Extremum Problems with Inequalities as Subsidiary Conditions

John, Fritz

doi:10.1007/978-1-4612-5412-6_25

Cited by 326 publications

(455 citation statements)

References 7 publications

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“…This contradicts Lemma 2 on the uniqueness of the Wu metric, which is in this context the uniqueness of the best fitting ellipsoid. (See also [10].) Therefore, combining this with the basic results on the Wu metric of the preceding section, we arrive at At this point, it is convenient to introduce the following terminology and notation for later purposes.…”

Section: Preliminaries For the Kobayashi-royden Metricmentioning

confidence: 95%

“…In this case, it is also possible to obtain an explicit expression for the Wu metric, which we give at the end of this section. However, we will take a more general approach based upon the study of extremal problems by Fritz John in [10]. We take such an approach here because this method can also be applied to the thin sets M 0 and Z.…”

Section: Wu Metric Atmentioning

confidence: 99%

“…In the following, we will follow Fritz John's method ( [10]), with a subtle difference that our choice for S b will also vary with b. This becomes important in the later analysis of the regularity of r(b).…”

Section: Wu Metric Atmentioning

confidence: 99%

“…While referring to [10] for a precise proof of this lemma, we give a rough but more intuitive recapitulation of the actual proof. In the statement of the lemma above, the vectors in the first set represent the directions along which the directional derivative is negative.…”

Section: Then the Intersection Of The Two Setsmentioning

confidence: 99%

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Analysis of the Wu metric. I: The case of convex Thullen domains

Cheung

Kim

1996

Trans. Amer. Math. Soc.

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Abstract. We present an explicit description of the Wu metric on the convex Thullen domains which turns out to be the first natural example of a purely Hermitian, non-Kählerian invariant metric. Also, we show that the Wu metric on these Thullen domains is in fact real analytic everywhere except along a lower dimensional subvariety, and is C 1 smooth overall. Finally, we show that the holomorphic curvature of the Wu metric on these Thullen domains is strictly negative where the Wu metric is real analytic, and is strictly negative everywhere in the sense of current.

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Section: Preliminaries For the Kobayashi-royden Metricmentioning

confidence: 95%

Section: Wu Metric Atmentioning

confidence: 99%

Section: Wu Metric Atmentioning

confidence: 99%

Section: Then the Intersection Of The Two Setsmentioning

confidence: 99%

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Analysis of the Wu metric. I: The case of convex Thullen domains

Cheung

Kim

1996

Trans. Amer. Math. Soc.

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“…[32], see also [26, Definition-Theorem 2.4]) associated to the norm · σ,sup . Recall that, for any s ∈ E D,σ , the following inequalities hold…”

Section: 2mentioning

confidence: 99%

Explicit uniform estimation of rational points I. Estimation of heights

Chen

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. -By using the slope method, we obtain an explicit uniform estimation for the density of rational points in an arithmetic projective variety with given degree and dimension, embedded in a given arithmetic projective space.Résumé. -On obtient, en utilisant la méthode de pentes, une estimation uniforme et explicite de la densité des points rationnels dans une variété arithmétique projective de degré et dimension fixés, plongée dans un espace projectif arithmétique donné.

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An optimization under uncertainty in system equations and applications to robust AC optimal power flow

Suzuki

Yasuda

Aiyoshi

2022

Elect Comm in Japan

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This paper presents a formulation of an optimization of uncertain systems and the solution technique. The optimization under uncertainties is formulated as an optimization problem which includes unknown variables in equality constraints called system equations as well as in an objective function and inequality constraints. As countermeasures against the uncertainties, a min-max criterion is applied to the objective function and robustness criteria are introduced to the inequality constraints. By differentiating state variables satisfying the equality constraints from the decision variables, we reformulate the optimization problem based on the worst-case scenario of the state variables corresponding to the uncertain variables and we propose a "constraints-relaxation procedure" based method considering the equality constraints to solve the reformulated problem. In this method, a constraints-relaxed problem is solved corresponding to a finite number of samples of uncertain variables. Furthermore, a new sample of uncertain variables is sequentially generated corresponding to the inequality constraint which the solution violates most, together with state variables satisfying the system equations at the new worst-case scenario. Finally, the effectiveness of the proposed method is illustrated by numerical examples including robust AC optimal power flow considering distributed energy resources.

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Extremum Problems with Inequalities as Subsidiary Conditions

Cited by 326 publications

References 7 publications

Analysis of the Wu metric. I: The case of convex Thullen domains

Analysis of the Wu metric. I: The case of convex Thullen domains

Explicit uniform estimation of rational points I. Estimation of heights

An optimization under uncertainty in system equations and applications to robust AC optimal power flow

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