PO.ID.Partial.Support­Function

Lower sandwich. For a feasible set C in a real inner-product space and a direction d, if a point x belongs to C and the linear functional ⟪d, ·⟫ is bounded above on C, then x's functional value is at most the support value: ⟪d, x⟫ ≤ supportFn C d.

Sharp identified interval (attainment form). For a convex feasible set C in a real inner-product space and a linear target direction d, assuming the target functional is bounded above on C in the direction d and in the opposite direction -d, and that the upper support value is attained by some feasible point and the lower support value -supportFn C (-d) is likewise attained, then the identified set of the target functional over C is exactly the closed interval [-supportFn C (-d), supportFn C d].

Sharp identified interval (compact convex form). For a linear target direction d and a feasible set that is compact, convex, and nonempty, the identified set of the target functional ⟪d, ·⟫ over C is exactly the closed interval [-supportFn C (-d), supportFn C d]: a continuous functional attains its supremum and infimum on a compact set and is automatically bounded there.

Point identification. For a feasible set C and direction d in a real inner-product space, assuming C is nonempty and the target functional ⟪d, ·⟫ is bounded above on C in the direction d and in the opposite direction -d, then the identified set collapses to a point (zero width) if and only if the target functional is constant on C, equal to its support value.

Bridge to the scalar engine. For an objective functional obj and a feasible set C, the scalar identified interval of obj relative to membership in C equals exactly the image of obj on C. Specialised to obj = ⟪d, ·⟫, this identifies the scalar engine's output with the linear image studied here.

Closed-form support value over the affine-ball fiber. Let A be a continuous linear map between real inner-product spaces, b a target value, B a radius, and c a direction. Suppose h₀ solves the linear constraint A h₀ = b, h₀ is orthogonal to the kernel of A (the minimum-norm solution), and h₀'s norm is at most B (h₀ fits the ball). Then the support value of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} equals ⟪c, h₀⟫ + √(B² − ‖h₀‖²)·‖P c‖, where P is the orthogonal projection onto the kernel of A.

Closed-form width of the affine-ball identified set. Under the same hypotheses as supportFn_affineBall_eq — h₀ solves A h₀ = b, h₀ is orthogonal to the kernel of A, and h₀'s norm is at most B — the width of the identified set of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} equals 2·√(B² − ‖h₀‖²)·‖P c‖, twice the residual radius times the norm of c's projection onto the kernel of A.

Point identification. Let A be a continuous linear map between real inner-product spaces with b in its range, B a radius, and c a target direction. Suppose h₀ solves the linear constraint A h₀ = b, h₀ is orthogonal to the kernel of A (the minimum-norm solution), and h₀'s norm is strictly less than B (strict slack in the norm bound). Then the identified set of ⟪c, ·⟫ over the fiber {h : A h = b, ‖h‖ ≤ B} collapses to a point if and only if the orthogonal projection of c onto the kernel of A vanishes, i.e. c is orthogonal to that kernel.

Subadditivity in the direction (sublinearity of the support function). For a feasible set C in a real inner-product space and directions d₁, d₂, assuming C is nonempty and the linear functionals ⟪d₁, ·⟫ and ⟪d₂, ·⟫ are each bounded above on C, then the support value in the combined direction is at most the sum of the support values in each direction: supportFn C (d₁ + d₂) ≤ supportFn C d₁ + supportFn C d₂.

No ambiguity implies point identification (upper). When the ambiguity set collapses to the singleton {w₀}, the upper robust bound for the linear functional c equals the identified value ⟪c, w₀⟫.

No ambiguity implies point identification (lower). When the ambiguity set collapses to the singleton {w₀}, the lower robust bound for the linear functional c equals the identified value ⟪c, w₀⟫.

The robust interval is the identified set. When the ambiguity set W is compact, convex, and nonempty, the set of reweighted means ⟪c, w⟫ attained as w ranges over W is exactly the closed interval [robustLower W c, robustUpper W c] — the robust bounds are sharp.

Reduction to the affine-ball engine. When the base direction has unit inner product with itself, ⟪e, e⟫ = 1, the χ²/L² ambiguity set l2Ball e ρ equals the translate by e of the affine ball of radius ρ for the linear functional ⟪e, ·⟫ centered at the origin — the substitution v = w - e turns the normalization constraint ⟪e, w⟫ = 1 into the linear constraint ⟪e, v⟫ = 0.

Closed-form worst case (mean plus ρ·SD). For a unit-norm base direction e and a nonnegative radius ρ, the largest reweighted value ⟪c, w⟫ attains over the χ²/L² ambiguity set l2Ball e ρ equals ⟪e, c⟫ + ρ · √(‖c‖² − ⟪e, c⟫²) — the base-direction mean plus ρ standard deviations of c.

Closed-form width. For a unit-norm base direction e and a nonnegative radius ρ, the width of the χ²/L² robust interval for the target c equals 2ρ · √(‖c‖² − ⟪e, c⟫²), twice the radius times the standard deviation of c.

Point identification. For a unit-norm base direction e and a strictly positive radius ρ, the χ²/L² robust interval for the target c collapses to a single point exactly when ‖c‖² equals ⟪e, c⟫², i.e. when c is collinear with e — equivalently, equality holds in the Cauchy–Schwarz inequality, meaning the represented outcome is almost-surely constant.