On the convexification procedure for nonconvex and nonsmooth infinite dimensional optimization problems. (English) Zbl 0755.49013

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 \(f\) to ensue by means of a strongly convex function \(g\), so that \(f+g\) 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.
Reviewer: H.Dietrich


49N15 Duality theory (optimization)
90C48 Programming in abstract spaces
49M05 Numerical methods based on necessary conditions
90C26 Nonconvex programming, global optimization
Full Text: DOI


[1] Bertsekas, D. P., Convexification procedures and decomposition methods for nonconvex optimization problems, J. Optim. Theory Appl., 29, 169-196 (1979) · Zbl 0389.90080
[2] Gfrerer, H., Globally convergent decomposition methods for nonconvex optimization problems, Computing, 32, 199-227 (1984) · Zbl 0529.65036
[3] Spingarn, J. E., Submonotone mappings and the proximal point algorithm, Numer. Funct. Anal. Optim., 4, No. 2, 123-150 (1981-1982) · Zbl 0495.49025
[4] Toland, J. F., Duality in nonconvex optimization, J. Math. Anal. Appl., 56, 399-415 (1978) · Zbl 0403.90066
[5] Zălinescu, Z., On uniformly convex functions, J. Math. Anal. Appl., 95, 344-374 (1983) · Zbl 0519.49010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.