# What is the difference between feasibility and admissibility?

In optimization and optimal control, I have seen the terms admissible and feasible used interchangeably to refer to solutions (or solution sets) that satisfy the problem constraints. Are ‘admissible solution’ and ‘feasible solution’ mathematically equivalent? To facilitate discussion, consider the following optimal control:

Objective: Minimize cost function $$J(\mathbf{x}(t), \mathbf{u}(t); t)$$

Subject to:

• Equality constraints: $$h(\mathbf{x}(t), \mathbf{u}(t); t) = 0$$
• Inequality constraints: $$g(\mathbf{x}(t), \mathbf{u}(t); t) ≤ 0$$

$$\mathbf{x}(t)$$ and $$\mathbf{u}(t)$$ are the state and control vectors, respectively. Let $$\mathbf{x}_\mathsf{cand}(t)$$ be a candidate solution.

To my knowledge, if $$\mathbf{x}_\mathsf{cand}(t)$$ satisfies the equality constraints $$h(\mathbf{x}(t), \mathbf{u}(t); t) = 0$$ and inequality constraints $$g(\mathbf{x}(t), \mathbf{u}(t); t) ≤ 0$$, then: $$\mathbf{x}_\mathsf{cand}(t)$$ is a feasible solution. Also, to my knowledge, I have been considering admissible solution to mean the same thing.

Uncertain claim: I have been informally told that a feasible solution is one that satisfies both the equality and inequality constraints, while an admissible solution is one that satisfies only the inequality constraints but may not necessarily satisfy the equality constraints. The implication here is that admissibility is more relaxed to allow for more potential solutions. In other words, the claim is that a feasible solution lies within the feasible region (intersection of all constraint sets), while admissible solutions form a superset of this region.

Given that I could not find an authoritative source on the above claim, I don't trust it and I am therefore reaching out to anyone who can help. Thank you!

• I’ve only heard of feasible solutions. Admissibility comes up in statistical decision theory but it refers to a function or rule, not a specific solution: en.m.wikipedia.org/wiki/…. Aug 19, 2023 at 16:17

according to our professor's convention:

• the inputs of a system that satisfy the constraints are generally referred to as "admissible,"
• and the states of a system that satisfy the constraints are called "feasible."

For example,

1. for given $$x_0$$, not all $$u \in \mathbb{U}$$ are admissible.
2. not all $$x_0$$ are feasible.

After some digging, it appears to be that the meanings of “feasible” and “admissible” are field-specific or at least context-specific. In optimal control, $$\mathbf{x}$$ is admissible is taken to mean $$\mathbf{x} \in \mathcal{X}$$ where $$\mathcal{X}$$ is the state's constraint set. Similarly, an admissible input $$\mathbf{u}$$ is taken to mean $$\mathbf{u} \in \mathcal{U}$$ where $$\mathcal{U}$$ is the input's constraint set. (For example, page 4 in these notes)

Annika also commented above that in statistical decision theory, ‘admissible’ has a specific meaning applied to decision rules, quoting from Wikipedia (link provided by Annika):

A decision rule is admissible (with respect to the loss function) if and only if no other rule dominates it; otherwise it is inadmissible.

Well, for now, all I can say is to check the specific subfield's authoritative sources for the precise meaning of these terms. Otherwise, if anyone else has any ideas, I'd be happy to know. Cheers.