Bifurcation and Persistance of Minima in Nonlinear Parametric Programming

Abstract

In this paper, the structure of solutions to the nonlinear parametric programming problem

$$\min \{ {\text{f(x,}}\alpha {\text{):h(x,}}\alpha {\text{) = 0,g(x,}}\alpha {\text{)}} \geqslant {\text{0\} }}$$

((1.1))

is studied as the parameter α ∈ Rvaries. Here f: R ⁿ⁺¹→ R, g: R ⁿ⁺¹← R ^p, h: R ⁿ⁺¹← R ^qwill be assumed at least twice continuously differentiable. Both the bifurcation of curves of critical points and the persistance of minima along these curves will be examined. There has been considerable recent interest [1,4,5,8,10–13,16,17,19–25] in this problem, stemming from the solution of parameter-free optimization problems by homotopy methods, from multi-objective optimization, and from applications requiring a parametric analysis of sensitivity and stability.