A strongly convergent norm-relaxed method of strongly sub-feasible direction for optimization with nonlinear equality and inequality constraints

In this paper, a class of optimization problems with nonlinear equality and inequality constraints is discussed. Firstly, the original problem is transformed to an associated simpler auxiliary optimization problem with only inequality constraints and a penalty parameter, and the later problem is showed to be equivalent to the original problem if the parameter is large enough (but finite). Then, combining the norm-relaxed Method of Feasible Direction (MFD) with the idea of Method of Strongly Sub-Feasible Direction (MSSFD), we present an algorithm with arbitrary initial point for the original problem. At each iteration of the auxiliary problem, an improved search direction is obtained by solving one Direction Finding Subproblem (DFS), i.e., a quadratic program, which always possesses a solution. In the process of iteration, the feasibility of the iteration points is monotone increasing. Furthermore, whenever an iteration point enters the feasible set, the proposed algorithm reduces to a feasible and decent method for the auxiliary problem. Under some suitable assumptions, the global and strong convergence of the proposed algorithm can be obtained. Finally, some elementary numerical experiments are reported.

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