# Interior-point polynomial algorithms in convex programming

@inproceedings{Nesterov1994InteriorpointPA, title={Interior-point polynomial algorithms in convex programming}, author={Yurii Nesterov and Arkadi Nemirovski}, booktitle={Siam studies in applied mathematics}, year={1994} }

Written for specialists working in optimization, mathematical programming, or control theory. The general theory of path-following and potential reduction interior point polynomial time methods, interior point methods, interior point methods for linear and quadratic programming, polynomial time methods for nonlinear convex programming, efficient computation methods for control problems and variational inequalities, and acceleration of path-following methods are covered. In this book, the…

## 3,768 Citations

### On Polynomial-time Path-following Interior-point Methods with Local Superlinear Convergence

- Mathematics
- 2016

Interior-point methods provide one of the most popular ways of solving convex optimization problems. Two advantages of modern interior-point methods over other approaches are: (i) robust global…

### Cones and interior-point algorithms for structured convex optimization involving powers andexponentials

- Computer Science, Mathematics
- 2009

It is demonstrated in this work that a large class of convex optimization problems are representable in a convex conic form based on the so-called power cone, which implies that all problems belonging to the aforementioned class are solvable in polynomial time.

### Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

- Computer Science, MathematicsFound. Comput. Math.
- 2001

It is shown that essentially all primal—dual interior-point algorithms for LP can be extended easily to the general setting of convex optimization problems, and makes a very strong connection to quasi-Newton methods by expressing the square of the symmetric primal-dual linear transformation as a quasi- newton update in the case of the least sophisticated framework.

### Interior-point methods for unconstrained geometric programming and scaling problems

- Mathematics, Computer ScienceArXiv
- 2020

The condition numbers are natural geometric quantities associated with the Newton polytope of the geometric program, and lead to diameter bounds on approximate minimizers, which generalize the iteration complexity of recent interior-point methods for matrix scaling and matrix balancing.

### An interior point-proximal method of multipliers for convex quadratic programming

- Computer Science, MathematicsComputational Optimization and Applications
- 2020

This paper combines an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and shows that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems.

### Optimization on linear matrix inequalities for polynomial systems control

- Mathematics, Computer Science
- 2013

This course describes semidefinite programming as an extension of linear programming to the cone of positive semideFinite matrices, and investigates the geometry of spectrahedra, convex sets defined by linear matrix inequalities or affine sections of the SDP cone.

### Interior-point methods

- Computer Science
- 2004

This work reviews some of the key developments in the modern era of interior-point methods, including comments on both the complexity theory and practical algorithms for linear programming, semide nite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.

### Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

- Computer ScienceMath. Program. Comput.
- 2019

The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms.

### A new perspective on the complexity of interior point methods for linear programming

- Mathematics
- 2007

In a dynamical systems paradigm, many optimization algorithms are equivalent to applying forward Euler method to the system of ordinary differential equations defined by the vector field of the…

### Interior Point Methods for Nonlinear Optimization

- Computer Science
- 2010

After more than a decade of turbulent research, the IPM community reached a good understanding of the basics of IPMs and several books were published that summarize and explore different aspects of IPM's.