I am working with the following definition for conic optimization problems (based on Alexander Barvinok's book "A Course in Convexity"):
Let $\mathbb{E}$ and $\mathbb{Y}$ be Euclidean spaces, let $K\subseteq\mathbb{E}$ and $L\subseteq\mathbb{Y}$ both be proper cones, let $c\in\mathbb{E}$ and $b\in\mathbb{Y}$, and consider a linear function $A\colon\mathbb{E}\to\mathbb{Y}$ .
A conic optimization problem over $K$ is defined as:
$$\inf\{\langle x,c\rangle\, \colon A(x)\succeq_{_L^{\ast}} b , x\in K \}.$$
However, I am having trouble defining conic problems over a specific cone. For example, i have the following definition for SOCP (which are conic problems over $\mathbb{L}_n=\{x\oplus t\in\mathbb{R}^n\oplus\mathbb{R}_+\,\colon\|x\|\leq t\}$):
Let $\mathbb{Y}$ be a Euclidean space, let $K\subseteq\mathbb{Y}$ be a proper cone, and consider $n\in\mathbb{N}$ and $\mathbb{L}_n\subseteq\mathbb{R}^n\oplus\mathbb{R}$. Also let a linear function $A\colon\mathbb{R}^n\oplus\mathbb{R}\to \mathbb{Y}$.
A second order conic problem (SOCP) is defined as: $$\inf\{\langle x,c\rangle\, \colon A(x)\succeq_{_K^{\ast}} b , x\in \mathbb{L}_n \}.$$
I see that my definition does not correspond to the definition of, for example, Wikipedia, because there is no 'extra' cone $K$ in other definitions. Where is the flaw with my definition? How could I correct it?
It seems like your question is about converting your SOCO problem with conic inequalities to the notation on Wikipedia with equality constraints. You can simply add an additional variable: $$Ax \succeq_{_K} b, \quad x \in L $$ is equivalent to: $$Ax - b = y, \quad (x,y) \in L \times K. $$ Note that $L \times K$ is a cone when $L$ and $K$ are cones.