey of major themes in both classical material and recent developments related to the theory and practice of interior methods for linear programming is presented.

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

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 than other methods considered, including Mehrotra predictor-corrector variants and issues of numerical stability.