## Journals

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### Open Access Journals

JIMO

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.

JIMO

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.

JIMO

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.

DCDS-B

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.

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