# Semidefinite programming in combinatorial optimization

@article{Goemans1997SemidefinitePI, title={Semidefinite programming in combinatorial optimization}, author={Michel X. Goemans}, journal={Mathematical Programming}, year={1997}, volume={79}, pages={143-161} }

We discuss the use of semidefinite programming for combinatorial optimization problems. The main topics covered include (i) the Lovász theta function and its applications to stable sets, perfect graphs, and coding theory, (ii) the automatic generation of strong valid inequalities, (iii) the maximum cut problem and related problems, and (iv) the embedding of finite metric spaces and its relationship to the sparsest cut problem.

## 335 Citations

Semidefinite Programming in Combinatorial and Polynomial Optimization

- Mathematics, Computer Science
- 2008

Some of the main mathematical tools that underlie semidefinite programming are sketched and their application to some graph problems dealing with maximum cuts, stable sets and graph coloring is illustrated.

Semidefinite Relaxations for Integer Programming

- Computer Science50 Years of Integer Programming
- 2010

A survey of recent developments in the area of semidefinite optimization applied to integer programming, and some more advanced modeling techniques, based on matrix relaxations leading to copositive matrices, are concluded.

Semidefinite programming and integer programming

- Computer Science, Mathematics
- 2002

The geometry of SDP-exactness in quadratic optimization

- Mathematics, Computer ScienceMath. Program.
- 2020

The semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation is studied and the algebraic boundary of this region is characterized and a formula for its degree is derived.

Foundations of Set-Semidefinite Optimization

- Computer Science, Mathematics
- 2010

In this paper, we present various foundations of a new field of research in optimization unifying semidefinite and copositive programming, which is called set-semidefinite optimization. A…

On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0–1 quadratic problems leading to quasi-Newton methods

- Mathematics, Computer ScienceMath. Program.
- 2013

This article presents a family of semidefinite programming bounds, obtained by Lagrangian duality, for 0–1 quadratic optimization problems with linear or quadratic constraints. These bounds have…

Lagrangian Quadratic Bounds in Polynomial Nonconvex and Boolean Models with Superfluous Constraints

- Mathematics
- 2001

The article is devoted to the Lagrangian methods of obtaining bounds for optimal value of nonlinear optimization problems, especially for that of quadratic type problems. Calculation of such bounds…

Semidefinite Programs and Association Schemes

- Computer Science, MathematicsComputing
- 2014

It is shown that in this case the semidefinite program can be solved through an ordinary linear program, where the underlying graph arises from an association scheme.

## References

SHOWING 1-10 OF 76 REFERENCES

Combining Semidefinite and Polyhedral Relaxations for Integer Programs

- Computer ScienceIPCO
- 1995

We present a general framework for designing semidefinite relaxations for constrained 0–1 quadratic programming and show how valid inequalities of the cut-polytope can be used to strengthen these…

Connections between semidefinite relaxations of the max-cut and stable set problems

- Computer ScienceMath. Program.
- 1997

It turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra.

Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

- MathematicsSIAM J. Optim.
- 1995

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.

Semidefinite Programming

- Computer Science, MathematicsSIAM Rev.
- 1996

A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.

The ellipsoid method and its consequences in combinatorial optimization

- Mathematics, Computer ScienceComb.
- 1981

The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.

Semidefinite programming for assignment and partitioning problems

- Computer Science
- 1998

An efficient "partial infeasible" primal-dual interior-point algorithm is developed by using a conjugate gradient method and by taking advantage of the special data structure of the SDP relaxation, which plays a significant role in simplifying the problem.

Quality of semidefinite relaxation for nonconvex quadratic optimization

- Mathematics, Computer Science
- 1997

In this paper we prove that the semidefinite relaxation of boolean quadratic maximization problem with indefinite matrix provides us with a fixed absolute accuracy estimate for the exact solution.

Quadratic Knapsack Relaxations Using Cutting Planes

- MathematicsIPCO
- 1996

It is shown that simple semidefinite relaxations are tighter than corresponding linear relaxations even in case of linear cost functions.

The Quadratic Assignment Problem: Theory and Algorithms

- Computer Science
- 1998

This work focuses on the Biquadratic Assignment Problem (BIQAP) and the applications of Heuristics and Asymptotic Behavior to QAPs Arising as Optimization Problems in Graphs.

Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems

- Computer ScienceSTOC '99
- 1999

Using outward rotations, an approximation algorithm is obtained for MAX CUT that performs better than the algorithm of Goemans and Williamson and an improved approximation algorithm for MAX NAE{3}-SAT is obtained.