Spaces:
Sleeping
Sleeping
Akshay Agrawal
commited on
Commit
·
07062d3
1
Parent(s):
f56200e
qp: level curves
Browse files- optimization/04_quadratic_program.py +140 -11
optimization/04_quadratic_program.py
CHANGED
@@ -3,7 +3,9 @@
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# dependencies = [
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# "cvxpy==1.6.0",
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# "marimo",
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# "numpy==2.2.2",
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# ]
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# ///
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@@ -74,25 +76,45 @@ def _():
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@app.cell(hide_code=True)
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def _(mo):
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mo.md("""First we generate synthetic data.""")
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return
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@app.cell
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def _(np):
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m =
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n =
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p = 5
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np.random.seed(1)
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P = np.random.randn(n, n)
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P = P.T @ P
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q = np.random.randn(n)
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G = np.random.randn(m, n)
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h = G @ np.random.randn(n)
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@app.cell(hide_code=True)
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@@ -102,12 +124,12 @@ def _(mo):
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@app.cell
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def _(
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x = cp.Variable(n)
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problem = cp.Problem(
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cp.Minimize((1 / 2) * cp.quad_form(x, P) + q.T @ x),
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[G @ x <= h
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)
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_ = problem.solve()
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return problem, x
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@@ -126,6 +148,113 @@ def _(mo, problem, x):
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return
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@app.cell
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def _():
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import marimo as mo
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# dependencies = [
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# "cvxpy==1.6.0",
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# "marimo",
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# "matplotlib==3.10.0",
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# "numpy==2.2.2",
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# "wigglystuff==0.1.9",
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# ]
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# ///
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@app.cell(hide_code=True)
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def _(mo):
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mo.md("""First we generate synthetic data. In this problem, we don't include equality constraints, only inequality.""")
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return
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@app.cell
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def _(np):
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m = 4
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n = 2
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np.random.seed(1)
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q = np.random.randn(n)
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G = np.random.randn(m, n)
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h = G @ np.random.randn(n)
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return G, h, m, n, q
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@app.cell(hide_code=True)
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def _(mo, np):
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import wigglystuff
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P_widget = mo.ui.anywidget(
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wigglystuff.Matrix(np.array([[4.0, -1.4], [-1.4, 4]]), step=0.1)
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)
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mo.md(
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f"""
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The quadratic form $P$ is equal to the symmetrized version of this
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matrix:
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{P_widget.center()}
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"""
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)
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return P_widget, wigglystuff
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@app.cell
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def _(P_widget, np):
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P = 0.5 * (np.array(P_widget.matrix) + np.array(P_widget.matrix).T)
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return (P,)
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@app.cell(hide_code=True)
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@app.cell
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def _(G, P, cp, h, n, q):
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x = cp.Variable(n)
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problem = cp.Problem(
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cp.Minimize((1 / 2) * cp.quad_form(x, P) + q.T @ x),
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[G @ x <= h],
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)
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_ = problem.solve()
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return problem, x
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return
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@app.cell
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def _(G, P, h, plot_contours, q, x):
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plot_contours(P, G, h, q, x.value)
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return
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@app.cell(hide_code=True)
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def _(mo):
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mo.md(
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r"""
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In this plot, the gray shaded region is the feasible region (points satisfying the inequality), and the ellipses are level curves of the quadratic form.
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**🌊 Try it!** Try changing the entries of $P$ above with your mouse. How do the
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level curves and the optimal value of $x$ change? Can you explain what you see?
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"""
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)
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return
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@app.cell(hide_code=True)
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def _(P, mo):
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mo.md(
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rf"""
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The above contour lines were generated with
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\[
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P= \begin{{bmatrix}}
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{P[0, 0]:.01f} & {P[0, 1]:.01f} \\
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{P[1, 0]:.01f} & {P[1, 1]:.01f} \\
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\end{{bmatrix}}
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\]
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"""
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)
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return
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@app.cell(hide_code=True)
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def _(np):
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def plot_contours(P, G, h, q, x_star):
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import matplotlib.pyplot as plt
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# Create a grid of x and y values.
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x = np.linspace(-5, 5, 400)
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y = np.linspace(-5, 5, 400)
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X, Y = np.meshgrid(x, y)
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# Compute the quadratic form Q(x, y) = a*x^2 + 2*b*x*y + c*y^2.
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# Here, a = P[0,0], b = P[0,1] (and P[1,0]), c = P[1,1]
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Z = (
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0.5 * (P[0, 0] * X**2 + 2 * P[0, 1] * X * Y + P[1, 1] * Y**2)
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+ q[0] * X
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+ q[1] * Y
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)
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# --- Evaluate the constraints on the grid ---
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# We stack X and Y to get a list of (x,y) points.
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points = np.vstack([X.ravel(), Y.ravel()]).T
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# Start with all points feasible
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feasible = np.ones(points.shape[0], dtype=bool)
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# Apply the inequality constraints Gx <= h.
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# Each row of G and corresponding h defines a condition.
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for i in range(G.shape[0]):
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# For a given point x, the condition is: G[i,0]*x + G[i,1]*y <= h[i]
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feasible &= points.dot(G[i]) <= h[i] + 1e-8 # small tolerance
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# Reshape the boolean mask back to grid shape.
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feasible_grid = feasible.reshape(X.shape)
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# --- Plot the feasible region and contour lines---
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plt.figure(figsize=(8, 6))
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# Use contourf to fill the region where feasible_grid is True.
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# We define two levels, so that points that are True (feasible) get one
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# color.
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plt.contourf(
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X,
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Y,
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feasible_grid,
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levels=[-0.5, 0.5, 1.5],
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colors=["white", "gray"],
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alpha=0.5,
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)
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contours = plt.contour(X, Y, Z, levels=10, cmap="viridis")
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plt.clabel(contours, inline=True, fontsize=8)
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plt.title("Feasible region and level curves")
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plt.xlabel("$x_1$")
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plt.ylabel("$y_2$")
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# plt.colorbar(contours, label='Q(x, y)')
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ax = plt.gca()
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# Optionally, mark and label the point x_star.
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ax.plot(x_star[0], x_star[1], "ko", markersize=5)
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ax.text(
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x_star[0],
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x_star[1],
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r"$\mathbf{x}^\star$",
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color="black",
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fontsize=12,
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verticalalignment="bottom",
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horizontalalignment="right",
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)
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return plt.gca()
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return (plot_contours,)
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@app.cell
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def _():
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import marimo as mo
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