# A proximal DC approach for quadratic assignment problem

@article{Jiang2021APD, title={A proximal DC approach for quadratic assignment problem}, author={Zhuoxuan Jiang and Xinyuan Zhao and Chao Ding}, journal={Computational Optimization and Applications}, year={2021}, pages={1-27} }

In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method. We show that the generated sequence converges to a stationary point of the… Expand

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#### References

SHOWING 1-10 OF 63 REFERENCES

Semidefinite Programming Relaxations for the Quadratic Assignment Problem

- Mathematics, Computer Science
- J. Comb. Optim.
- 1998

These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods. Expand

Alternating direction augmented Lagrangian methods for semidefinite programming

- Mathematics, Computer Science
- Math. Program. Comput.
- 2010

Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that the algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems. Expand

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

- Mathematics, Computer Science
- SIAM J. Optim.
- 2010

This work considers a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods and shows that the positive definiteness of the generalized Hessian of the objective function in these inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems. Expand

On Lagrangian Relaxation of Quadratic Matrix Constraints

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2000

It is shown that the Lagrangian dual based on relaxing the constraints XXT=I and the seemingly redundant constraints XT X=I has a zero duality gap, which has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. Expand

Bounds for the quadratic assignment problem using the bundle method

- Mathematics, Computer Science
- Math. Program.
- 2007

Some SDP relaxations of QAP are recalled and solved approximately using a dynamic version of the bundle method, finding their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree. Expand

Copositive and semidefinite relaxations of the quadratic assignment problem

- Computer Science, Mathematics
- Discret. Optim.
- 2009

It is shown that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices and several of the well-studied models are in fact equivalent. Expand

Proximal alternating linearized minimization for nonconvex and nonsmooth problems

- Mathematics, Computer Science
- Math. Program.
- 2014

A self-contained convergence analysis framework is derived and it is established that each bounded sequence generated by PALM globally converges to a critical point. Expand

On optimization over the doubly nonnegative cone

- Mathematics, Computer Science
- 2010 IEEE International Symposium on Computer-Aided Control System Design
- 2010

It is shown that the doubly nonnegative relaxation gives significantly tight bounds for a class of quadratic assignment problems, while the computational time may not be affordable as long as the authors solve the relaxation as usual. Expand

Recent advances in the solution of quadratic assignment problems

- Mathematics, Computer Science
- Math. Program.
- 2003

This work describes the developments of a number of long-open QAPs, including those posed by Steinberg (1961), Nugent et al. (1968) and Krarup (1972), as well as recent work which is likely to result in the solution of even more difficult instances. Expand

A Sequential Semismooth Newton Method for the Nearest Low-rank Correlation Matrix Problem

- Mathematics, Computer Science
- SIAM J. Optim.
- 2011

This work forms the nearest low-rank correlation matrix problem as a nonconvex SDP and proposes a numerical method that solves a sequence of least-square problems. Expand