Discuss extended formulations in docs (#396)

* Discuss extended formulations in docs

* hardwrap and format

Author: ericphanson

docs/src/index.md

@@ -3,26 +3,85 @@ Convex.jl - Convex Optimization in Julia
 
 Convex.jl is a Julia package for [Disciplined Convex
 Programming](http://dcp.stanford.edu/) (DCP). Convex.jl makes it easy to
-describe optimization problems in a natural, mathematical syntax, and to
-solve those problems using a variety of different (commercial and
-open-source) solvers. Convex.jl can solve
+describe optimization problems in a natural, mathematical syntax, and to solve
+those problems using a variety of different (commercial and open-source)
+solvers. Convex.jl can solve
 
 -   linear programs
--   mixed-integer linear programs and mixed-integer second-order cone
-    programs
+-   mixed-integer linear programs and mixed-integer second-order cone programs
 -   dcp-compliant convex programs including
     -   second-order cone programs (SOCP)
     -   exponential cone programs
     -   semidefinite programs (SDP)
 
 Convex.jl supports many solvers, including
+[COSMO](https://github.com/oxfordcontrol/COSMO.jl),
 [Mosek](https://github.com/JuliaOpt/Mosek.jl),
 [Gurobi](https://github.com/JuliaOpt/gurobi.jl),
 [ECOS](https://github.com/JuliaOpt/ECOS.jl),
 [SCS](https://github.com/karanveerm/SCS.jl) and
 [GLPK](https://github.com/JuliaOpt/GLPK.jl), through
 [MathOptInterface](https://github.com/JuliaOpt/MathOptInterface.jl).
 
-Note that Convex.jl was previously called CVX.jl. This package is under
-active development; we welcome bug reports and feature requests. For
-usage questions, please contact us via the [Julia Discourse](https://discourse.julialang.org/c/domain/opt).
+Note that Convex.jl was previously called CVX.jl. This package is under active
+development; we welcome bug reports and feature requests. For usage questions,
+please contact us via the [Julia
+Discourse](https://discourse.julialang.org/c/domain/opt).
+
+## Extended formulations and the DCP ruleset
+
+Convex.jl works by transforming the problem—which possibly has nonsmooth,
+nonlinear constructions like the nuclear norm, the log determinant, and so
+forth—into a linear optimization problem subject to conic constraints. This
+reformulation often involves adding auxiliary variables, and is called an
+"extended formulation", since the original problem has been extended with
+additional variables. These formulations rely on the problem being modelled by
+combining Convex.jl's "atoms" or primitives according to certain rules which
+ensure convexity, called the [disciplined convex programming (DCP)
+ruleset](http://cvxr.com/cvx/doc/dcp.html). If these atoms are combined in a way
+that does not ensure convexity, the extended formulations are often invalid. As
+a simple example, consider the problem
+
+```julia
+minimize( abs(x), x >= 1, x <= 2)
+```
+
+Obviously, the optimum occurs at `x=1`, but let us imagine we want to solve this
+problem via Convex.jl using a linear programming (LP) solver. Since `abs` is a
+nonlinear function, we need to reformulate the problem to pass it to the LP
+solver. We do this by introducing an auxiliary variable `t` and instead solving
+
+```julia
+minimize(t, x >= 1, x <= 2, t >= x, t >= -x)
+```
+
+That is, we add the constraints `t >= x` and `t >= -x`, and replace `abs(x)` by
+`t`. Since we are minimizing over `t` and the smallest possible `t` satisfying
+these constraints is the absolute value of `x`, we get the right answer. That
+is, this reformulation worked because we were minimizing `abs(x)`, and that is a
+valid way to use the primitive `abs`.
+
+If we were maximizing `abs`, Convex.jl would print
+
+> Warning: Problem not DCP compliant: objective is not DCP
+
+Why? Well, let us consider the same reformulation for a maximization problem.
+The original problem is now
+
+```julia
+maximize( abs(x), x >= 1, x <= 2)
+```
+
+and trivially the optimum is 2, obtained at `x=2`. If we do the same
+replacements as above, however, we arrive at the problem
+
+```julia
+maximize(t, x >= 1, x <= 2, t >= x, t >= -x)
+```
+
+whose solution is infinity. In other words, we got the wrong answer by using the
+reformulation, since the extended formulation was only valid for a minimization
+problem. Convex.jl always performs these reformulations, but they are only
+guaranteed to be valid when the DCP ruleset is followed. Therefore, Convex.jl
+programatically checks the whether or not these rules were satisfied and warns
+if they were not. One should not take these DCP warnings lightly!