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@@ -728,19 +728,19 @@ <h2>Remarks</h2>
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A key technique underlying the success of NEWUOA, BOBYQA, and LINCOA is the least Frobenius norm updating of quadratic models elaborated in [3] and [4].
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A key technique underlying the success of NEWUOA, BOBYQA, and LINCOA is the least Frobenius norm updating of quadratic models elaborated in [4] and [5].
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The idea comes from the <aclass="theme-link" href="https://www.jstor.org/stable/2030103?seq=1" target="_blank">least change update</a> for <aclass="theme-link" href="https://epubs.siam.org/doi/abs/10.1137/1019005" target="_blank">quasi-Newton methods</a>, a vast research area initiated by the <aclass="theme-link" href="https://academic.oup.com/comjnl/article/6/2/163/364776" target="_blank">DFP algorithm</a>, where P stands for Powell.
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The least Frobenius norm updating is a quadratic programming problem, whose constraints correspond to the interpolation conditions.
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At each iteration of Powell's algorithms, only one of the constraints is different from the previous iteration.
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To solve this problem efficiently and stably, Powell designed a procedure to update the inverse of its KKT matrix along the iterations.
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Such a procedure is detailed in [5], and it is indispensable for the remarkable numerical stability of NEWUOA, BOBYQA, and LINCOA.
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Such a procedure is detailed in [6], and it is indispensable for the remarkable numerical stability of NEWUOA, BOBYQA, and LINCOA.
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LINCOA seeks the least value of a nonlinear function subject to linear inequality constraints without using derivatives of the objective function.
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Professor Powell did not publish a paper to introduce the algorithm.
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The paper [10] discusses how LINCOA solves its trust-region subproblems.
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The paper [11] discusses how LINCOA solves its trust-region subproblems.
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Different from PDFO, which provides interfaces for Powell's code,
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