Generalized notions of solutions in various areas of mathematics In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are also other reasons to generalize the notion of a solution in various contexts.) 
I would like to collect a list of "generalized solutions" concepts in various areas of mathematics, hoping that looking at these various concepts side-by side can be useful and interesting.
Let me demonstrate what I mean by an example from graph theory: A perfect matching in a graph is a set of disjoint edges such that every vertex is included in precisely one edge. A fractional perfect matching is an assignment of non negative weights to the edges so that for every vertex, the sum of weights is 1. In combinatorics, moving from a notion described by a 0-1 solution for a linear programming problem to the solution over the reals is called LP relaxation of a problem and it is quite important in various contexts.
(There are, of course, useful papers or other resources on generalized solutions in specific areas. It will be useful to have links to those but not as a substitute for actual answers with some details.)
 A: Formal solutions to partial differential relations
Given a partial differential relation, that is, a subset $\mathcal{R} \subset J^k(\mathbb{R}^n, \mathbb{R}^m)$ of the space of $k$-jets of smooth maps $\mathbb{R}^n \to \mathbb{R}^m$, one can consider the space of smooth (say) maps $f$ from an $n$-manifold $N$ to an $m$-manifold $M$ such that $J^k(f) \in \mathcal{R}$, i.e. so that the $k$-jet of the function lies in the subspace $\mathcal{R}$ at each point. Call the space of such maps $\mathrm{Sol}_\mathcal{R}(N, M)$.
On the other hand, we can consider the bundle $J^k(N, M) \to N$ of $k$-jets of maps from $N$ to $M$, and the associated subbundle $\mathcal{R}(N, M) \to N$, and call the space of sections of this last bundle $\mathrm{FSol}_\mathcal{R}(N,M)$, the space of formal solutions. This space is far easier to analyse, for example because constructing sections of a bundle is a purely homotopy-theoretic problem.
Taking derivatives gives a comparison map
$$\mathrm{Sol}_\mathcal{R}(N, M) \to \mathrm{FSol}_\mathcal{R}(N, M).$$
If $\mathcal{R}$ is open in $J^k(\mathbb{R}^n, \mathbb{R}^m)$ and  the manifold $N$ is open, Gromov showed that the comparison map is a homotopy equivalence. In particular, if the space of formal solutions is non-empty, so is the space of actual solutions.
A: Moduli problem: find a good parametrization of geometric objects of some type; parametrization should form a collection equipped with some natural geometric structure, therefore being a geometric object in its own right. While naive "parameter space" is a set, in structured formulation it is replaced by a moduli space which classifies the geometric objects we started with. In the simplest case, the moduli problem is representable by a space in a usual sense, an object in more or less the same category in which the original geometric object was. For example a manifold or a scheme where the original objects were manifolds or schemes. With harder problems the moduli lead to more and more general kinds of objects. This motivated new types of spaces as stacks, higher stacks, derived stacks and so on. 
It appears that starting with original geometric category, most of the generalized objects needed to solve the moduli problem live in some nice geometric subcategory (e.g. algebraic stacks) of the category of (possibly categorified) presheaves or sheaves on the original category, including higher versions like simplicial presheaves and so on. The original category embeds by the corresponding version of Yoneda embedding into the category of (pre)sheaves. The new ambient category of presheaves not only more generically has a solution to the moduli problem, but also has many other improved natural properties like closedness under limits. 
Cohomology theories, various generalized cocycles and so on, generalized smoothness notions and so on, can also be accomodated after Yoneda embedding into a homotopy correct version of presheaf category, like in the emerging subject of derived geometry. In the original terms of non-generalized spaces, one would need to use all kinds of difficult and dirty technique to define and study the generalized notions, for example introducing various piecewise-continuous cocycles, multivalued or infinite-dimensional models and so on. Methods depending on Yoneda philosophy give rather universal setting to attack moduli problems and many other problems (like deformation theory), allowing to often eliminate construction of very elaborate but ad hoc modifications of original concepts. Inside the bigger category it may be easier to cut out some nice geometric subcategory of geometric spaces which include the solutions to the moduli problem than constructing some similar category in terms of original geometry. Of course, sometimes the difficult elementary models have their own specific strengths, which do not follow from the application of general methods. 
A: Affine schemes:
Given any ring $R$, try to find a map from it into local ring $L$ which is initial among maps to local rings (i.e. any other map from $R$ into a local ring should factor through this one, followed by a map of local rings, i.e. one such that the preimage of the maximal ideal is the maximal ideal). Such a thing does not exist, unless $R$ is already local.
But if we allow $L$ to be a ring object living in a different topos than that of sets, then it exists: It is the local ring object living in $Sh(Spec R)$ given by the structure sheaf $\mathcal{O}_{Spec R}$ (see also my post here)
A: Given a set of polynomial diophantine equations, it is useful to study solutions in any ring, instead of just studying integer solutions.
(This is the "functorial point of view" of a scheme over $\mathbb Z$.)
A: Well, then I'll start with the most obvious generalized solutions:


*

*weak solutions to PDEs

*Schwartz's generalized Functions aka Distributions,

*Colombeau's algebra(s) of generalized functions and

*various other kinds of generalized functions

*Quasi-Minima in functional analysis: A quasi-minimum of a functional $\mathcal{F}$ is a $u$ such that $\mathcal{F}u\leq Q\mathcal{F}v$ for all $v$ (with some constant $Q\geq 1$)

*Every solution of an polynomial equation within $\mathbb{C}$ can be a generalized solution if you're problem is something that has only real (maybe some geometric problem) or only integer or even only natural (maybe something from number theory) solutions. But considering all complex solution to your particular equation often gives a very elegant treatment of the problem.

