A Rank-Two Feasible Direction Algorithm for the Binary Quadratic Programming

Based on the semidefinite programming relaxation of the binary quadratic programming, a rank-two feasible direction algorithm is presented. The proposed algorithm restricts the rank of matrix variable to be two in the semidefinite programming relaxation and yields a quadratic objective function with simple quadratic constraints. A feasible direction algorithm is used to solve the nonlinear programming. The convergent analysis and time complexity of the method is given. Coupled with randomized algorithm, a suboptimal solution is obtained for the binary quadratic programming. At last, we report some numerical examples to compare our algorithm with randomized algorithm based on the interior point method and the feasible direction algorithm on max-cut problem. Simulation results have shown that our method is faster than the other two methods.

Constrained Variable Metric Method With Feasible Directions

This paper proposes a new feasible direction algorithm based on the constrained variable metric method of Powell in order to handle the design optimization problmes which demand that all iterative points are feasible. The algorithm retains many advantages of the constrained variable metric method, makes use of the properties of the solutions of quadratic programming problems and information of iterative points to define feasible directions, and uses the monotonicity analysis to establish the linesearch strategy which is especially suitable for feasible direction algorithms and a simple and efficient method for finding feasible initial points. The numerical results presented in the paper demonstrate that its rate of convergence is faster than those of Powell’s method and another feasible direction algorithm of Herskovits and its iterative procedure avoids Maratos effect.