Canonical dual approach to solving the maximum cut problem
@article{Wang2012CanonicalDA, title={Canonical dual approach to solving the maximum cut problem}, author={Zhenbo Wang and S. C. Fang and David Yang Gao and Wenxun Xing}, journal={Journal of Global Optimization}, year={2012}, volume={54}, pages={341-351} }
This paper presents a canonical dual approach for finding either an optimal or approximate solution to the maximum cut problem (MAX CUT). We show that, by introducing a linear perturbation term to the objective function, the maximum cut problem is perturbed to have a dual problem which is a concave maximization problem over a convex feasible domain under certain conditions. Consequently, some global optimality conditions are derived for finding an optimal or approximate solution. A gradient…
46 Citations
Improved Canonical Dual Algorithms for the Maxcut Problem
- MathematicsArXiv
- 2012
By introducing a quadratic perturbation to the canonical dual of the maxcut problem, we transform the integer programming problem into a concave maximization problem over a convex positive domain…
Global Optimal Solution to Quadratic Discrete Programming Problem with Inequality Constraints
- Mathematics
- 2017
This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. By using a linear transformation, the problem is first…
Global Optimal Solution to Discrete Value Selection Problem with Inequality Constraints
- MathematicsArXiv
- 2012
This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints and a maximization problem of a concave function over a convex continuous space.
Four strategies to develop canonical dual algorithms for global optimization problems
- Mathematics, Computer Science
- 2012
This paper shows that under certain conditions, this canonical dual problem is equivalent to the standard semi-definite programming (SDP) problem, which can be solved by well-developed software packages.
Canonical Dual Solutions to Quadratic Optimization over One Quadratic Constraint
- MathematicsAsia Pac. J. Oper. Res.
- 2015
Under the dual Slater's condition, it is shown that the canonical dual has a smooth concave objective function over a convex feasible domain, and thisDual has a finite supremum unless the original quadratic optimization problem is infeasible.
Canonical dual method for mixed integer fourth-order polynomial minimization problems with fixed cost terms
- Mathematics
- 2016
We study a canonical duality method to solve a mixed-integer nonconvex fourth-order polynomial minimization problem with fixed cost terms. This constrained nonconvex problem can be transformed into a…
Canonical duality for solving general nonconvex constrained problems
- MathematicsOptim. Lett.
- 2016
It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem can be obtained in a unified form with global optimality conditions provided.
Unified Interior Point Methodology for Canonical Duality in Global Optimization
- Mathematics
- 2017
We propose an interior point method to solve instances of the nonconvex optimization problems reformulated with canonical duality theory. To this aim we propose an interior point potential reduction…
Global Solutions to Large-Scale Spherical Constrained Quadratic Minimization via Canonical Dual Approach
- Mathematics
- 2013
This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied…
CANONICAL DUALITY AND CONIC PROGRAMMING FOR SOLVING 0-1 QUADRATIC PROGRAMMING PROBLEMS
- Mathematics
- 2010
An extended canonical dual approach for solving 0-1 quadratic programming problems is introduced. We derive the relationship between the optimal solutions to the extended canonical dual problem and…
References
SHOWING 1-10 OF 26 REFERENCES
Canonical dual approach to solving 0-1 quadratic programming problems
- Mathematics
- 2008
By using the canonical dual transformation developed
recently, we derive a pair of canonical dual problems for 0-1
quadratic programming problems in both minimization and maximization
form.…
GLOBAL EXTREMAL CONDITIONS FOR MULTI-INTEGER QUADRATIC PROGRAMMING
- Mathematics
- 2008
This paper presents a canonical duality approach to solve an
integer quadratic programming problem, in which the objective
function is quadratic and each variable may assume the value of one
of…
Solutions to quadratic minimization problems with box and integer constraints
- MathematicsJ. Glob. Optim.
- 2010
Results show that under Gao and Strang’s general global optimality condition, these well-known nonconvex and discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without duality gap.
Solving the canonical dual of box- and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm
- MathematicsOptim. Methods Softw.
- 2013
This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or±1 integer constraints using the canonical dual transformation to solve these dual problems to obtain global minimizers.
CANONICAL DUALITY AND CONIC PROGRAMMING FOR SOLVING 0-1 QUADRATIC PROGRAMMING PROBLEMS
- Mathematics
- 2010
An extended canonical dual approach for solving 0-1 quadratic programming problems is introduced. We derive the relationship between the optimal solutions to the extended canonical dual problem and…
Lagrange-type Functions in Constrained Non-Convex Optimization
- Computer Science, Mathematics
- 2003
The question arises how to generalize classical Lagrange and penalty functions, in order to obtain an appropriate scheme for reducing constrained optimization problems to unconstrained ones that will be suitable for sufficiently broad classes of optimization problems from both the theoretical and computational viewpoints.
A projected gradient algorithm for solving the maxcut SDP relaxation
- Computer Science, Mathematics
- 2001
A projected gradient algorithm for solving the semidefinite programming (SDP) relaxation of the maximum cut (maxcut) problem is presented and combined with a randomized method this gives a very efficient approximation algorithm for the maxcut problem.
On the stability of a dual weak vector variational inequality problem
- Mathematics
- 2008
In this paper, we obtain some stability results for the dual
problem of a weak vector variational inequality problem. We
establish the upper semicontinuity property of the solution set
for a…
Perfect duality theory and complete solutions to a class of global optimization problems
- Mathematics
- 2003
This article presents a complete set of solutions for a class of global optimization problems. These problems are directly related to numericalization of a large class of semilinear nonconvex partial…
An improved lower bound and approximation algorithm for binary constrained quadratic programming problem
- Computer Science, MathematicsJ. Glob. Optim.
- 2010
An improved lower bound and an approximation algorithm based on spectral decomposition for the binary constrained quadratic programming problem and a worst case performance analysis for the algorithm are provided.