Duality in Nonconvex Optimization

1.2 Preliminaries

• $\bar{\mathbb{R}}$,
  • extended real number
• $V, V^{*}$,
  • real vector space
• $\langle , \rangle: V \times V^{*} \rightarrow \mathbb{R}$,
  • bilinear form

Definition Separating

$V$ and $V^{*}$ is said to be vector spaces put in duality if there exists bilinear form $\langle , \rangle$.

Duality $\langle , \rangle$ is said to be separating if

\[u \neq 0 \in V, \ \exists u^{*} \in V^{*}, \text{ s.t. } \langle u, u^{*} \rangle \neq 0,\]

and

\[u^{*} \neq 0 \in V^{*}, \ \exists u \in V, \text{ s.t. } \langle u, u^{*} \rangle \neq 0.\]

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Definition Weak topology

• $(\mathbb{R}, \mathcal{O})$,
  • standard topogical sp.
• $f_{u^{*}}: V \rightarrow \mathbb{R}$,
  • \(f_{u^{*}}(u) := \langle u, u^{*} \rangle\),

We denote the weak topology on $V$ by $\sigma(V, V^{*})$.

\[\sigma(V, V^{*}) := \mathfrak{M} \{ f_{u^{*}}^{-1}(O) \mid u^{*} \in V^{*}, \ O \in \mathcal{O} \} .\]

Analogousy we define

\[\sigma(V, V^{*}) := \mathfrak{M} \{ f_{u}^{-1}(O) \mid u \in V, \ O \in \mathcal{O} \},\]

where \(f_{u}(u^{*}) := \langle u, u^{*} \rangle\) and $\mathfrak{M}(\mathcal{A})$ is the smallest topology containing $\mathcal{A} \subseteq 2^{\mathbb{R}}$.

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Proposition Equivalent condition for Hausdorff

The following statements are equivalent:

• (a) $\sigma(V, V^{*})$ is Hausdorff,
• (b) $\sigma(V^{*}, V)$ is Hausdorff,
• (c) the duality between $V$ and $V^{*}$ is separating.

proof

(a) $\Rightarrow$ (b)

(b) $\Rightarrow$ (c)

$\Box$

We shall consider the spaces $V$ and $V^{*}$ in separating duality endowed with the topologies \(\sigma(V, V^{*})\) and \(\sigma(V^{*}, V)\). All statements concerning continuity, lower semicontinuity, convergence, closure, etc. will refer to continuity, lower semicontinuity, convergence, closure etc. in these topology.

Definition lower semicontinuity

• $F: V \rightarrow \bar{\mathbb{R}}$,
  • functional

$F$ is said to be lower semicontinuity (l.s.c.) if

\[\forall u \in V, \ u_{n} \rightarrow u, \ F(u) \le \lim \inf F(u_{n}) .\]

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Definition convex

• $F: V \rightarrow \bar{\mathbb{R}}$,
  • functional

$F$ is said to be convex if

\[\forall w, u \in V, \ F(\lambda u + (1 - \lambda) w) \le \lambda F(u) + (1 - \lambda)) F(w) \quad \lambda \in (0, 1) .\]

$F$ is said to be strictly convex if the above inequality is strict inequality for all $u \neq w$.

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Proposition

• $F:V \rightarrow \bar{\mathbb{R}}$,
  • convex, l.s.c.

\[\exists u^{\prime} \in V, \ F(u^{\prime}) = - \infty .\]

Then

\[\forall u \in V, \ F(u) = -\infty .\]

proof

$\Box$

Definition affine continuous

• $f: V \rightarrow \bar{\mathbb{R}}$,
  • function

$f$ is said to be affine continuous on $V$ if there exists continuous linear functional $l:V \rightarrow \mathbb{R}$ and $\alpha \in \mathbb{R}$ such that

\[f(u) = l(u) + \alpha .\]

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Proposition

• $V$, $V^{*}$
  • in separating duality

If $f$ is affine continuous functions on $V$, then

\[\exists u^{*} \in V^{*}, \text{ s.t. } f(u) = \langle u, u^{*} \rangle + \alpha .\]

proof

$\Box$

We shall denote by $\Gamma(V)$

\[\Gamma(V) := \left\{ f(u) := \sup_{ i \in \Lambda } f_{i}(u) \mid \{ f_{i} \}_{i \in \Lambda} \text{ a set of affine continuous funcitons on } V \right\}\]

Reference

• Toland, J. F. (1978). Duality in nonconvex optimization. Journal of Mathematical Analysis and Applications. https://doi.org/10.1016/0022-247X(78)90243-3