Question

Write out the bordered Hessian for a constrained optimization problem with four choice variables and two constraints. Then state specifically the second- order sufficient condition for a maximum and for a minimum of \(z,\) respectively.

Short answer

Construct the bordered Hessian with sign checks for sufficiency.

Step-by-step solution

Understanding the Problem

We are dealing with a constrained optimization problem with four variables and two constraints. The problem involves maximizing or minimizing a function subject to these constraints.

Define the Objective and Constraints

Let the objective function be denoted as \( z = f(x_1, x_2, x_3, x_4) \) and the two constraints be \( g_1(x_1, x_2, x_3, x_4) = 0 \) and \( g_2(x_1, x_2, x_3, x_4) = 0 \).

Formulate the Lagrangian

Write the Lagrangian function as \( \, \mathcal{L} = f(x_1, x_2, x_3, x_4) + lambda_1 g_1(x_1, x_2, x_3, x_4) + lambda_2 g_2(x_1, x_2, x_3, x_4) \).

Construct the Bordered Hessian

The bordered Hessian matrix includes the second partial derivatives and bordered by the gradients of the constraints. Construct it as follows:\[\begin{bmatrix}0 & (abla g_1)^T & (abla g_2)^T \abla g_1 & H_f - \lambda_1 H_{g_1} - \lambda_2 H_{g_2}\end{bmatrix}\]where \( abla g_1 \) and \( abla g_2 \) are the gradients of the constraint functions, and \( H_f \) is the Hessian matrix of the objective function.

Second-Order Sufficient Conditions for Maximum

For a maximum, the leading principal minors of the bordered Hessian should alternate in sign, starting with a negative sign.

Second-Order Sufficient Conditions for Minimum

For a minimum, the leading principal minors of the bordered Hessian should all be positive.

Key concepts

Bordered Hessian

In constrained optimization, the bordered Hessian is a crucial tool that helps us verify solutions to optimization problems when subject to constraints. It is a matrix constructed to encapsulate both the objective function and the constraints.

When dealing with multiple choice variables and constraints (like in the problem where we have four choice variables and two constraints), we form the bordered Hessian by combining partial derivatives from both the objective function and the constraints.

• The first row and the first column of this matrix are made up of zeros alongside the gradients of the constraints.
• The remaining square matrix is a combination of the Hessian of the objective function modified by the Hessians of the constraints, weighted by Lagrange multipliers.

This matrix structure is essential in discerning the nature (maximum or minimum) of the solution under given constraints.

Second-Order Sufficient Conditions

Once we have formulated the bordered Hessian, determining the nature of our solution requires checking the second-order sufficient conditions. These conditions provide a way to confirm whether a critical point of our constrained function is a maximum or a minimum.

• For achieving a local maximum, the leading principal minors of the bordered Hessian need to alternate in sign. The pattern should start with a negative sign.
• For a local minimum, all the leading principal minors must be positive.

Understanding these conditions ensures that the solutions are not just critical points but are indeed extrema, located at peaks or valleys, respectively.

Lagrangian Function

The Lagrangian function is a pivotal concept in constrained optimization. It creatively combines the objective function with the constraints by introducing new variables called Lagrange multipliers.

• The Lagrangian is expressed as: \( \mathcal{L} = f(x_1, x_2, x_3, x_4) + \lambda_1 g_1(x_1, x_2, x_3, x_4) + \lambda_2 g_2(x_1, x_2, x_3, x_4) \).
• Here, the functions \( f \) and \( g \) are the objective and constraint functions respectively, while \( \lambda_1 \) and \( \lambda_2 \) are Lagrange multipliers.

By solving \( \mathcal{L} \), we find not only the extremum of the objective function under constraints but also ensure those solutions respect the constraint boundaries. The Lagrangian provides a path to bridge the objective and constraints in our optimization study.

Principal Minors

Principal minors are the determinants of the square submatrices formed within the bordered Hessian matrix. These minors play an instrumental role in applying the second-order sufficient conditions for maximization or minimization.

• To derive them, you calculate the determinant of each leading square submatrix within the overall bordered Hessian matrix.
• Their signs help us determine whether we have found a local maximum or minimum in the constrained optimization problem.

By evaluating these minors, we employ a systematic approach to check whether our solution obeys the required second-order conditions, ensuring robust optimization solutions.