VMLS: Constrained least squares

The data for this plot comes from Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares (Boyd and Vandenberghe, 2018).

#| '!! shinylive warning !!': |
#|   shinylive does not work in self-contained HTML documents.
#|   Please set `embed-resources: false` in your metadata.
#| standalone: true
#| viewerHeight: 600

## file: app.py

from shiny import App, render, ui, reactive
import numpy as np
import matplotlib.pyplot as plt
import plotly.graph_objects as go
from shinywidgets import render_widget, output_widget
from pathlib import Path

def Z_func(x, y):
    return x ** 2 + y ** 2
C = np.ones((1, 2))
d = np.ones((1, 1))

app_ui = ui.page_fluid(
    ui.layout_columns(
        output_widget('curve'),
        output_widget('contourplot')
    ),
)


def server(input, output, session):

    @render_widget
    def curve():

        X_lin  = np.arange(-1.5, 2.01, 0.05)
        Y_lin = np.arange(-1.5, 2.01, 0.05)
        X, Y = np.meshgrid(X_lin, Y_lin)
        Z = Z_func(X, Y)

        Y_constr = 1 - X_lin
        Z_constr = Z_func(X_lin, Y_constr)

        fig = go.Figure(data=[go.Surface(x=X, y=Y, z=Z, name='Curve'),
                                go.Scatter3d(
                                    x=X_lin, 
                                    y=Y_constr, 
                                    z=Z_constr, 
                                    mode='lines',
                                    name='Feasible region',
                                    marker=dict(color='White', size=20, line=dict(color='Black', width=40)),
                                    showlegend=False)])
    
        fig = fig.update_layout(
            autosize=True,
            height=600,
            margin=dict(l=65, r=50, b=65, t=90),
            scene = dict(xaxis = dict(range=[-1.5, 2], title=dict(text='x_1')),
                         yaxis = dict(range=[-1.5, 2], title=dict(text='x_2')),
                         zaxis = dict(range=[np.min(Z) - (np.max(Z) - np.min(Z)) * 0.05, np.max(Z) + (np.max(Z) - np.min(Z)) * 0.05], title=dict(text=''))
                         )
            )
        return fig

    # ========================================================================

    @render_widget
    def contourplot():
        X_lin  = np.arange(-1.5, 2.01, 0.05)
        Y_lin = np.arange(-1.5, 2.01, 0.05)
        X, Y = np.meshgrid(X_lin, Y_lin)
        Z = Z_func(X, Y)

        X_constr  = np.arange(-1, 1.96, 0.05)
        Y_constr = 1 - X_constr
        Z_constr = Z_func(X_constr, Y_constr)

        fig = go.Figure(data =
            [go.Contour(
                z=Z,
                x=X_lin,
                y=Y_lin
            ),
            go.Scatter(
                x=X_constr, 
                y=Y_constr,
                mode='lines',
                name='Feasible region',
                marker=dict(color='White', size=20, line=dict(color='Black', width=40)),
                showlegend=False)])

        fig = fig.update_layout(
        autosize=True,
        height=600,
        margin=dict(l=65, r=50, b=65, t=90),
        scene = dict(xaxis = dict(range=[-1, 2], title=dict(text='x_1')),
                        yaxis = dict(range=[-1, 2], title=dict(text='x_2')),
                        zaxis = dict(range=[np.min(Z) - (np.max(Z) - np.min(Z)) * 0.05, np.max(Z) + (np.max(Z) - np.min(Z)) * 0.05], title=dict(text=''))
                        )
        )
        return fig

    # ========================================================================


app = App(app_ui, server)

References

Boyd, S. P., & Vandenberghe, L. (2018). Introduction to Applied Linear Algebra : Vectors, Matrices, and Least Squares (First edition.). Cambridge University Press.