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Oct 19, 2020
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[robustness] update rst to remove problematic whitespace for conversion using tomyst #146

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update rst to remove problematic whitespace for conversion using tomyst
mmcky Oct 19, 2020
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Merge branch 'master' into robustness-rstadjust
mmcky Oct 19, 2020
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29 changes: 15 additions & 14 deletions source/rst/robustness.rst
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Expand Up @@ -100,9 +100,9 @@ Here

* *Entropy* is a non-negative number that measures the size of a set of models surrounding the decision-maker's approximating model.

* Entropy is zero when the set includes only the approximating model, indicating that the decision-maker completely trusts the approximating model.
* Entropy is zero when the set includes only the approximating model, indicating that the decision-maker completely trusts the approximating model.

* Entropy is bigger, and the set of surrounding models is bigger, the less the decision-maker trusts the approximating model.
* Entropy is bigger, and the set of surrounding models is bigger, the less the decision-maker trusts the approximating model.

The shaded region indicates that for **all** models having entropy less than or equal to the number on the horizontal axis, the value obtained will be somewhere within the indicated set of values.

Expand Down Expand Up @@ -132,9 +132,9 @@ Below we'll explain in detail how to construct these sets of values for a given

Here is a hint about the *secret weapons* we'll use to construct these sets

* We'll use some min problems to construct the lower bounds
* We'll use some min problems to construct the lower bounds

* We'll use some max problems to construct the upper bounds
* We'll use some max problems to construct the upper bounds

We will also describe how to choose :math:`F` to shape the sets of values.

Expand Down Expand Up @@ -540,15 +540,16 @@ In particular, the lower bound on the left side of :eq:`rob_bound` is attained w

To construct the *lower bound* on the set of values associated with all perturbations :math:`{\mathbf w}` satisfying the entropy constraint :eq:`rb_dec` at a given entropy level, we proceed as follows:

* For a given :math:`\theta`, solve the minimization problem :eq:`rb_a2o`.
* For a given :math:`\theta`, solve the minimization problem :eq:`rb_a2o`.

* Compute the minimizer :math:`R_\theta(x_0, F)` and the associated entropy using :eq:`rb_pdt22`.
* Compute the minimizer :math:`R_\theta(x_0, F)` and the associated entropy using :eq:`rb_pdt22`.

* Compute the lower bound on the value function :math:`R_\theta(x_0, F) - \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`.
* Compute the lower bound on the value function :math:`R_\theta(x_0, F) - \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`.

* Repeat the preceding three steps for a range of values of :math:`\theta` to trace out the lower bound.
* Repeat the preceding three steps for a range of values of :math:`\theta` to trace out the lower bound.

.. note::

This procedure sweeps out a set of separating hyperplanes indexed by different values for the Lagrange multiplier :math:`\theta`.

The Upper Bound
Expand Down Expand Up @@ -608,13 +609,13 @@ The upper bound on the left side of :eq:`robboundmax` is attained when

To construct the *upper bound* on the set of values associated all perturbations :math:`{\mathbf w}` with a given entropy we proceed much as we did for the lower bound

* For a given :math:`\tilde \theta`, solve the maximization problem :eq:`rba2omax`.
* For a given :math:`\tilde \theta`, solve the maximization problem :eq:`rba2omax`.

* Compute the maximizer :math:`V_{\tilde \theta}(x_0, F)` and the associated entropy using :eq:`rbpdt223`.
* Compute the maximizer :math:`V_{\tilde \theta}(x_0, F)` and the associated entropy using :eq:`rbpdt223`.

* Compute the upper bound on the value function :math:`V_{\tilde \theta}(x_0, F) + \tilde \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`.
* Compute the upper bound on the value function :math:`V_{\tilde \theta}(x_0, F) + \tilde \theta \ {\rm ent}` and plot it against :math:`{\rm ent}`.

* Repeat the preceding three steps for a range of values of :math:`\tilde \theta` to trace out the upper bound.
* Repeat the preceding three steps for a range of values of :math:`\tilde \theta` to trace out the upper bound.


Reshaping the Set of Values
Expand Down Expand Up @@ -922,9 +923,9 @@ Here is a brief description of the methods of the class
* ``robust_rule()`` and ``robust_rule_simple()`` both solve for the
triple :math:`\hat F, \hat K, \hat P`, as described in equations :eq:`rb_oc_ih` -- :eq:`rb_kd` and the surrounding discussion

* ``robust_rule()`` is more efficient
* ``robust_rule()`` is more efficient

* ``robust_rule_simple()`` is more transparent and easier to follow
* ``robust_rule_simple()`` is more transparent and easier to follow

* ``K_to_F()`` and ``F_to_K()`` solve the decision problems
of :ref:`agent 1 <rb_a1>` and :ref:`agent 2 <rb_a2>` respectively
Expand Down