# Interior-point methods

@inproceedings{Boyd2004InteriorpointM, title={Interior-point methods}, author={Stephen Boyd and Lieven Vandenberghe}, year={2004} }

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…

## 468 Citations

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

- Mathematics, Computer Science
- 2017

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

- Mathematics, Computer ScienceComput. Optim. Appl.
- 2004

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

- Mathematics
- 2017

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.

- Mathematics
- 2018

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

- Mathematics, Computer Science
- 2020

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.

### ON INTERIOR-POINT METHODS AND SIMPLEX METHOD IN LINEAR PROGRAMMING

- Mathematics, Computer Science
- 2005

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

- MathematicsJ. 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

- MathematicsJournal 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

- Computer Science, Mathematics
- 2012

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

- Mathematics2016 American Control Conference (ACC)
- 2016

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.

## References

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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.