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Coupling optimization methods and variational convergence. (English) Zbl 0633.49010
Trends in mathematical optimization, 4th French-German Conf., Irsee/FRG 1986, ISNM 84, 163-179 (1988).
[For the entire collection see Zbl 0626.00020.]
Minimization problems of the form $(1)\quad \min \{J(u)+\phi(u): u\in X\}$ are considered, where X is a Hilbert space, $$J: X\to {\mathbb{R}}$$ is a Gâteaux differentiable convex function, and $$\phi: X\to ]- \infty,+\infty]$$ is a proper l.s.c. convex function. Since J and $$\phi$$ are supposed convex, problem (1) is equivalent to the monotone inclusion $(2)\quad 0\in J'(u)+\partial \phi(u)$ where J’ is the Gâteaux derivative of J and $$\partial \phi$$ is the subdifferential of $$\phi$$. To solve problems (1) or (2) the author uses an approximation method, obtained by coupling a standard iterative method with approximation by the variational Moscow convergence which is defined as follows: $$\phi_ n\to^{M}\phi$$ if and only if (i) For all $$u\in X$$ and for all $$(u_ n)_{{\mathbb{N}}}$$ such that $$u_ n\to u$$ weakly in x there holds $$\phi(u)\leq \liminf \phi_ n(u_ n)$$, and (ii) For all $$u\in X$$ there exists $$(u_ n)_{{\mathbb{N}}}$$ such that $$u_ n\to u$$ strongly in X with $$\phi(u)\geq \limsup \phi_ n(u_ n).$$
In the last section of the paper, some applications are given, to penalty methods in convex programming.
Reviewer: G.Buttazzo

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49M30 Other numerical methods in calculus of variations (MSC2010) 90C55 Methods of successive quadratic programming type 90C52 Methods of reduced gradient type 90C25 Convex programming