In this paper, we consider the socalled k-coloring problem in general case.Firstly, a special quadratic 0-1 programming is constructed to formulate k-coloring problem. Secondly, by use of the equivalence between above...In this paper, we consider the socalled k-coloring problem in general case.Firstly, a special quadratic 0-1 programming is constructed to formulate k-coloring problem. Secondly, by use of the equivalence between above quadratic0-1 programming and its relaxed problem, k-coloring problem is converted intoa class of (continuous) nonconvex quadratic programs, and several theoreticresults are also introduced. Thirdly, linear programming approximate algorithmis quoted and verified for this class of nonconvex quadratic programs. Finally,examining problems which are used to test the algorithm are constructed andsufficient computation experiments are reported.展开更多
In this paper we study a Class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Str...In this paper we study a Class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.展开更多
Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a noncon...Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.展开更多
This paper indicates the possible difficulties for applying the interior point method to NPcomplete problems,transforms an NP-complete problem into a nonconvex quadratic program and then develops some convexity theori...This paper indicates the possible difficulties for applying the interior point method to NPcomplete problems,transforms an NP-complete problem into a nonconvex quadratic program and then develops some convexity theories for it. Lastly it proposes an algorithm which uses Karmarkar's algorithm as a subroutine. The finite convergence of this algorithm is also proved.展开更多
The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on ti...The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations(SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method,quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K^(-1/4)) rate of gradient convergence with K being the total number of iterative steps.展开更多
This paper develops new semidefinite programming(SDP)relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs and analyzes their approximation performance.The first class of ...This paper develops new semidefinite programming(SDP)relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs and analyzes their approximation performance.The first class of problems finds two minimum norm vectors in N-dimensional real or complex Euclidean space,such that M out of 2M concave quadratic constraints are satisfied.By employing a special randomized rounding procedure,we show that the ratio between the norm of the optimal solution of this model and its SDP relaxation is upper bounded by 54πM2 in the real case and by 24√Mπin the complex case.The second class of problems finds a series of minimum norm vectors subject to a set of quadratic constraints and cardinality constraints with both binary and continuous variables.We show that in this case the approximation ratio is also bounded and independent of problem dimension for both the real and the complex cases.展开更多
文摘In this paper, we consider the socalled k-coloring problem in general case.Firstly, a special quadratic 0-1 programming is constructed to formulate k-coloring problem. Secondly, by use of the equivalence between above quadratic0-1 programming and its relaxed problem, k-coloring problem is converted intoa class of (continuous) nonconvex quadratic programs, and several theoreticresults are also introduced. Thirdly, linear programming approximate algorithmis quoted and verified for this class of nonconvex quadratic programs. Finally,examining problems which are used to test the algorithm are constructed andsufficient computation experiments are reported.
基金Supported by the fundamental research funds for the central universities under grant YWF-10-02-021 and by National Natural Science Foundation of China under grant 11001006 The author is very grateful to all the three anonymous referees for their constructive criticisms and useful suggestions that help to improve the paper.
文摘In this paper we study a Class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.
基金supported by US Army Research Office Grant(No.W911NF-04-D-0003)by the North Carolina State University Edward P.Fitts Fellowship and by National Natural Science Foundation of China(No.11171177)。
文摘Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.
文摘This paper indicates the possible difficulties for applying the interior point method to NPcomplete problems,transforms an NP-complete problem into a nonconvex quadratic program and then develops some convexity theories for it. Lastly it proposes an algorithm which uses Karmarkar's algorithm as a subroutine. The finite convergence of this algorithm is also proved.
基金This work was supported by the National Key R&D Program of China(No.2018YFA0703900)the National Natural Science Foundation of China(No.11631004)。
文摘The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations(SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method,quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the O(K^(-1/4)) rate of gradient convergence with K being the total number of iterative steps.
基金the National Natural Science Foundation of China(No.11101261).
文摘This paper develops new semidefinite programming(SDP)relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs and analyzes their approximation performance.The first class of problems finds two minimum norm vectors in N-dimensional real or complex Euclidean space,such that M out of 2M concave quadratic constraints are satisfied.By employing a special randomized rounding procedure,we show that the ratio between the norm of the optimal solution of this model and its SDP relaxation is upper bounded by 54πM2 in the real case and by 24√Mπin the complex case.The second class of problems finds a series of minimum norm vectors subject to a set of quadratic constraints and cardinality constraints with both binary and continuous variables.We show that in this case the approximation ratio is also bounded and independent of problem dimension for both the real and the complex cases.