A: Ideals in rings of integers of number fields arose as "ideal numbers"...
A: Grothendieck topologies (or: toposes as generalized spaces):
There is no topology on a general scheme which is e.g. fine enough to give back the cohomological dimensions expected from geometry, but with a more general notion of covering (or: of space) this works out.
A: Quotient "spaces":
While quotients, e.g. of group actions, in geometry often are degenerate, several generalized notions of quotient space help here: Sheaf quotients, Orbifolds, Algebraic Spaces, Stack quotients, Homotopy quotients, Non-commutative quotients, GIT quotients, ...
(similarly with moduli spaces)
A: Complex numbers arose as ideal solutions of polynomial equations with real coefficients, I guess.
A: Generalized Eigenvector
I'm surprised no one has yet mentioned the first example an undergraduate is likely to see.  Suppose $A$ is a linear map from a finite dimensional vector space $V$ to $V$, with eigenvalue $\lambda$.  Any nonzero vector in $\text{ker}(A-\lambda I)^k$ for some $k\ge1$ is called a generalized eigenvector for eigenvalue $\lambda$.  These are used in proving the existance of the Jordan Canonical Form.
A: In linear algebra (linear inverse problems) one generalizes the notion of a solution of a linear operator equation $Ax=y$ to


*

*"best approximation" if there is no solution, i.e. minimizing the functional $\|Ax-y\|$,

*"Minimum-norm solution" if there is a subspace of solutions, i.e. taking that solution of $Ax=y$ which has minimal norm,

*both (if the best approximation is not unique) leading to the Moore-Penrose inverse.

A: Virtual knots
Louis Kauffman generalized the knots by introducing virtual crossings, virtual knots and virtual Reidemeister moves. He obtained some interesting developments in knot theory.
A: Partial Differential Equations (PDE) is a topic where generalizing the notion of solutions is a daily activity. 
The most obvious generalization has been the notion of weak solutions, which means that a solution $u$ is not necessarily differentiable enough times for the derivatives involved in the equation to make sense; but an integration against test functions, followed by an integration by parts, cures the problem. The most known example is that of the Laplace equation
$$\Delta u=f\qquad\hbox{over }\Omega,$$
where it is enough for $u$ to have locally integrable first-order derivatives, by rewriting the equation as a variational formulation (Dirichlet principle)
$$\int_\Omega \nabla u\cdot\nabla vdx=-\int_\Omega fvdx$$
for every $v\in{\mathcal C}^1_c(\Omega)$ (subscript $c$ means compact support).
What is important in this process is to satisfy the rule
If $u$ has enough derivatives that the equation makes sense pointwise, then it is a weak solution if and only if it is a classical solution.
Let us mention in passing that in order to use the full strength of functional analysis and operator theory, this weak notion of solutions led to the birth of Sobolev spaces and Distribution theory (L. Schwartz).
This framework has been used for nonlinear equations and systems too, for instance for the Navier-Stokes, Euler, Schrödinger equations, ... An important question is whether this framework is accurate or not. By accurate, we mean that boundary and/or initial data yield a unique solution, which depends continuously on the data. This is the question of well-posedness. In many cases, functional analysis, sometimes associated to topological arguments, yield an existence theorem. A celebrated one is J. Leray's existence result to the Navier-Stokes equation of an incompressible fluid. However, uniqueness is often an other matter, a difficult one. For a $3$-dimensional fluid, the uniqueness to Navier-Stokes is a $1$M US Dollar open question.
Uniqueness is often (but not always) associated to regularity. In many situations, there are weak-strong uniqueness result, which state that if a classical, or a regular enough solution exists, then there does not exist any other weak solution (say in a class where we do have an existence result). It is an if-theorem, in the absence of an existence result of strong solutions. For elliptic and parabolic equations, the regularity theory is a topic of its own.
Whereas regularity is often expected in elliptic or parabolic equations and systems, it is not for hyperbolic ones, because we know that singularities do propagate, and that they can even be created in finite time thanks to nonlinear effects. Then the notion of weak solutions becomes meaningful, in that it translates in mathematical terms the physical notion of conserved quantities. It gives algebraic relation for the jump of the solution and its derivatives across discontinuities (Rankine-Hugoniot relations).
Finally, I like a lot the way the theory of nonlinear elliptic equations, and of Hamilton-Jacobi equations have develloped in the past decades. At the beginning, it was observed that the maximum principle, known for classical solutions, remains valid for weak ones. This suggested, when the nonlinearity is so strong that a variational formulation is not available, that the maximum principle itself be used to define a notion of viscosity solution. The idea is to test at $x_0$ the PDE with a test function $\phi$ being comparable to $u$ (either $\phi\le u$ or $\phi\ge u$ locally) and touching $u$ at $x_0$. This has been extremely powerfull.
A: One of the most fruitful notion of generalized solution in optimization and combinatorics is linear programming relaxation. Quoting from the wikipedia article: In mathematics, the linear programming relaxation of a 0-1 integer program is the problem that arises by replacing the constraint that each variable must be 0 or 1 by a weaker constraint, that each variable belong to the interval [0,1].
A: A form of "generalized solution" which I saw in various areas like for combinatorial optimization problems, for diophanine equations, for computational complexity purposes, and others is "statistical physics relaxation". You regard your original problem as a "temperature 0" case of a more general problem and try to gain insight on the original problem based on statistical-physics insights for the generalized problem. I am not sure what is the general recipe for this apprach and I will be happy to see an edited version with further explanation and links. 
A: I think that such an example is the use of the Residue theorem to calculate contour integrals. You use complex analysis to solve a problem in real analysis.
