Convex Optimization · Problem 2.33

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · monotone-nonnegative-cone

Problem

The monotone nonnegative cone. We define the monotone nonnegative cone as Km+={x∈Rn∣x1≥x2≥⋯≥xn≥0}.K_{\mathrm{m+}} = \{x \in \mathbf{R}^n \mid x_1 \geq x_2 \geq \cdots \geq x_n \geq 0\}. i.e., all nonnegative vectors with components sorted in nonincreasing order. ((a)) Show that Km+K_{\mathrm{m+}} is a proper cone. ((b)) Find the dual cone Km+∗K^*_{\mathrm{m+}}. Hint. Use the identity ∑i=1nxiyi=(x1−x2)y1+(x2−x3)(y1+y2)+(x3−x4)(y1+y2+y3)+⋯+(xn−1−xn)(y1+⋯+yn−1)+xn(y1+⋯+yn).\sum_{i=1}^n x_i y_i = (x_1 - x_2)y_1 + (x_2 - x_3)(y_1 + y_2) + (x_3 - x_4)(y_1 + y_2 + y_3) + \cdots + (x_{n-1} - x_n)(y_1 + \cdots + y_{n-1}) + x_n(y_1 + \cdots + y_n).

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