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.

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 |