CVX(matlab) syntax note
===
###### tags: `Math` `matlab`

* cvx relies on external solver to solve problems.
* ex: Mosek, Gurobi, SDTP3, SeDuMi
* note: Mosek can solve MISOC-problem, while Gurobi cannot.(in 2, 3 years ago.)
* CVX's DCP ruleset is **sufficient** to check if a problem is convex or not, but **not necessary**.

## cvx variables
* `cvx_optval`: the optimal solution of the objective function, should be a number.
* `cvx_status`: the solver status.
* ex: Solved, Unbouned, Infeasible....
* note: Infeasible appears very often when the DCP rules is violated.
* `cvx_where`: the path of the cvx directory.
* `cvx_begin ... cvx_end`: the problem should be declared in this block.
* `cvx_begin quiet`: ask cvx to be quiet.
* `cvx_begin sdp`: tell cvx the following problem is a semidefinite problem.

## quick start examples
### 1. LP problem
$$
\max\ c^Tx \\
\text{s.t.}\ Ax<=b \\
x >= 0
$$

- in cvx:
- ```=
cvx_begin
variable x(n)
maximize (c’ * x)
subject to
A * x <= b
x >= 0
cvx_end

- w/o CVX:
    - `linprog(c, A, b, [], [], zeros(n,1), zeros(n,1)+Inf)`