[docs] goal programming write-up.

docs/source/commands.rst

@@ -0,0 +1,23 @@
+More advanced commands
+======================
+
+**robust** has a variety of tools beyond the basics
+to aid engineering design under uncertainty.
+
+All of these methods have been implemented in the `robustSPpaper`_
+code repository, which defines the models used in [Ozturk, 2019].
+
+.. _robustSPpaper: https://github.com/1ozturkbe/robustSPpaper/tree/master/code
+
+Choosing between RGP approximation methods
+------------------------------------------
+
+We prefer to define the methods in a dict
+
+.. code-blocK:: python
+
+    methods = [{'name': 'Best Pairs', 'twoTerm': True, 'boyd': False, 'simpleModel': False},
+               {'name': 'Linearized Perturbations', 'twoTerm': False, 'boyd': False, 'simpleModel': False},
+               {'name': 'Simple Conservative', 'twoTerm': False, 'boyd': False, 'simpleModel': True}]
+
+and choose the appropriate

docs/source/goal.rst

@@ -1,4 +1,54 @@
 Goal programming
 ****************
 
-Work in progress...
+Recall the standard RO form below
+
+.. math::
+
+    \begin{split}
+        \text{min} &~~f_0(x) \\
+    \text{s.t.}     &~~\underset{u}{\text{max}}~f_i(x,u) \leq 0,~i = 1,\ldots,n \\
+                    &~~\left\lVert u \right\rVert \leq \Gamma \\
+        \end{split}
+
+where we attempt to minimize :math:`f_0` for the worst-case realization of
+uncertain parameters in the set. We can flip this on its head, and
+solve the following problem
+
+.. math::
+
+    \begin{split}
+    \text{max}~~\Gamma \\
+    \text{s.t.}~~f_i(x,u) &\leq 0,~i = 1,\ldots,n \\
+                    \left\lVert u \right\rVert &\leq \Gamma \\
+                    f_0(x) &\leq (1+\delta)f_0^*,~\delta \geq 0
+    \end{split}
+
+where :math:`f_0^*` is the optimum of the nominal problem and :math:`\delta`
+is a fractional penalty on the objective that we are willing to sacrifice for robustness, which
+gives :math:`(1+\delta)f_0^*` as the upper bound on the objective value.
+
+The benefit of this goal programming form is that we can start to use risk as a global
+design variable against with all objectives can be weighed.
+
+Implementation
+--------------
+
+To use the goal programming functions in **robust**, you can use the ```simulate.variable_goal_results```
+function, which has the same inputs as ```simulate.variable_gamma_result``` except for
+having the penalty parameter :math:`\delta` instead of the uncertainty set size :math:`\Gamma` as its input.
+
+What occurs to the solved nominal model under the hood is the following:
+
+.. code-block:: python
+
+    Gamma = Variable('\\Gamma', '-', 'Uncertainty bound')
+    delta = Variable(value, '1+\\delta', '-', 'Acceptable optimal solution bound', fix = True)
+    origcost = model.cost
+    mGoal = Model(1 / Gamma, [model, origcost <= Monomial(nominal_solution(origcost)) * delta, Gamma <= 1e30, delta <= 1e30],
+                  model.substitutions)
+    robust_goal_model = RobustModel(mGoal, uncertainty_set, gamma=Gamma)
+    sol = robust_goal_model.robustsolve()
+
+This is an exact formulation of the aforementioned risk minimization problem!
+

docs/source/index.rst

@@ -26,11 +26,11 @@ Table of contents:
    robust101
    installation
    gettingstarted
+   commands
    whyro
    math
    methods
    rspapproaches
-   simulation
    goal
    references