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However, even when this is not the case, non-degeneracy conditions can be applied a posteriori to determine whether the results of the method can be trusted. Otherwise, the method continues without any changes. Expanded stopping criterion Step 2. For instance, when using the nondegeneracy condition 2. Chapter 3 Scenario Analysis We analyze seven different scenarios that cover the possibilities for any run of a method within the framework.
Stationarity and Convergence in Reduce-or-Retreat Minimization
We have already analyzed the convergence of the iterates xk in the reduceor-retreat framework, and we have investigated how an approach to stationarity is connected to the convergence of the target-gaps. When a reduce-or-retreat method terminates, we completely understand what eventually happens to the trial-sizes and the iterate-sets; and our interest now is to develop our understanding of these elements when the method does not terminate.
In order to present the most complete analysis, we consider the reduce-or-retreat framework augmented by the expanded stopping criterion with degeneracy check , which we give here for convenience. The other six other possible scenarios for a run of a method can be seen by following the branches to the six terminal nodes on the tree pictured in Fig.
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For simplicity, we will refer to each of these six scenarios by their terminal node numbers. The scenario analysis of a method then refers to the analysis of its convergence properties in a each of the seven scenarios. This kind of scenario analysis allows us to provide customized results that may not be revealed by the usual analysis that lumps all of the scenarios together.
Our approach thus expands the utility of the convergence analysis because different scenarios may occur for different runs of the same method in practice. Scenario analysis also encourages us to look for conditions guaranteeing a certain scenario for a reduce-or-retreat method since then we can conclude the customized convergence properties for that scenario.
For example, [5, Lemma 3. Theorem 3. The sequence of trial-sizes converges to zero in any of the following cases: Case i: Scenario I. Case ii: Scenario II. Case iv: Scenario IV, and reduction eventually does not increase the trial-size. Case v: Scenario V, and degeneracy-resets eventually do not increase the trialsize.
Case vii: Scenario VII, and together reductions and degeneracy-resets eventually do not increase the trial-size. Case i: The trial-sizes are eventually all zero. Proceeding in this manner successively through all m component n-vectors xki gives the result. We will show below that the lower-collapse of the iterate-sets can be characterized by lower-diminishing iterate-diameters.
We use the term iterate-radius because radk is the radius of the smallest closed ball centered at the iterate xk and containing the entire iterate-set. The next proposition establishes some convergence relationships between iteratesets, iterate-diameters, and iterate-radii. If we also know that the iterates are bounded or convergent, most of the implications in Proposition 15 become equivalences.
The rightward implications in all cases follow immediately from Proposition 15, so we only need to show that the final property implies the first in each case. See Proposition 1 for relatively minor assumptions that ensure bounded iterates.
Proposition 16 does not include an analogue to the pair of implications ii from Proposition 15, because those implications cannot necessarily be reversed, even under the stronger additional assumption of convergent iterates. Notice that by the definition of the maximum value f maxk , the iterate-set X k is always contained in the level set lev f maxk f. Every iteration inducing a reduction in a reduce-or-retreat method is f -stable. However, iterations inducing retreat are not necessarily f -stable without additional conditions being met.
The following lemma gives one pair of such conditions that apply to the Nelder—Mead method.
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Lemma 5. The containment 3.
Lemma 6. The formula in Eq. The result follows from Proposition 14 after we show that the iterate-sets X k are eventually contained in B. The desired containment 3. The condition 3. The assumption b of bounded level sets is a very natural one in the present context of minimizing f. Rockafellar and Wets [7, Theorem 1. We now summarize the results of this section to deduce when each of the scenarios associated with a reduce-or-retreat method is certain to generate lowerconvergent iterate-sets.
Theorem 4. Case iii: This follows from Proposition 17 since every reduction iteration is f stable. Case v: This follows from Proposition 17 since every reduction iteration is f -stable. The generalized line-search method is covered by option c in Case ii , the generalized trust-region method is covered by options b and c, the general pattern-search method is covered by option c, and Nelder—Mead is covered by option a. References 1. Audet, C. SIAM J. Ciarlet, P.
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Rockafellar, R. Springer, Berlin 8. Torczon, V. Levy Coupling, Stationarity, and Regeneration. Read more. Convergence in shareholder law. Convergence in Shareholder Law.
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