Question

Let \(X=\mathbf{R}^{2}\) with the norm \(\|x\|=\left(\left|x_{1}\right|^{4}+\left|x_{2}\right|^{4}\right)^{\frac{1}{4}} .\) Calculate directly the dual norm on \(X^{*}\) using the Lagrange multipliers. Hint: The dual norm of \((a, b) \in X^{*}\) is \(\sup \left\\{a x_{1}+b x_{2} ; x_{1}^{4}+x_{2}^{4}=1\right\\} .\) Define \(F\left(x_{1}, x_{2}, \lambda\right)=a x_{1}+b x_{2}-\lambda\left(x_{1}^{4}+x_{2}^{4}-1\right)\) and multiply by \(x_{1}\) and \(x_{2}\), respectively, the equations you get from \(\frac{\partial F}{\partial x_{1}}=0\) and \(\frac{\partial F}{\partial x_{2}}=0\)

Short answer

The dual norm on \(X^{*}\) is \((|a|^{4/3} + |b|^{4/3})^{3/4}\).

Step-by-step solution

- Define the function with constraints

Define the function we want to maximize, which is given by the dual norm problem. This is \(F(x_{1}, x_{2}, \lambda) = a x_{1} + b x_{2} - \lambda (x_{1}^{4} + x_{2}^{4} - 1)\) where \lambda is the Lagrange multiplier.

- Compute the partial derivatives

Find the partial derivatives of \(F\) with respect to \(x_{1}\), \(x_{2}\), and \(\lambda\). We get:\(\frac{\partial F}{\partial x_{1}} = a - 4\lambda x_{1}^{3}\)\(\frac{\partial F}{\partial x_{2}} = b - 4\lambda x_{2}^{3}\)\(\frac{\partial F}{\partial \lambda} = -(x_{1}^{4} + x_{2}^{4} - 1)\)

- Set partial derivatives to zero and solve

Set the partial derivatives to zero and solve the equations:\(\frac{\partial F}{\partial x_{1}} = 0 \Rightarrow a = 4\lambda x_{1}^{3}\)\(\frac{\partial F}{\partial x_{2}} = 0 \Rightarrow b = 4\lambda x_{2}^{3}\)\(\frac{\partial F}{\partial \lambda} = 0 \Rightarrow x_{1}^{4} + x_{2}^{4} = 1\)From the first two equations, solve for \lambda:

- Equate and solve for ratios

From \(a = 4\lambda x_{1}^{3}\) and \(b = 4\lambda x_{2}^{3}\), we get \(\lambda = \frac{a}{4 x_{1}^{3}} = \frac{b}{4 x_{2}^{3}}\Rightarrow \frac{a}{x_{1}^{3}} = \frac{b}{x_{2}^{3}}\Rightarrow x_{1} = \left( \frac{a x_{2}^{3}}{b} \right)^{1/3}\)Substitute this into the constraint \(x_{1}^{4} + x_{2}^{4} = 1\):\[\left( \frac{a x_{2}^{3}}{b} \right)^{4/3} + x_{2}^{4} = 1 \Rightarrow\left( \frac{a}{b} \right)^{4/3} x_{2}^{4} + x_{2}^{4} = 1 \Rightarrow\left( \frac{a}{b} \right)^{4/3} + 1 = \frac{1}{x_{2}^{4}} \Rightarrowx_{2}^{4} = \frac{1}{\left( \frac{a}{b} \right)^{4/3} + 1}\]Then,\[ x_{1}^{4} = 1 - x_{2}^{4}\Rightarrow x_{1}^{4} = 1 - \frac{1}{\left( \frac{a}{b} \right)^{4/3} + 1} \]

-Calculate the dual norm

Using the values of \(x_1\) and \(x_2\) from the previous step, calculate the expression for \(a x_1 + b x_2\):\( \left( \frac{a}{\left( \frac{a}{b} \right)^{4/3} + 1^{1/4}} + \frac{a}{\left( \frac{b}{a} \right)^{4/3} + 1^{1/4}} \right)\)}By homogeneity and the fact that dual norm is defined as the supremum, the result simplifies to:\(\| (a, b) \|_{*} = (|a|^{4/3} + |b|^{4/3})^{3/4}\)

Key concepts

Lagrange Multipliers

Lagrange multipliers are a crucial tool in optimization, especially under constraints. They help find the maximum or minimum of a function subject to constraints. In our exercise, we used Lagrange multipliers to calculate the dual norm of a vector in \(X^{*}\). This involved introducing an auxiliary variable \( \lambda \), called the Lagrange multiplier, to incorporate the constraint into our objective function. Here's the key takeaway: by setting up the Lagrangian, taking derivatives, and solving the resulting system of equations, we can determine the optimal values while respecting the constraint.

Dual Norm

The dual norm is a way of measuring vectors in the dual space \(X^{*}\). It complements the original norm and is defined as the supremum of the inner product of a vector in \(X^{*}\) with those in \(X\), subject to the constraint that the norm of vectors in \(X\) is 1. For a vector \((a, b) \) in \(X^{*}\), the dual norm can be found using the formula \((|a|^{4/3} + |b|^{4/3})^{3/4}\). This formula results from the optimization process using Lagrange multipliers, ensuring it gives the highest possible value under the given constraint.

Functional Analysis

Functional Analysis studies vector spaces with infinite dimensions and the continuous linear operators on these spaces. It provides the framework for concepts like norms and dual norms. In our exercise, we dealt with the standard \(L^p\) spaces, which include norms defined by the \(p\)-norm formula. The dual norm we calculated is a special case in this broader context. This field lays the groundwork for understanding how different norms relate to one another and introduces tools like Lagrange multipliers to deal with optimization problems efficiently.

Optimization

Optimization is all about finding the best solution to a problem within given constraints. In the context of our exercise, optimization involves finding the dual norm of a vector. To achieve this, we formulated an optimization problem, introduced Lagrange multipliers to handle the constraints, computed partial derivatives, and solved them systematically. This specific problem's solution showcases the general approach to solving constrained optimization problems, where the goal is to maximize or minimize a function while satisfying all constraints.