The paper gives a generalization and modification of a convexification procedure of D. P. Bertsekas
[J. Optimization Theory Appl. 29, 169- 197 (1979; Zbl 0389.90080
)] for infinite dimensional, nonsmooth and nonconvex optimization problems. A nonconvex minimization problem is being transformed into a convex parametrical minimization problem and an additional minimization of an everywhere Fréchet differentiable nonconvex function. The convexification of the nonconvex cost function
to ensue by means of a strongly convex function
, so that
is also a strongly convex function. One can prove a connection with the duality theory of J. F. Toland
[J. Math. Anal. Appl. 66, 399-415 (1978; Zbl 0403.90066
)] and give a descent method for the original problem.