Duan Li's Research Interest
My primary research interest have been the theoretical and practical aspects of optimization and control.
My current research is focussed on
Nonlinear Integer Programming
Nonconvex Optimization and Global Optimization
Dynamic Portfolio Selection with a Mean-Variance Formulation
Nonlinear Integer Programming
The research goal is to establish convergent duality theory and to develop efficient solution algorithms for large-scale nonlinear integer programming problems. The fundamental target underlying our theoretical development is to eliminate duality gap in the classical Lagrangian dual formulation. We have developed nonlinear Lagrangian theory that has yielded several new dual formulations with asymptotic zero duality gap. The key concept is the construction of a nonlinear support for a nonconvex piecewise-constant perturbation function. Our numerical implementation of a duality-gap reduction process relies on some novel cutting procedures. Performing objective-level cut, objective contour cut or domain cut reshapes the perturbation function, thus exposing eventually an optimal solution to the convex hull of a revised perturbation function and guaranteeing a zero duality gap for a convergent Lagrangian method. Applications include nonlinear knapsack problems, constrained redundancy optimization in reliability networks, and optimal control problems with integer constraints.
Nonconvex Optimization and Global Optimization
The research goal is to develop equivalent transformations for generating a saddle point for nonconvex optimization problems. A saddle point condition is a sufficient condition for optimality. A saddle point can be generated in an equivalent representation space for nonconvex optimization problems that do not have a saddle point in their original settings. Certain equivalent transformations may convexify the perturbation function and a zero duality gap can be thus achieved. This investigation would lead to some efficient dual search algorithms that ensure the global optimality for a class of nonconvex optimization problems.
The research goal is to develop sufficient conditions to identify hidden convex minimization problems. A nonconvex minimization problem is called a hidden convex minimization problem if there exists an equivalent transformation such that the transformed minimization problem is convex. Sufficient conditions that are independent of transformations can be derived for identifying such class of seemingly nonconvex minimization problems that are equivalent to convex minimization problems. A global optimality can be thus achieved for this class of hidden convex optimization problems by using local search methods.
Dynamic Portfolio Selection with a Mean-Variance Formulation
The research goal is to seek optimal investment strategies for dynamic portfolio selection problems with a mean-variance formulation. To seek an optimal dynamic portfolio policy within a mean-variance framework implies to achieve a dual balance between the expected return and the risk and between the short term and long term benefits. Variance minimization is a notorious problem in stochastic control due to its associated property of nonseparability. Separation schemes can be developed to overcome this difficulty of nonseparability. Further research efforts are needed to improve the portfolio selection models and to derive full feedback optimal investment policies.
Except for a few ideal situations, an optimal control usually pursues two often conflicting objectives: To drive the system toward a desired state, and to perform active learning to reduce the systems uncertainty. The dual roles of an optimal control, optimization and estimation, in general situations, cannot be separated. This coupling between optimization and estimation makes an analytical form of optimal control, in most situations, unattainable. The research goal is to develop some embedding schemes in order to achieve optimal control laws with an active learning property for certain classes of dual control problems.
