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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

MSC:

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

[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
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