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### Vector Composition

**Composition of Functions:**

- \( g : \mathbb{R}^n \rightarrow \mathbb{R}^k \) and \( h : \mathbb{R}^k \rightarrow \mathbb{R} \)
- \( f(x) = h(g(x)) = h(g_1(x), g_2(x), \ldots, g_k(x)) \)

**Convexity Conditions:**

- \( f \) is convex if:
  - \( g_i \) are convex, \( h \) is convex, and \( \tilde{h} \) is nondecreasing in each argument
- \( f \) is concave if:
  - \( g_i \) are concave, \( h \) is convex, and \( \tilde{h} \) is nonincreasing in each argument

**Proof (for \( n = 1 \), differentiable \( g, h \)):**

- \( f''(x) = g'(x)^T \nabla^2 h(g(x)) g'(x) + \nabla h(g(x))^T g''(x) \)

**Examples:**

- \( \sum_{i=1}^m \log g_i(x) \) is concave if \( g_i \) are concave and positive
- \( \log \sum_{i=1}^m \exp g_i(x) \) is convex if \( g_i \) are convex

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*A. d'Aspremont. Convex Optimization M2.*  
Page 20/67
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