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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 {\it 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 {\it 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 in calculus of variations 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. 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