Clean up maximization of minimum tutorial

docs/src/tutorials/Polynomial Optimization/extracting_minimizers.jl → docs/src/tutorials/Polynomial Optimization/min_univariate.jl

@@ -1,35 +1,54 @@
-# # Extracting minimizers
+# # Maximizing as minimum
 
-#md # [![](https://mybinder.org/badge_logo.svg)](@__BINDER_ROOT_URL__/generated/Polynomial Optimization/extracting_minimizers.ipynb)
-#md # [![](https://img.shields.io/badge/show-nbviewer-579ACA.svg)](@__NBVIEWER_ROOT_URL__/generated/Polynomial Optimization/extracting_minimizers.ipynb)
-# **Adapted from**: Example 6.23 of [L09]
+#md # [![](https://mybinder.org/badge_logo.svg)](@__BINDER_ROOT_URL__/generated/Polynomial Optimization/min_univariate.ipynb)
+#md # [![](https://img.shields.io/badge/show-nbviewer-579ACA.svg)](@__NBVIEWER_ROOT_URL__/generated/Polynomial Optimization/min_univariate.ipynb)
+# **Adapted from**: Section 4.10 of [F99], Example 6.23 of [L09] and Table 5.1 of [Las09]
+#
+# [F99] Floudas, Christodoulos A. et al.
+# *Handbook of Test Problems in Local and Global Optimization.*
+# Nonconvex Optimization and Its Applications (NOIA, volume 33).
 #
 # [L09] Laurent, Monique.
 # *Sums of squares, moment matrices and optimization over polynomials.*
 # Emerging applications of algebraic geometry (2009): 157-270.
+#
+# [Las09] Lasserre, J. B.
+# *Moments, positive polynomials and their applications*
+# World Scientific, **2009**.
 
 # ## Introduction
 
-# Consider the polynomial optimization problem [L09, Example 6.23] of
-# minimizing the linear function $-x_1 - x_2$
+# Consider the polynomial optimization problem [F99, Section 4.10]
+# of minimizing the linear function $-x_1 - x_2$
 # over the basic semialgebraic set defined by the inequalities
 # $x_2 \le 2x_1^4 - 8x_1^3 + 8x_1^2 + 2$,
 # $x_2 \le 4x_1^4 - 32x_1^3 + 88x_1^2 - 96x_1 + 36$ and the box constraints
 # $0 \le x_1 \le 3$ and $0 \le x_2 \le 4$,
-# World Scientific, **2009**.
 
 using Test #src
 using DynamicPolynomials
 @polyvar x[1:2]
 p = -sum(x)
 using SumOfSquares
-K = @set x[1] >= 0 && x[1] <= 3 && x[2] >= 0 && x[2] <= 4 && x[2] <= 2x[1]^4 - 8x[1]^3 + 8x[1]^2 + 2 && x[2] <= 4x[1]^4 - 32x[1]^3 + 88x[1]^2 - 96x[1] + 36
+f1 = 2x[1]^4 - 8x[1]^3 + 8x[1]^2 + 2
+f2 = 4x[1]^4 - 32x[1]^3 + 88x[1]^2 - 96x[1] + 36
+K = @set x[1] >= 0 && x[1] <= 3 && x[2] >= 0 && x[2] <= 4 && x[2] <= f1 && x[2] <= f2
+
+# As we can observe below, the bounds on `x[2]` could be dropped and
+# optimization problem is equivalent to the maximization of `min(f1, f2)`
+# between `0` and `3`.
+
+xs = range(0, stop = 3, length = 100)
+using Plots
+plot(xs, f1.(xs), label = "f1")
+plot!(xs, f2.(xs), label = "f2")
+plot!(xs, 4 * ones(length(xs)), label = nothing)
 
 # We will now see how to find the optimal solution using Sum of Squares Programming.
 # We first need to pick an SDP solver, see [here](https://jump.dev/JuMP.jl/v1.12/installation/#Supported-solvers) for a list of the available choices.
 
-import CSDP
-solver = optimizer_with_attributes(CSDP.Optimizer, MOI.Silent() => true)
+import Clarabel
+solver = Clarabel.Optimizer
 
 # A Sum-of-Squares certificate that $p \ge \alpha$ over the domain `S`, ensures that $\alpha$ is a lower bound to the polynomial optimization problem.
 # The following function searches for the largest lower bound and finds zero using the `d`th level of the hierarchy`.
@@ -74,3 +93,7 @@ model7 = solve(7)
 x_opt = η.atoms[1].center
 @test x_opt ≈ [2.3295, 3.1785] rtol=1e-4 #src
 p(x_opt)
+
+# We can see visualize the solution as follows:
+
+scatter!([x_opt[1]], [x_opt[2]], markershape = :star, label = nothing)