In the hyperclip formulation, the inequality involves which combination of A, X, and R?

In the hyperclip formulation, the inequality involves which combination of A, X, and R?

• A. A^T X + R ≤ 0
• B. A^T X ≥ 0
• C. A X + R ≤ 0
• D. X^T A + R ≤ 0

Answer: (A)

In this formulation, the inequality is written to encode many linear constraints in one compact vector form. Each constraint is a linear combination of the decision variable X, with the coefficients coming from A, and an offset from R. Using A^T X + R ≤ 0 means you take the inner products of X with each constraint normal (the columns or rows of A give these normals), add the corresponding offset from R, and require every result to be nonpositive. This directly enforces that every constraint is satisfied simultaneously. Why this form is the natural choice: A^T X produces a vector where each entry is the value of one linear constraint evaluated at X. Adding R aligns each constraint with its offset, and the ≤ 0 enforces feasibility for all constraints at once. The other options would either flip the inequality direction, mix dimensions in a less standard way, or represent the same idea only if you reinterpret orientation and dimensions, which isn’t the typical, straightforward way the hyperclip formulation expresses the constraints.

In this formulation, the inequality is written to encode many linear constraints in one compact vector form. Each constraint is a linear combination of the decision variable X, with the coefficients coming from A, and an offset from R. Using A^T X + R ≤ 0 means you take the inner products of X with each constraint normal (the columns or rows of A give these normals), add the corresponding offset from R, and require every result to be nonpositive. This directly enforces that every constraint is satisfied simultaneously.

Why this form is the natural choice: A^T X produces a vector where each entry is the value of one linear constraint evaluated at X. Adding R aligns each constraint with its offset, and the ≤ 0 enforces feasibility for all constraints at once. The other options would either flip the inequality direction, mix dimensions in a less standard way, or represent the same idea only if you reinterpret orientation and dimensions, which isn’t the typical, straightforward way the hyperclip formulation expresses the constraints.