圓錐上的軌跡函數的凸性
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2015
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Abstract
在Using Schur Complement Theorem to prove convexity of some SOC-functions這篇論文中，證明了在二階錐(Second order cone)上由凸函數(Convex function)所生成的軌跡函數(Trace function)亦為凸函數，我們試著將這個結果擴展到所有圓錐(Circular cone)上。但是，根據圓錐角度的不同，一個凸函數所生成的軌跡函數不一定會是凸函數。在本篇論文中找出一組充份條件，使凸函數所生成的軌跡函數會是凸函數。同時也給出幾個不滿足條件下軌跡函數不是凸函數的情況。這組條件是：圓錐角度大於45○且函數同時是遞增函數，或圓錐角度小於45○且函數同時是遞減函數。
Uusing the Schur complement theorem[1][2] to nd the sucient conditions which make the Hessian of circular cone trace function positive semi-denite or positive denite, and proved that these monotone conditions can make the trace function convex or strictly convex. After that, some examples are given to explain why we have to satisfy these monotone conditions.
Uusing the Schur complement theorem[1][2] to nd the sucient conditions which make the Hessian of circular cone trace function positive semi-denite or positive denite, and proved that these monotone conditions can make the trace function convex or strictly convex. After that, some examples are given to explain why we have to satisfy these monotone conditions.
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Circular cone, Trace, Convexity, Monotone, Schur