Exact Penalties for Local Minima
Szymon Dolecki, Stefan Rolewicz · SIAM Journal on Control and Optimization · 1979
We provide a sufficient condition for the exact equivalence of constrained minimization problems and the minimization of associated generalized Lagrangians with respect to a perturbing class $\Phi _1 $. Exact equivalence amounts to equality of the sets of local solutions restricted to some region. The sufficient condition is expressed in terms of certain semicontinuity properties of objective functions and constraint multifunctions; for Banach spaces it becomes local controllability. The requirement is made more specific for mathematical programming. In this context we discuss properties of inner derivatives and approximations of multifunctions and we present a considerable extension of the Lusternik theorem.