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I am studying a theorem on the characterization of solutions in nondifferentiable convex problems.

Say that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and $f: \mathbb{R}^n \to \mathbb{R}_{\infty}$ is convex ($\mathbb{R}_\infty = \mathbb{R} \cup \{\infty\})$. A point $\bar x \in (\text dom f)^o$ is a solution of the minimization problem:

$$\text{minimise} \space \space f(x), \space \space \text{subject to} \space \space \delta_c(x) \leq 0$$

if and only if $\bar x \in C$ and $\exists w \in \partial f(\bar x)$ such that $\langle w,y-\bar x\rangle \geq 0, \space \space\forall y \in C$

If you need clarification on any notation please let me know, but briefly $\partial f(\bullet)$ means the boundary of... and $(\bullet)^o$ means interior of.

I want to know the interpretation of this line: $\langle w,y-\bar x\rangle \geq 0, \space \space\forall y \in C$

Specifcally the meaning of the dot product. I have a feeling it related to hyperplanes but am not sure of the significance. Geometrically, what does it convey?

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  • $\begingroup$ $\partial f$ certainly does not refer to the "boundary", it refers to the subgradient. $\endgroup$ Feb 2, 2015 at 14:53
  • $\begingroup$ oops, thanks for the correction $\endgroup$
    – yaweh
    Feb 3, 2015 at 3:12

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First if $\partial f(\bar{x})=0$ you have a necessary and sufficient condition for $\bar{x}$ to be a minimum of f and you satisfy your relation.

The line $⟨w,y−x¯⟩≥0, ∀y∈C$ means that in C there is no descent direction for the function f. If this relation is not respected then it means there exists an $x \in C$ such that f decrease strictly which contradict the optimality of $\bar{x}$.

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