tutorials: Import SOS tutorial from colab #12642

EricCousineau-TRI · 2020-01-27T22:42:53Z

Resolves #12634

This is just a quick import.

This change is

EricCousineau-TRI

+@hongkai-dai for feature review, please.
+@RussTedrake for platform review, please.

Reviewable status: LGTM missing from assignees hongkai-dai,RussTedrake(platform) (waiting on @hongkai-dai and @RussTedrake)

hongkai-dai

thanks for bringing it into Drake.

Reviewed 3 of 3 files at r1.
Reviewable status: LGTM missing from assignee RussTedrake(platform) (waiting on @RussTedrake)

jwnimmer-tri · 2020-01-27T23:54:59Z

I suggest amending the commit message to use Co-authored-by.

EricCousineau-TRI

Done!

Reviewable status: LGTM missing from assignee RussTedrake(platform) (waiting on @RussTedrake)

RussTedrake

Reviewable status: 6 unresolved discussions, LGTM missing from assignee RussTedrake(platform) (waiting on @EricCousineau-TRI)

tutorials/sum_of_squares_optimization.ipynb, line 25 at r2 (raw file):

    "p(x) = \\sum_i q_i(x)^2\n",
    "\\end{align}\n",
    "where each $q_i(x)$ is also a polynomial of the indeterminates $x$. Apparently $p(x)$ being sum-of-squares is a *sufficient* condition for $p(x)$ being non-negative for all $x$. In some cases it is also a necessary and sufficient condition. For more details on the relationship between sum-of-squares and non-negativity of a polynomial, you could refer to Chapter 3 of [Semidefinite Optimization and Convex Algebraic Geometry](http://www.mit.edu/~parrilo/sdocag/).\n",

Remove "Apparently"

tutorials/sum_of_squares_optimization.ipynb, line 55 at r2 (raw file):

    "In sum-of-squares problem, often we want to find a sum-of-squares polynomial that satisfies certain additional properties. Note that there are two different types of variables in sum-of-squares problems, the *indeterminates* and *decision variables*.\n",
    "* Decision variables: the coefficients of the polynomial are affine functions of our decision variables. We are optimizing over the decision variables.\n",
    "* Indeterminates: \"dummy variables\", in the sense that we are *not* optimizing over them.\n",

I would replace this definition... perhaps with:
variables that can take any value. In the example above, $x$ is an indeterminate -- the constraints that we write must hold $\forall x$. We are not optimizing over these variables.

tutorials/sum_of_squares_optimization.ipynb, line 102 at r2 (raw file):

    "in this case, $x$ is the indeterminate, and $a$ is the decision variables.\n",
    "\n",
    "The code below shows how to model and solve this sos problem in drake's MathematicalProgram."

SOS

tutorials/sum_of_squares_optimization.ipynb, line 126 at r2 (raw file):

    "# Declare p2(x) = (a0 - a1 + 1) * pow(x, 2) + a1 * x + 1\n",
    "p2 = sym.Polynomial((a[0] - a[1] + 1) * x[0] * x[0] + a[1] * x[0] + 1, sym.Variables(x))\n",
    "# Add the constraint that p1(x) and p2(x) are sos\n",

SOS
(please search and replace through the doc?)

tutorials/sum_of_squares_optimization.ipynb, line 152 at r2 (raw file):

    "To solve this problem, we use the following theorem:\n",
    "\n",
    "**Theorem**: An even degree univariate polynomial $p(x)$ is nonnegative on interval $[0, 1]$ if and only if it can be written as\n",

I would probably not call this version the theorem. The theorem is for the more general S-procedure, and using it for the interval [0,1] is a special case...?

tutorials/sum_of_squares_optimization.ipynb, line 192 at r2 (raw file):

    "# Declares sos polynomial t(x), such that s(x) = p(x) - x(1-x) * t(x) is sos.\n",
    "# The second return argument of NewSosPolynomial is the positive-semidefinite\n",
    "# Hessian matrix of poly_s. We ignore this matrix here.\n",

"Hessian" => "Gramian"
btw -- in the NewSosPolynomial method doc string, "Grammian" => "Gramian"

EricCousineau-TRI

Pulling in Hongkai's review responses.

Reviewable status: 6 unresolved discussions, LGTM missing from assignee RussTedrake(platform), commits need curation (https://drake.mit.edu/reviewable.html#curated-commits) (waiting on @EricCousineau-TRI, @hongkai-dai, and @RussTedrake)

tutorials/sum_of_squares_optimization.ipynb, line 25 at r2 (raw file):