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Zbl 0633.49010
Lemaire, B.
Coupling optimization methods and variational convergence. (English)
[A] Trends in mathematical optimization, 4th French-German Conf., Irsee/FRG 1986, ISNM 84, 163-179 (1988).

[For the entire collection see Zbl 0626.00020.] \par Minimization problems of the form $$ (1)\quad \min \{J(u)+\phi(u): u\in X\} $$ are considered, where X is a Hilbert space, $J: X\to {\bbfR}$ 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: $\phi\sb n\to\sp{M}\phi$ if and only if (i) For all $u\in X$ and for all $(u\sb n)\sb{{\bbfN}}$ such that $u\sb n\to u$ weakly in x there holds $\phi(u)\le \liminf \phi\sb n(u\sb n)$, and (ii) For all $u\in X$ there exists $(u\sb n)\sb{{\bbfN}}$ such that $u\sb n\to u$ strongly in X with $\phi(u)\ge \limsup \phi\sb n(u\sb n).$ \par In the last section of the paper, some applications are given, to penalty methods in convex programming.
[G.Buttazzo]

MSC 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence
49M30 Methods of successive approximation, not based on necessary cond.
90C55 Methods of successive quadratic programming type
90C52 Methods of reduced gradient type
90C25 Convex programming

Keywords: Minimization problems; monotone inclusion; Moscow convergence; penalty methods; convex programming

Citations: Zbl 0626.00020

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