The Optimal Value Function in Quasidifferentiable Programming
Bernd Luderer · 1992
This paper is dedicated to the study of the nonconvex nonsmooth mathematical programming problem $$(P)\left\{ \begin{gathered} f(x,y) \to \inf ; \hfill \\ g(x,y) \leqslant 0, \hfill \\ \end{gathered} \right.$$ where the functions f:Rn × Rm → R1 and g:Rn × Rm → Rp are assumed to be continuous and quasidifferentiable in the sense of Dem’ya-nov and Rubinov [1] In our approach, the rectors x and y are regarded as two groups of variables, and we decompose problem (P) by fixing the vector y. In doing so, at the lower level we get smaller and possibly easier subproblems to be solved. Let S(y) and P(y) be the sets of feasible and optimal, resp., solutions in the lower-level problem for fixed y. At the upper level we obtain the implicitly defined optimal value function $$p(y) = \mathop {\inf }\limits_x \{ f(x,y)\left| {g(x,y) \leqslant 0} \right.\} $$ (p(y)=+ ∞ if ∄ x: g(x, y) ≤ 0). Now one has to solve the problem $$p(y) \to \mathop {\inf }\limits_x $$ .