Quantitative stability of variational systems. II. A framework for nonlinear conditioning.

*(English)*Zbl 0793.49005[For part I see the authors, Trans. Am. Math. Soc. 328, No. 2, 695-729 (1991; Zbl 0753.49007).]

Some quantitative results are proved concerning the continuity properties of the optimal value and of the set of the optimal solutions of optimization problems with respect to perturbations of global character. One of the tools used to achieve such results is epi-distance, a concept introduced by the authors in a previous paper and that, in some sense, allows to measure the distance between optimization problems. First of all the notion of epi-distance is recalled and its main properties are proved. Then the basic stability results are proved, Lipschitz continuity of optimal values with respect to epi-distance is established, moreover, after having introduced conditioning functions, the continuity in the norm topology of optimal solutions with respect to epi-distance is obtained.

Some applications of the above results are made to get Hölder estimates of the distance between the projections of a fixed point in a Hilbert space on two convex sets in terms of the epi-distance of the sets. Another application concerns the slow convergence rate of penalization methods in nonlinear programming: given a particular function \(f\) to be minimized and a certain family \(\{f_ \theta\}_{\theta>0}\) of penalizations of \(f\), estimates of the epi-distance between \(f\) and \(f_ \theta\) and between the optimal solutions of \(f\) and of \(f_ \theta\) are established. The estimates on the sets of the optimal solutions rely on the existence and on the properties of conditioning functions. Finally some examples show that the estimates obtained are sharp. Some applications of \(L^ p\) spaces are also given.

Some quantitative results are proved concerning the continuity properties of the optimal value and of the set of the optimal solutions of optimization problems with respect to perturbations of global character. One of the tools used to achieve such results is epi-distance, a concept introduced by the authors in a previous paper and that, in some sense, allows to measure the distance between optimization problems. First of all the notion of epi-distance is recalled and its main properties are proved. Then the basic stability results are proved, Lipschitz continuity of optimal values with respect to epi-distance is established, moreover, after having introduced conditioning functions, the continuity in the norm topology of optimal solutions with respect to epi-distance is obtained.

Some applications of the above results are made to get Hölder estimates of the distance between the projections of a fixed point in a Hilbert space on two convex sets in terms of the epi-distance of the sets. Another application concerns the slow convergence rate of penalization methods in nonlinear programming: given a particular function \(f\) to be minimized and a certain family \(\{f_ \theta\}_{\theta>0}\) of penalizations of \(f\), estimates of the epi-distance between \(f\) and \(f_ \theta\) and between the optimal solutions of \(f\) and of \(f_ \theta\) are established. The estimates on the sets of the optimal solutions rely on the existence and on the properties of conditioning functions. Finally some examples show that the estimates obtained are sharp. Some applications of \(L^ p\) spaces are also given.

Reviewer: R.De Arcangelis (Napoli)

##### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

65K10 | Numerical optimization and variational techniques |

90C30 | Nonlinear programming |

49J40 | Variational inequalities |

58E35 | Variational inequalities (global problems) in infinite-dimensional spaces |

65K05 | Numerical mathematical programming methods |