An Exact Potential Method for Constrained Maxima

The main result of the paper consists of the theorem that under certain, natural assumptions the local conditional maximum x₀ of the function f on the set \[ A = \{ {x ∉ R^n |φ _i (x) 0,ψ _j (x) = 0,i = 1, ⋯ ,k,j = 1, ⋯ ,l} \} \] is identical with the unconditional maximum of the potential function \[ p(x,μ ) = μ f(x) + ∑ i = 1^k {{neg}(φ _i (x))} - ∑ j = 1^l {| {ψ (x)} |} , x ∈ R^n , μ 0, \] for μ sufficiently small. There is also provided a draft of a modified gradient procedure for maximizing the potential p(x,μ ) since it is generally nonsmooth even for differentiable f, φ ᵢ and ψ ⱼ.