JIMO
Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved
Yang Li Yonghong Ren Yun Wang Jian Gu
Journal of Industrial & Management Optimization 2015, 11(1): 65-81 doi: 10.3934/jimo.2015.11.65
In this paper, we analyze the convergence properties of a nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming (NCSDP) problems. It is different from other convergence analysis, because the subproblem in our algorithm is inexactly solved. Under the constraint nondegeneracy condition, the strict complementarity condition and the second order sufficient conditions, it is obtained that the nonlinear Lagrangian algorithm proposed is locally convergent by choosing a proper stopping criterion and the error bound of solution is proportional to the penalty parameter when the penalty parameter is less than a threshold.
keywords: nonconvex semidefinite programming Löwner operator nonlinear Lagrangian method Log-Sigmoid function. Inexact solution
JIMO
A nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming
Yang Li Liwei Zhang
Journal of Industrial & Management Optimization 2009, 5(3): 651-669 doi: 10.3934/jimo.2009.5.651
We present a nonlinear Lagrangian method for nonconvex semidefinite programming. This nonlinear Lagrangian is generated by a Löwner operator associated with Log-Sigmoid function. Under a set of assumptions, we prove a convergence theorem, which shows that the nonlinear Lagrangian algorithm is locally convergent when the penalty parameter is less than a threshold and the error bound of the solution is proportional to the penalty parameter.
keywords: Löwner operator. nonlinear Lagrangian nonconvex semidefinite programming
JIMO
A differential equation method for solving box constrained variational inequality problems
Li Wang Yang Li Liwei Zhang
Journal of Industrial & Management Optimization 2011, 7(1): 183-198 doi: 10.3934/jimo.2011.7.183
In this paper, we discuss a system of differential equations based on the projection operator for solving the box constrained variational inequality problems. The equilibrium solutions to the differential equation system are proved to be the solutions of the box constrained variational inequality problems. Two differential inclusion problems associated with the system of differential equations are introduced. It is proved that the equilibrium solution to the differential equation system is locally asymptotically stable by verifying the locally asymptotical stability of the equilibrium positions of the differential inclusion problems. An Euler discrete scheme with Armijo line search rule is introduced and its global convergence is demonstrated. The numerical experiments are reported to show that the Euler method is effective.
keywords: global convergence. differential inclusion asymptotical stability differential equations Box constrained variational inequality problem
DCDS-B
A generalized $\theta$-scheme for solving backward stochastic differential equations
Weidong Zhao Yang Li Guannan Zhang
Discrete & Continuous Dynamical Systems - B 2012, 17(5): 1585-1603 doi: 10.3934/dcdsb.2012.17.1585
In this paper we propose a new type of $\theta$-scheme with four parameters ($\{\theta_i\}_{i=1}^4$) for solving the backward stochastic differential equation $-dy_t=f(t,y_t,z_t) dt - z_t dW_t$. We rigorously prove some error estimates for the proposed scheme, and in particular, we show that accuracy of the scheme can be high by choosing proper parameters. Various numerical examples are also presented to verify the theoretical results.
keywords: error estimate second order numerical tests. Backward stochastic differential equations $\theta$-scheme

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