If the primal problem is constrained by equations instead of...

If the primal problem is constrained by equations instead of inequalities-Minimize cx subject to $A x=b$ and $x \geq 0$-then the requirement $y \geq 0$ is left out of the dual: Maximize yb subject to $y A \leq c$. Show that the one-sided inequality $y b \leq c x$ still holds. Why was $y \geq 0$ needed in equation (1) but not here? This weak duality can be completed to full duality.

Key Concepts

Linear Programming Duality

This concept refers to the correspondence between a primal linear programming problem and its associated dual problem. In duality, every constraint in the primal gives rise to a variable in the dual, and vice versa. The relationship between the two formulations ensures that the value of any feasible solution to the dual provides a bound on the value of any feasible solution of the primal, and under certain conditions, the optimal values of both problems coincide.

Equality vs Inequality Constraints

In linear programming, the form of the constraints affects the structure of the dual problem. When the primal is formulated with equality constraints (Ax = b) rather than inequality constraints, it leads to a dual formulation that does not require the dual variables to be nonnegative. This distinction is crucial because, with inequality constraints, the corresponding dual variables must be nonnegative to maintain the validity of the weak duality proof.

Weak Duality Theorem

The weak duality theorem in linear programming states that for any feasible solution of the primal and any feasible solution of the dual, the value of the dual objective is less than or equal to the value of the primal objective. This property holds regardless of whether the constraints are equalities or inequalities, and it ensures that bounds can be established between the two problems leading the way to proofs of strong duality.

Role of Nonnegativity Conditions in the Dual

The nonnegativity conditions imposed on dual variables are directly linked to the type of constraints in the primal problem. When the primal contains inequality constraints, the corresponding dual variables must be nonnegative to ensure that the weak duality inequality holds. However, when the primal is constrained purely by equalities, the sign restrictions on the corresponding dual variables are not needed because the proof of weak duality can be established from the structure of the equality without relying on nonnegativity, thus allowing the dual variables to take on any sign.

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