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Conditional finite-mixture envelope for an additive ordered representation

Statement

Assume the strict-reference representation conclusion of the draft supplier Additive extensive representation of compound comparison for its nonempty state set $X$ and stated comparison hypotheses. Let $d \geq 1$ and $m \geq 1$ be integers, and take a finite list of distinct states $s_{1}, \dots, s_{m} \in X$ . Let $q_{1}, \dots, q_{m} \in R^{d}$ and set $a_{i} : = S (s_{i})$ for $1 \leq i \leq m$ . Let $e_{s}$ be the unit-state compound defined in Tagged scalable compounds; it is a mass-one point mass by Compound mass is well-defined and additive. For $θ \in Δ_{m}$ , put $M_{θ} = \sum_{i} θ_{i} e_{s_{i}}$ and set $V (θ) : = \sum_{i} θ_{i} S (s_{i})$ . Suppose a map $χ : {M_{θ} : θ \in Δ_{m}} \to R^{d}$ satisfies $χ (M_{θ}) = \sum_{i} θ_{i} q_{i}$ for every $θ \in Δ_{m}$ . For each $x \in R^{d}$ , define the value to be $+ \infty$ if no $θ \in Δ_{m}$ has coordinate $x$ . Then the minimum represented value among those finite mixtures, with that empty-feasible-set convention, is exactly the finite-mixture envelope $f_{m i x} (x)$ of Finite-mixture convex envelope and its supporting-face condition, with all its attainment, convexity, lower-semicontinuity, biconjugation, and supporting-face conclusions. This is the complete conditional transfer under the representation conclusion and affine coordinate assignment; the supplier definition and mass-one fact are provided locally.

Facts & Assumptions

Given: A finite family of states and coordinate data as in the Statement.

[L1]

Additive extensive representation of compound comparison: under its strict-reference hypotheses the normalized $S : X \to R$ assigns each compound $\sum_{i} a_{i} e_{x_{i}}$ the finite additive value $\sum_{i} a_{i} S (x_{i})$ and represents all equal-mass comparisons. Its normalized reference values are $0$ and $1$ . The supplier remains draft; this use records no independent proof acceptance.

[A1]

Conditional premise: assume the strict-reference representation conclusion [L1] for the chosen state set and its listed comparison hypotheses.

[A3]

The coordinate assignment $χ$ is affine on the finite mixtures of the fixed states, exactly as specified in the Statement.

[L2]

Tagged scalable compounds defines $e_{s} (x) = 1$ when $x = s$ and $0$ otherwise, nonnegative pointwise scaling with $0 e_{s} = 0$ , and finite pointwise composition. Compound mass is well-defined and additive proves $m (e_{s}) = 1$ , finite mass additivity and homogeneity for every nonnegative scalar, and $m (C) = 0$ exactly for the empty compound.

[L3]

Finite-mixture convex envelope and its supporting-face condition: for finite data $(q_{i}, a_{i})$ the minimized weighted value is the proper closed convex envelope with the stated dual and support properties.

Proof

technique · direct

1.1A1A3L1L2L3algebra

By [A1] and [L1], each $a_{i} = S (s_{i})$ is finite. For any $θ \in Δ_{m}$ , [L2] gives $m (M_{θ}) = \sum_{i} θ_{i} m (e_{s_{i}}) = \sum_{i} θ_{i} = 1$ . If $θ_{i} = 0$ , its term is the empty compound and contributes zero; equivalently omit such terms, leaving only positive coefficients and never applying a positive-scaling comparison axiom at zero. The representation assigns $M_{θ}$ the value $V (θ) = \sum_{i} θ_{i} a_{i}$ . By [A3], its coordinate is $\sum_{i} θ_{i} q_{i}$ . Since the states are distinct, the coefficient of $s_{i}$ in $M_{θ}$ is exactly $θ_{i}$ , so this parametrization is injective. Hence mixtures in this fixed family with coordinate $x$ correspond exactly to $θ \in Θ_{x}$ from [L3].

2.1A1A3L3step 1.1∎

The minimum of $V (θ)$ over those mixtures is therefore $\min_{θ \in Θ_{x}} \sum_{i} θ_{i} a_{i} = f_{m i x} (x)$ by [L3], with $+ \infty$ when $Θ_{x}$ is empty. Every further conclusion in the Statement follows by applying the corresponding conclusion of [L3] to the same finite data. The point-mass and zero-coefficient cases are supplied by [L2]; the transfer remains conditional on [A1] and [A3] and records no independent acceptance of the draft representation.

Conditional finite-mixture envelope for an additive ordered representation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels