Analysing And Solving Distributed Optimization Problem With Typical Nonlinear Cost Functions

Distributed optimization has attracted the attention of scholars from various fields including mathematics,computer science,and automation.The limited task allocation problem is an important problem in distributed optimization,and cost function is the core index of the limited task allocation problem.It is worth mentioning that there is an optimal solution when the cost function is strictly convex.However,until now few works have studied the optimal solution to limited task allocation problem with typical nonlinear cost functions.Therefore,we study the limited task allocation problem with two typical nonlinear cost functions in this thesis.The main work and results are as follows:First,a mathematical model of limited task allocation is given in this thesis,and the optimal solution of limited task allocation problem is studied by potential game.We prove that when the Nash equilibrium point in the potential game is in the feasible solutions for the limited task allocation problem,the Nash equilibrium is the unique globally optimal solution,and then prove that the Nash equilibrium point is globally asymptotically stable.Otherwise,we also derive analytically the unique globally optimal solution.Furthermore,we study the limited task allocation problem with exponential cost function.We demonstrate the convexity of the exponential cost function.In addition,distributed replication dynamics equation,Lagrange interpolation,and consensus algorithms are proposed to seek the optimal solution to the limited task problem with exponential cost function.Meanwhile,we provide some examples and verify the validity of the algorithms by Monte Carlo simulations.We also study the limited task allocation problem with quadratic cost function.We demonstrate the convexity of the quadratic cost function.Meanwhile,the above algorithms are used to seek the optimal solution to the limited task problem with quadratic cost function.In addition,we provide some examples and check the validity of the algorithms by Monte Carlo simulations.