关于不可行问题的对偶/拉格朗日变量取值及原对偶问题无界性与不可行性关系的技术问询

Hey there, let's break down your questions step by step—this is a super common point of confusion in convex optimization, so great call asking for clarification!

First: The relationship between primal/dual unboundedness and infeasibility #

The two statements you encountered are actually consistent, not conflicting—they just differ in how explicitly they state the optimization direction:

• The general rule you found is a simplified version, assuming the standard convex optimization setup: the primal problem is a minimization (so "unbounded" means the objective can decrease without bound, i.e., unbounded below), and the dual problem is a maximization (so "unbounded" means the objective can increase without bound, i.e., unbounded above).
• Boyd's Convex Optimization just makes these direction assumptions explicit to avoid ambiguity. For example:

  If the primal problem is unbounded below, the Lagrangian dual problem is infeasible. If the dual problem is unbounded above, the primal problem is infeasible.

This is exactly the same as the simplified rule—Boyd just removes any room for misinterpreting what "unbounded" means for each problem type. And you’re right that it’s possible for both primal and dual to be infeasible: for example, take the primal problem min x with constraints x ≥ 1 and x ≤ 0—there’s no feasible x, and its corresponding dual problem will also have no feasible dual variables.

Second: Does primal infeasibility mean dual variables are infinite? #

No, that’s not correct. Here’s the breakdown:
When the primal problem is infeasible, the dual problem has exactly two possible outcomes:

1. The dual problem is also infeasible: As in the example above, there are no valid dual variables that satisfy the dual feasibility conditions.
2. The dual problem is unbounded: This means we can find a sequence of valid dual variables where the dual objective value trends toward positive or negative infinity (depending on whether the dual is a max or min problem). Crucially, though, there’s no such thing as an "infinite" dual variable—infinity isn’t a valid, finite value for a dual variable. We’re talking about a limit of a sequence, not a single variable taking an infinite value.

In short, primal infeasibility doesn’t equate to dual variables being infinite; it just means the dual is either infeasible or unbounded (with variable sequences approaching infinity in objective value, not variable values themselves).

备注：内容来源于stack exchange，提问作者Ruihao Wang