A method of multipliers for mathematical programming problems with equality and inequality constraints

Abstract

This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0.

The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ^Tφ(x)+kφ^T(x)φ(x) −μ^T $\tilde \psi$(x)+k $\tilde \psi$ ^T(x)$\tilde \psi$(x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and$\tilde \psi$(x), and the elements of$\tilde \psi$(x) are defined by$\tilde \psi$ _(j)(x)=min[ψ_(j)(x), (1/2k) μ_(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm.

Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.