Then the coupling cycle joint method and a heuristic algorithm for the growth of P-cycles and network protection resource deployment based on this method are proposed.

By using it,we give a necessary and sufficient condition of which a cyCle is a self-dual cycle,and we also present a structure theorem of self-dual cycled.

Lagrangian duality for vector optimization of set-valued maps;

Study on risk measures-duality method;

Optimality and duality for minimax fractional problems;

Fuzzy subgradient algorithm for solving Lagrangian relaxation dual problem;

A dual characterization for Benson proper efficiency;

The generalization of a dual J.M.Child inequality;