[docs} More math, and references.

docs/source/math.rst

@@ -10,4 +10,85 @@ from linear robust optimization to geometric and signomial programming.
 - GPs have robust formulations.
 - RSPs can be represented as sequential RGPs.
 
+As a quick demonstration, we paraphrase the foundational works of mathematics,
+in order of application,
+that allow us to come up with robust counterparts for GPs and SPs.
+
+Linear programs (LPs) have tractable robust counterparts.
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+This is a seminal finding by [Ben-Tal, 1999] that derives the robust counterpart for
+a linear program given different types of bounded uncertainty sets.
+One of the problems they detail is below,
+which is a linear program in which the coefficients :math:`\mathbf{a}_i` are subject
+to an affine perturbation by uncertain parameters :math:`u_i`, which are contained in an ellipsoidal
+uncertainty set.
+
+.. math::
+
+    \text{min} &~~\mathbf{c}^T\mathbf{x} \\
+    \text{s.t.}     &~~\mathbf{a}_i\mathbf{x} \leq b_i,~\forall \mathbf{a}_i \in \mathcal{U}_i,~i = 1,\ldots,m, \\
+                    &~~\mathcal{U} = \{(\mathbf{a}_1, \ldots, \mathbf{a}_m): \mathbf{a}_i = \mathbf{a}_i^0 + \Delta_i u_i, ~i = 1,\ldots,m,
+                    ~~~\left\lVert u \right\rVert_2 \leq \rho\}
+
+The robust counterpart for the above linear program is given by the following:
+
+.. math::
+
+    \text{min} &~~\mathbf{c}^T\mathbf{x} \\
+    \text{s.t.}&~~\mathbf{a}_i^0 \leq b_i - \rho\left\lVert \Delta_i\mathbf{x} \right\rVert_2,~\forall i = 1,\ldots,m, \\
+
+This is tractable second-order cone program.
+
+Two-term posynomials are LP-approximable.
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+We were not the first people interested in robust GPs formulations. Folks from
+Stephen Boyd's group at Stanford took the first concrete steps to combine principles of robust linear programming
+with GPs. Work described in [Hsiung, 2008] paves the way for low-error piecewise-linear (PWL) approximations
+of posynomials.
+
+|picHsiung|
+
+.. |picHsiung| image:: picHsiung.png
+        :width: 70%
+
+For derivation of robust GPs, the central finding is in Corollary 1 of the paper,
+which asserts that there is an analytical solution for the lowest-error
+lower and upper approximation of a two-term posynomial in log-space. An image of the corollary
+of the paper is above; the proof can be found in the paper.
+
+All posynomials are LP-approximable.
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+We use the PWL two-term posynomial approximation above to approximate any posynomial
+with PWL approximations of two-term posynomials. The full recipe is described by [Saab, 2018];
+here we demonstrate with a simple example from the paper. The following problem
+
+.. math::
+
+    \text{min} &~~f \\
+    \text{s.t.}&~~\text{max}\{M_1 + M_2 + M_3 + M_4\} &\leq 1 \\
+               &~~\text{max}\{M_5 + M_6\} & \leq 1
+
+is equivalent to
+
+.. math::
+
+    \text{min} &~~f \\
+    \text{s.t.}&~~\text{max}\{M_1 +e^{t_1}\} &\leq 1 \\
+               &~~\text{max}\{M_2 +e^{t_1}\} &\leq e^{t_1} \\
+               &~~\text{max}\{M_3 + M_4\} &\leq e^{t_2} \\
+               &~~\text{max}\{M_5 + M_6\} & \leq 1
+
+by adding auxiliary variables and using properties of inequalities.
+
+GPs have robust formulations.
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+
+RSPs can be represented as sequential RGPs.
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+
 Work in progress...

docs/source/references.rst

@@ -1,8 +1,15 @@
 References
 **********
 
-*[Ozturk, 2019] Ozturk, B. and Saab, A., "Optimal Aircraft Design Decisions Under Uncertainty via Robust Signomial Programming", AIAA Aviation 2019 Conference Proceedings.
+[Ben-Tal, 1999] Ben-Tal, A., and Nemirovski, A., “Robust solutions of uncertain linear programs,” Operations Research Letters, 1999.
+
+[Hsiung, 2008] Hsiung, K. L., Kim, S. J., and Boyd, S., “Tractable approximate robust geometric programming,” Optimization and Engineering, vol. 9, 2008, pp. 95–118.
+
+[Ozturk, 2019] Ozturk, B. and Saab, A., "Optimal Aircraft Design Decisions Under Uncertainty via Robust Signomial Programming", AIAA Aviation 2019 Conference Proceedings.
+
+[Saab, 2018] Saab, A., Burnell, E., and Hoburg, W., "Robust Designs via Geometric Programming", arXiv:1808.07192v1.
+
+[Soyster, 1974] Soyster, A.L., "A Duality Theory for Convex Programming with Set-Inclusive Constraints", Operations Research, Vol. 22, No. 4, August 1974.
 
-*[Saab, 2018] Saab, A., Burnell, E., and Hoburg, W., "Robust Designs via Geometric Programming", arXiv:1808.07192v1.
 
 Work in progress...

docs/source/robust101.rst

@@ -9,8 +9,8 @@ unlike general stochastic optimization methods which optimize statistics of the
 of the objective over probability distributions of uncertain parameters. As such, RO
 sacrifices generality for tractability, probabilistic guarantees and engineering intuition.
 
-Basic mathematical principle
-----------------------------
+Basic mathematical principles
+-----------------------------
 
 [*paraphrased from Ozturk and Saab, 2019*]