17.5 Constrained problems[2D5]
[1F6]Let now \(A ⊆ ℝ^ n\) an open non-empty set, let \(f,𝜑:A\to {\mathbb {R}}\) be real functions of class \(C^ 1\) on \(A\). Having fixed \(a∈ℝ\) we then define the level set
we assume that \(E_ a\) is non-empty, and that \(∇ 𝜑(x) ≠ 0\) for each \(x ∈ E_ a\).
We call local minimum point of \(f\) bound to \(E_ a\) a point of \(E_ a\) that is a local minimum for \(f|_{E_ a}\); and similarly for maxima.
To solve the following exercises it may be useful to apply the results seen in 6, 6, 3.
- E402
[1H8] Prerequisites:402,3. Let \(f,𝜑\) be class \(C^ 1\) in the open set \(A\), and let \(\overline x\) be a local minimum point for \(f\) bound to \(E_ a\) (so \(𝜑(x)=a\)). Show that \(𝜆∈ℝ\) exists such that \(∇ f(\overline x)+𝜆 ∇𝜑(\overline x)=0\); this \(𝜆\) is called the Lagrange multiplier.
Hidden solution: [UNACCESSIBLE UUID ’1H9’]
- E402
[1HB] Prerequisites:402,6,1.Let \(f,𝜑\) be of class \(C^ 2\) in the open set \(A\), and let \(\overline x\) be a minimum point for \(f\) constrained to \(E_ a\); let \(𝜆\) be the Lagrange multiplier; let’s define \(h=f(x)+𝜆𝜑(x)\), then
\[ ∀ v, v⋅ ∇ 𝜑(x)=0⟹ v⋅ H v≥ 0 \]where \(H\) is the Hessian matrix of \(h\).
Hidden solution: [UNACCESSIBLE UUID ’1HC’]
- E402
[1HD] In the same hypotheses, we see a "vice versa". Let \(f,𝜑:A\to {\mathbb {R}}\) be of class \(C^ 2\) in the open set \(A\), and let \(\overline x∈ E_ a\) and \(𝜆∈ℝ\) be such that \(∇ f(\overline x)+𝜆 ∇𝜑(\overline x)=0\); suppose that
\[ ∀ v, v⋅ ∇ 𝜑(x)=0⟹ v⋅ H v {\gt} 0 \]where
\[ h(x)=f(x)+𝜆𝜑(x) \]and \(H\) is the Hessian matrix of \(h\) in \(\overline x\). Show that \(\overline x\) is a local minimum point for \(f\) bound to \(E_ a\).
Hidden solution: [UNACCESSIBLE UUID ’1HF’]
Constraints with inequalities
Now let’s consider a different kind of constraint.
[2BH]Let
we always assume that \(F_ a\) is non-empty and that \(∇ 𝜑(x) ≠ 0\) for each \(x ∈ E_ a\).
We call local minimum point of \(f\) bound to \(F_ a\) a point of \(F_ a\) that is of local minimum for \(f|_{F_ a}\); and similarly for maxima.
- E403
[1HG]Prerequisites:403,402.Show that \(∂ F_ a = E_ a\) and that \(F_ a\) coincides with the closure of its interior. (Topological operations must be performed within \(A\), seen as a topological space!)
- E403
[1HH]Prerequisites:403.Show that a necessary condition for \(x∈ A\) to be a local minimum of \(f\) bound to \(F_ a\), is that,
either \(𝜑(x){\lt}a\) and \(∇ f(x)=0\),
or \(𝜑(x)=a\) and \(∇ f(x)+𝜆 ∇𝜑(x)=0\) with \(𝜆≥ 0 \).
These are the Karush–Kuhn–Tucker conditions.
Hidden solution: [UNACCESSIBLE UUID ’1HJ’]
- E403
[1HK]Prerequisites:403.In the case \(n=1\), suppose \(A\) is an open interval, show that if \(𝜑(x)=a\) and \( f'(x) 𝜑'(x){\lt} 0\) then the point \(x\) is a local minimum point for \(f\) bound to \(F_ a\).
- E403
[1HM]Prerequisites:403.Find a simple example in the case \(n=2\) where the point \(x\) is not a local minimum for \(f\) bound to \(F_ a\), but \(𝜑(x)=a\) and \(∇ f(x)+𝜆 ∇ 𝜑(x)=0\) with \(𝜆 {\gt}0\).
Hidden solution: [UNACCESSIBLE UUID ’1HN’]